# nLab geometry of physics -- categories and toposes

Contents

This entry is one chapter of geometry of physics.

next chapters: smooth sets, supergeometry

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category theory

topos theory

## Theorems

Category theory and topos theory concern the general abstract structure underlying algebra, geometry and logic. They are ubiquituous in and indispensible for organizing conceptual mathematical frameworks.

We give here an introduction to the basic concepts and results, aimed at providing background for the synthetic higher supergeometry of relevance in formulations of fundamental physics, such as used in the chapters on perturbative quantum field theory and on fundamental super p-branes. For quick informal survey see Introduction to Higher Supergeometry.

This makes use of the following curious dictionary between category theory/topos theory and the geometry of generalized spaces, which we will explain in detail (following Grothendieck 65, Lawvere 86, p. 17, Lawvere 91):

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$\phantom{A}$category theoryRmk. $\phantom{A}$geometry of generalized spaces
$\phantom{A}$presheafExpl. $\phantom{A}$generalized space
$\phantom{A}$representable presheaf$\phantom{A}$Expl. $\phantom{A}$$\phantom{A}$model space
$\phantom{A}$regarded as generalized space
$\phantom{A}$Yoneda lemmaProp. $\phantom{A}$$\phantom{A}$sets of probes of generalized spaces
$\phantom{A}$are indeed
$\phantom{A}$sets of maps from model spaces $\phantom{A}$
$\phantom{A}$Yoneda embedding $\phantom{A}$Prop. $\phantom{A}$$\phantom{A}$nature of model spaces is preserved when
$\phantom{A}$regarding them as generalized spaces $\phantom{A}$
$\phantom{A}$Yoneda embedding is$\phantom{A}$
$\phantom{A}$free co-completion$\phantom{A}$
Prop. $\phantom{A}$generalized spaces really are$\phantom{A}$
$\phantom{A}$glued from ordinary spaces$\phantom{A}$
$\phantom{A}$topos theoryRmk. $\phantom{A}$local-global principle for generalized spaces$\phantom{A}$
$\phantom{A}$coverageDefn. $\phantom{A}$notion of locality
$\phantom{A}$sheaf conditionDefn. $\phantom{A}$
Prop.
$\phantom{A}$plots of generalized spaces
$\phantom{A}$satisfy local-to-global principle $\phantom{A}$
$\phantom{A}$comparison lemmaProp. $\phantom{A}$notion of generalized spaces
$\phantom{A}$independent under change of model space
$\phantom{A}$gros topos theoryRmk. $\phantom{A}$generalized spaces at the foundations
$\phantom{A}$cohesionDefn. $\phantom{A}$generalized spaces obey
$\phantom{A}$principles of differential topology
$\phantom{A}$differential cohesionDefn. $\phantom{A}$generalized spaces obey
$\phantom{A}$principles of differential geometry
$\phantom{A}$super cohesion$\phantom{A}$Defn. $\phantom{A}$generalized spaces obey
$\phantom{A}$principles of supergeometry

The perspective is that of functorial geometry (Grothendieck 65). (For more exposition of this point see also at motivation for sheaves, cohomology and higher stacks.) This dictionary implies a wealth of useful tools for handling and reasoning about geometry:

We discuss below that sheaf toposes, regarded as categories of generalized spaces via the above disctionary, are “convenient contexts” for geometry (Prop. below), in the technical sense that they provide just the right kind of generalization that makes all desireable constructions on spaces actually exist:

$\phantom{A}$sheaf topos$\phantom{A}$$\phantom{A}$as category of generalized spaces $\phantom{A}$
$\phantom{A}$Yoneda embedding: $\phantom{A}$$\phantom{A}$contains and generalizes ordinary spaces $\phantom{A}$
$\phantom{A}$has all limits: $\phantom{A}$$\phantom{A}$contains all Cartesian products and intersections $\phantom{A}$
$\phantom{A}$has all colimits: $\phantom{A}$$\phantom{A}$contains all disjoint unions and quotients
$\phantom{A}$cartesian closure: $\phantom{A}$$\phantom{A}$contains all mapping spaces$\phantom{A}$
$\phantom{A}$local cartesian closure: $\phantom{A}$$\phantom{A}$contains all fiber-wise mapping spaces $\phantom{A}$

Notably mapping spaces play a pivotal role in physics, in the guise of spaces of field histories, but fall outside the applicability of traditional formulations of geometry based on just manifolds. Topos theory provides their existence (Prop. below) and the relevant infrastructure, for example for the construction of transgression of differential forms to mapping spaces of smooth sets, that is the basis for sigma-model-field theories. This is discussed in the following chapters on smooth sets and on supergeometry.

In conclusion, one motivation for category theory and topos theory is a posteriori: As a matter of experience, there is just no other toolbox that allows to deeply understand and handle the geometry of physics. Similar comments apply to a wealth of other topics of mathematics.

We offer also an a priori motivation:

Category theory is the theory of duality. Duality is of course an ancient notion in philosophy. At least as a term, it makes a curious re-appearance in the conjectural theory of fundamental physics formerly known as string theory, as duality in string theory. In both cases, the literature left some room in delineating what precisely is meant. But the philosophically inclined mathematician could notice (see Lambek 82) that an excellent candidate to make precise the idea of duality is the mathematical concept of adjunction, from category theory. This is particularly pronounced for adjoint triples (Remark below) and their induced adjoint modalities (Lawvere 91, see Def. below), which exhibit a given “mode of being” of any object $X$ as intermediate between two dual opposite extremes (Prop. below):

$\Box X \overset{\phantom{AAAA}}{\longrightarrow} X \overset{\phantom{AAAA}}{\longrightarrow} \bigcirc X$

For example, cohesive geometric structure on generalized spaces is captured, this way, as modality in between the discrete and the codiscrete (Example , and Def. below).

Historically, category theory was introduced in order to make precise the concept of natural transformation: The concept of functors was introduced just so as to support that of natural transformations, and the concept of categories only served that of functors (see e.g. Freyd 65, Part II). But natural transformations are, in turn, exactly the basis for the concept of adjoint functors (Def. below), equivalently adjunctions between categories (Prop. below), shown on the left. All universal constructions, the heart of category theory, are special cases of adjoint functors – hence of dualities, if we follow Lambek 82: This includes the concepts of limits and colimits (Def. below), ends and coends (Def. below) Kan extensions (Prop. below), and the behaviour of these constructions, such as for instance the free co-completion nature of the Yoneda embedding (Prop. below).

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Therefore it makes sense to regard category theory as the theory of adjunctions,
hence the theory of duality:

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$\phantom{A}$hierarchy of concepts$\phantom{A}$$\phantom{A}$category theory$\phantom{A}$$\phantom{A}$enriched$\phantom{A}$$\phantom{A}$homotopical$\phantom{A}$
$\phantom{A}$adjunction of adjunctions$\phantom{A}$
$\phantom{AA}$duality of dualities$\phantom{A}$
$\phantom{A}$Def. $\phantom{A}$$\phantom{A}$Def. $\phantom{A}$
$\phantom{A}$ adjoint equivalence$\phantom{A}$
$\phantom{AA}$dual equivalence $\phantom{AA}$
$\phantom{A}$ Def. $\phantom{A}$$\phantom{A}$ Def. $\phantom{A}$$\phantom{A}$Def.
$\phantom{A}$ adjunction$\phantom{A}$
$\phantom{AA}$duality$\phantom{A}$
$\phantom{A}$ Def. $\phantom{A}$$\phantom{A}$ Def. $\phantom{A}$$\phantom{A}$Def.
$\phantom{A}$ natural transformation $\phantom{A}$$\phantom{A}$ Def. $\phantom{A}$$\phantom{A}$ Def. $\phantom{A}$
$\phantom{A}$ functor $\phantom{A}$$\phantom{A}$ Def. $\phantom{A}$$\phantom{A}$ Def. $\phantom{A}$
$\phantom{A}$ category $\phantom{A}$$\phantom{A}$ Def. $\phantom{A}$$\phantom{A}$ Def. $\phantom{A}$$\phantom{A}$ Def.

The pivotal role of adjunctions in category theory (Lawvere 08) and in the foundations of mathematics (Lawvere 69, Lawvere 94 ) was particularly amplified by F. W. Lawvere1. Moreover, Lawvere saw the future of category theory (Lawvere 91) as concerned with adjunctions expressing systems of archetypical dualities that reveal foundations for geometry (Lawvere 07) and physics (Lawvere 97, see Def. and Def. below). He suggested (Lawvere 94) this as a precise formulation of core aspects of the theory of everything of early 19th century philosophy: Hegel‘s Science of Logic.

These days, of course, theories of everything, such as string theory, are understood less ambitiously than Hegel’s ontological process, as mathematical formulations of fundamental theories of physics, that could conceptually unify the hodge-podge of currently available “standard models” of particle physics and of cosmology to a more coherent whole.

The idea of duality in string theory refers to different perspectives on physics that appear dual to each other while being equivalent. But one of the basic results of category theory (Prop. , below) is that equivalence is indeed a special case of adjunction. This allows to explore the possibility that there is more than a coincidence of terms.

Of course the usage of the term duality in string theory is too loose for one to expect to be able to refine each occurrence of the term in the literature to a mathematical adjunction. However, we will see mathematical formalizations of core aspects of key string-theoretic dualities, such as topological T-duality and the duality between M-theory and type IIA string theory, in terms of adjunctions. Indeed, at the heart of these dualities in string theory is the phenomenon of double dimensional reduction, which turns out to be formalized by one of the most fundamental adjunctions in (higher) category theory: base change along the point inclusion into a classifying space. All this is discussed in the chapter on fundamental super p-branes.

This suggests that there may be a deeper relation here between the superficially alien uses of the word “duality”, that is worth exploring.

In this respect it is worth noticing that core structure of string/M-theory arises via universal constructions from the superpoint (as explained in the chapter on fundamental super p-branes), while the superpoint itself arises, in a sense made precise by category theory, “from nothing”, by a system of twelve adjunctions (explained in the chapter on supergeometry).

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Here we introduce the requisites for understanding these statements.

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# Contents

## Basic notions of Category theory

We introduce here the basic notions of category theory, along with examples and motivation from geometry:

This constitutes what is sometimes called the language of categories. While we state and prove some basic facts here, notably the notorious Yoneda lemma (Prop. below), what makes category theory be a mathematical theory in the sense of a coherent collection of non-trivial theorems is all concerned with the topic of universal constructions, which may be formulated (only) in this language. This we turn to further below.

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### Categories and Functors

The notion of a category (Def. below) embodies the idea of structuralism applied to concepts in mathematics: it collects, on top of the set (or generally: class) of mathematical objects that belong to it, also all the structure-preserving maps between them, hence the homomorphisms in the case of Bourbaki-style mathematical structures.

The first achievement of the notion of a category is to abstract away from such manifestly concrete categories (Examples , below) to more indirectly defined mathematical objects whose “structure” is only defined, after the fact, by which maps, now just called morphisms, there are between them.

This structuralism-principle bootstraps itself to life by considering morphisms between categories themselves to be those “maps” that respect their structuralism, namely the connectivity and composition of the morphisms between their objects: These are the functors (Def. below).

For the purpose of geometry, a key class of examples of functors are the assignments of algebras of functions to spaces, this is Example below.

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###### Definition

(category)

A category $\mathcal{C}$ is

1. a class $Obj_{\mathcal{C}}$, called the class of objects;

2. for each pair $X,Y \in Obj_{\mathcal{C}}$ of objects, a set $Hom_{\mathcal{C}}(X,Y)$, called the set of morphisms from $X$ to $Y$, or the hom-set, for short.

We denote the elements of this set by arrows like this:

$X \overset{f}{\longrightarrow} Y \;\;\in Hom_{\mathcal{C}}(X,Y) \,.$
3. for each object $X \in Obj_{\mathcal{C}}$ a morphism

$X \overset{id_X}{\to} X \;\; \in Hom_{\mathcal{C}}(X,X)$

called the identity morphism on $X$;

4. for each triple $X_1, X_2, X_3 \in Obj$ of objects, a function

$\array{ Hom_{\mathcal{C}}(X_1, X_2) &\times& Hom_{\mathcal{C}}(X_2, X_3) &\overset{\circ_{X_1,X_2,X_3}}{\longrightarrow}& Hom_{\mathcal{C}}(X_1, X_3) \\ X_1 \overset{f}{\to} X_2 &,& X_2 \overset{f}{\to} X_3 &\mapsto& X_1 \overset{ g \circ f }{\longrightarrow} X_3 }$

called composition;

such that:

1. for all pairs of objects $X,Y \in Obj_{\mathcal{C}}$ unitality holds: given

$X \overset{f}{\to} Y \;\;\in Hom_{\mathcal{C}}(X,Y)$

then

$X \overset{id_Y \circ f}{\longrightarrow} Y \;=\; X \overset{f}{\longrightarrow} Y \;=\; X \overset{f \circ id_X }{\longrightarrow} Y \,;$
2. for all quadruples of objects $X_1, X_2, X_3, X_4 \in Obj_{\mathcal{C}}$ composition satifies associativity: given

$X_1 \overset{f_{12}}{\to} X_2 \overset{f_{23}}{\to} X_3 \overset{f_{34}}{\to} X_4$

then

$X_1 \overset{f_{34} \circ (f_{23} \circ f_{12})}{\longrightarrow} X_4 \;\;=\;\; X_1 \overset{(f_{34} \circ f_{23}) \circ f_{12}}{\longrightarrow} X_4 \,.$

The archetypical example of a category is the category of sets:

###### Example

(category of all sets)

The class of all sets with functions between them is a category (Def. ), to be denoted Set:

• $Obj_{Set} = \text{class of all sets}$;

• $Hom_{Set}(X,Y) = \text{set of functions from set X to set Y}$;

• $id_X \in Hom_{Set}(X,X) =$ identity function on set $X$;

• $\circ_{X_1,X_2,X_3} = \text{ordinary composition of functions}$.

More generally all kind of sets with structure, in the sense going back to Bourbaki, form categories, where the morphisms are the homomorphisms (whence the name “morphism”!). These are called concrete categories (we characterize them precisely in Example , further below):

###### Example

(basic examples of concrete categories)

For $\mathcal{S}$ a kind of mathematical structure, there is the category (Def. ) $\mathcal{S}Set$ whose objects are the corresponding structured sets, and whose morphisms are the corresponding structure homomorphisms, hence the functions of underlying sets which respect the given structure.

Basic examples of concrete categories include the following:

$\phantom{A}$concrete category$\phantom{A}$$\phantom{A}$objects$\phantom{A}$$\phantom{A}$morphisms$\phantom{A}$
$\phantom{A}$Set$\phantom{A}$sets$\phantom{A}$functions
$\phantom{A}$Top$\phantom{A}$topological spaces$\phantom{A}$$\phantom{A}$continuous functions$\phantom{A}$
$\phantom{A}$Mfd${}_{k}$$\phantom{A}$differentiable manifolds$\phantom{A}$$\phantom{A}$differentiable functions$\phantom{A}$
$\phantom{A}$Vect$\phantom{A}$vector spaces$\phantom{A}$$\phantom{A}$linear functions$\phantom{A}$
$\phantom{A}$Grp$\phantom{A}$groups$\phantom{A}$$\phantom{A}$group homomorphisms$\phantom{A}$
$\phantom{A}$Alg$\phantom{A}$algebras$\phantom{A}$$\phantom{A}$algebra homomorphism$\phantom{A}$

This is the motivation for the terminology “categories”, as the examples in Example are literally categories of mathematical structures. But not all categories are “concrete” in this way.

Some terminology:

###### Definition

(commuting diagram)

Let $\mathcal{C}$ be a category (Def. ), then a directed graph with edges labeled by morphisms of the category is called a commuting diagram if for any two vertices any two ways of passing along edges from one to the other yields the same composition of the corresponding morphisms.

For example, a commuting triangle is

$f = h \circ g \phantom{AAAAAA} \array{ && X \\ & {}^{\mathllap{ g }}\swarrow && \searrow^{ \mathrlap{ f } } \\ Y && \underset{\phantom{A}h\phantom{A}}{\longrightarrow} && Z }$

while a commuting square is

$g_1 \circ f_1 \;=\; g_2 \circ f_2 \phantom{AAAAAA} \array{ X &\overset{\phantom{A}f_1\phantom{A}}{\longrightarrow}& Y_1 \\ {}^{ \mathllap{f_2} }\big\downarrow && \big\downarrow^{\mathrlap{ g_1 }} \\ Y_2 &\underset{\phantom{A}g_2\phantom{A}}{\longrightarrow}& Z }$
###### Definition

(initial object and terminal object)

Let $\mathcal{C}$ be a category (Def. ). Then

1. an object $\ast \in \mathcal{C}$ is called a terminal object if for every other object $c \in \mathcal{C}$, there is a unique morphism from $c$ to $\ast$

$c \overset{\exists!}{\longrightarrow} \ast$

hence if the hom-set is a singleton $\ast \in Set$:

$Hom_{\mathcal{C}}(c,\ast) \;\simeq\; \ast \,.$
2. an object $\emptyset \in \mathcal{C}$ is called an initial object if for every other object $c \in \mathcal{C}$, there is a unique morphism from $\emptyset$ to $c$

$\emptyset \overset{\exists!}{\longrightarrow} c$

hence if the hom-set is a singleton $\ast \in Set$:

$Hom_{\mathcal{C}}(\emptyset,c) \;\simeq\; \ast \,.$
###### Definition

(small category)

If a category $\mathcal{C}$ (Def. ) happens to have as class $Obj_{\mathcal{C}}$ of objects an actual set (i.e. a small set instead of a proper class), then $\mathcal{C}$ is called a small category.

As usual, there are some trivial examples, that are however usefully made explicit for the development of the theory:

###### Example

(initial category and terminal category)

1. The terminal category $\ast$ is the category (Def. ) whose class of objects is the singleton set, and which has a single morphism on this object, necessarily the identity morphism.

2. The initial category or empty category $\emptyset$ is the category (Def. ) whose class of objects is the empty set, and which, hence, has no morphism whatsoever.

Clearly, these are small categories (Def. ).

###### Example

(preordered sets as thin categories)

Let $(S, \leq)$ be a preordered set. Then this induces a small category whose set of objects is $S$, and which has precisely one morphism $x \to y$ whenever $x \leq y$, and no such morphism otherwise:

(1)$x \overset{\exists !}{\to} y \phantom{AAA} \text{precisely if} \phantom{AAA} x \leq y$

Conversely, every small category with at most one morphism from any object to any other, called a thin category, induces on its set of objects the structure of a partially ordered set via (1).

Here the axioms for preordered sets and for categories match as follows:

$\phantom{A}$reflexivity$\phantom{A}$$\phantom{A}$transitivity$\phantom{A}$
$\phantom{A}$partially ordered sets$\phantom{A}$$\phantom{A}$ $x \leq x$ $\phantom{A}$$\phantom{A}$ $(x \leq y \leq z) \Rightarrow (x \leq z)$ $\phantom{A}$
$\phantom{A}$thin categories$\phantom{A}$$\phantom{A}$identity morphisms$\phantom{A}$$\phantom{A}$composition$\phantom{A}$
###### Definition

(isomorphism)

For $\mathcal{C}$ a category (Def. ), a morphism

$X \overset{f}{\to} Y \;\;\in Hom_{\mathcal{C}}(X,Y)$

is called an isomorphism if there exists an inverse morphism

$Y \overset{f^{-1}}{\longrightarrow} Y \;\; \in Hom_{\mathcal{C}}(Y,X)$

namely a morphism such that the compositions with $f$ are equal to the identity morphisms on $X$ and $Y$, respectively

$f^{-1} \circ f \;=\; id_X \phantom{AAA} f \circ f^{-1} \;=\; id_Y$
###### Definition

(groupoid)

If $\mathcal{C}$ is a category in which every morphism is an isomorphism (Def. ), then $\mathcal{C}$ is called a groupoid.

###### Example

(delooping groupoid)

For $G$ a group, there is a groupoid (Def. ) $\mathbf{B}G$ with a single object, whose single hom-set is $G$, with identity morphism the neutral element and composition the group operation in $G$:

• $Obj_{\mathbf{B}G} = \ast$

• $Hom_{\mathcal{C}}(\ast,\ast) \;=\; G$

In fact every groupoid with precisely one object is of the form.

###### Remark

(groupoids and homotopy theory)

Even though groupoids (Def. ) are special cases of categories (Def. ), the theory of groupoids in itself has a rather different flavour than that of category theory: Part of the homotopy hypothesis-theorem is that the theory of groupoids is really homotopy theory for the special case of homotopy 1-types.

(In applications in homotopy theory, groupoids are considered mostly in the case that the class $Obj_{\mathcal{C}}$ of objects is in fact a set: small groupoids, Def. ).

For this reason we will not have more to say about groupoids here, and instead relegate their discussion to the section on homotopy theory, further below.

There is a range of constructions that provide new categories from given ones:

###### Example

(opposite category and formal duality)

Let $\mathcal{C}$ be a category. Then its opposite category $\mathcal{C}^{op}$ has the same objects as $\mathcal{C}$, but the direction of the morphisms is reversed. Accordingly, composition in the opposite category $\mathcal{C}^{op}$ is that in $\mathcal{C}$, but with the order of the arguments reversed:

• $Obj_{\mathcal{C}^{op}} \;\coloneqq\; Obj_{\mathcal{C}}$;

• $Hom_{\mathcal{C}^{op}}(X,Y) \;\coloneqq\; Hom_{\mathcal{C}}(Y,X)$.

Hence for every statementa about some category $\mathcal{C}$ there is a corresponding “dual” statement about its opposite category, which is “the same but with the direction of all morphisms reversed”. This relation is known as formal duality.

###### Example

(product category)

Let $\mathcal{C}$ and $\mathcal{D}$ be two categories (Def. ). Then their product category $\mathcal{C} \times \mathcal{D}$ has as objects pairs $(c,d)$ with $c \in Obj_{\mathcal{C}}$ and $d \in Obj_{\mathcal{D}}$, and as morphisms pairs $(c_1 \overset{f}{\to} c_2) \in Hom_{\mathcal{C}}(c_1,c_2)$, $(d_1 \overset{g}{\to} d_2) \in Hom_{\mathcal{D}}(d_1,d_2)$, and composition is defined by composition in each entry:

• $Obj_{\mathcal{C} \times \mathcal{D}} \coloneqq Obj_{\mathcal{C}} \times Obj_{\mathcal{D}}$;

• $Hom_{\mathcal{C} \times \mathcal{D}}( (c_1,d_1), (c_2,d_2) ) \coloneqq Hom_{\mathcal{C}}(c_1,c_2) \times Hom_{\mathcal{D}}( d_1, d_2 )$

• $(f_2, g_2) \circ (f_1, g_1) \;\coloneqq\; ( f_2 \circ f_1, g_2 \circ g_1 )$

###### Definition

(functor)

Let $\mathcal{C}$ and $\mathcal{D}$ be two categories (Def. ). A functor from $\mathcal{C}$ to $\mathcal{D}$, to be denoted

$\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D}$

is

1. a function between the classes of objects:

$F_{Obj} \;\colon\; Obj_{\mathcal{C}} \longrightarrow Obj_{\mathcal{D}}$
2. for each pair $X,Y \in Obj_{\mathcal{C}}$ of objects a function

$F_{X,Y} \;\colon\; Hom_{\mathcal{C}}(X,Y) \longrightarrow Hom_{\mathcal{D}}(F_{Obj}(X), F_{Obj}(Y))$

such that

1. For each object $X \in Obj_{\mathcal{C}}$ the identity morphism is respected:

$F_{X,X}(id_X) \;=\; id_{F_{Obj}(X)} \,;$
2. for each triple $X_1, X_2, X_3 \in Obj_{\mathcal{C}}$ of objects, composition is respected: given

$X_1 \overset{f}{\longrightarrow} X_2 \overset{g}{\longrightarrow} X_3$

we have

$F_{X_1, X_3}(g \circ f ) \;=\; F_{X_2, X_3}(g) \circ F_{X_1,X_2}(f) \,.$
###### Example

(categories of small categories and of small groupoids)

It is clear that functors (Def. ) have a composition operation given componentwise by the composition of their component functions. Accordingly, this composition is unital and associative. This means that there is

1. the category (Def. ) Cat whose objects are small categories (Def. ) and whose morphisms are functors (Def. ) between these

2. the category (Def. ) Grpd whose objects are small groupoids (Def. ) and whose morphisms are functors (Def. ) between these.

###### Example

(hom-functor)

Let $\mathcal{C}$ be a category (Def. ). Then its hom-functor

$Hom_{\mathcal{C}} \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Set$

is the functor (Def. ) out of the product category (Def. ) of $\mathcal{C}$ with its opposite category to the category of sets, which sends a pair $X,Y \in \mathcal{C}$ of objects to the hom-set $Hom_{\mathcal{C}}(X,Y)$ between them, and which sends a pair of morphisms, with one of them into $X$ and the other out of $Y$, to the operation of composition with these morphisms:

$Hom_{\mathcal{C}} \;\;\colon\;\;\; \array{ X_1 & Y_1 \\ {}^{\mathllap{g}}\big\uparrow & \big\downarrow^{\mathrlap{h}} \\ X_2 & Y_2 } \;\;\mapsto\;\; \array{ Hom_{\mathcal{C}}(X_1, Y_1) \\ \big\downarrow^{ \mathrlap{ f \mapsto h \circ f \circ g } } \\ Hom_{\mathcal{C}}(X_2, Y_2) }$
###### Definition

(monomorphism and epimorphism)

Let $\mathcal{C}$ be a category (Def. ). Then a morphism $X \overset{f}{\to } Y$ in $\mathcal{C}$ is called

• a monomorphism if for every object $Z \in \mathcal{C}$ the hom-functor (Example ) out of $Z$ takes $f$ to an injective function of hom-sets:

$Hom_{\mathcal{C}}(Z,f) \;\colon\; Hom_{\mathcal{C}}(Z,X) \overset{\phantom{AAA}}{\hookrightarrow} Hom_{\mathcal{C}}(Z,Y) \,;$
• an epimorphism if for every object $Z \in \mathcal{Z}$ the hom-functor (Example ) into $Z$ takes $f$ to an injective function:

$Hom_{\mathcal{C}}( f,Z ) \;\colon\; Hom_{\mathcal{C}}(Y, Z) \overset{\phantom{AAA}}{\hookrightarrow} Hom_{\mathcal{C}}(X, Z) \,.$
###### Definition

(full, faithful and fully faithful functors)

A functor $F \;\colon\; \mathcal{C} \to \mathcal{D}$ (Def. ) is called

• a full functor if all its hom-functions are surjective functions

$Hom_{\mathcal{C}}(X,Y) \underoverset{surj}{F_{X,Y}}{\longrightarrow} Hom_{\mathcal{D}}(F(X), F(Y))$
• a faithful functor if all its hom-functions are injective functions

$Hom_{\mathcal{C}}(X,Y) \underoverset{inj}{F_{X,Y}}{\longrightarrow} Hom_{\mathcal{D}}(F(X), F(Y))$
• a fully faithful functor if all its hom-functions are bijective functions

$Hom_{\mathcal{C}}(X,Y) \underoverset{bij}{F_{X,Y}}{\longrightarrow} Hom_{\mathcal{D}}(F(X), F(Y))$

A fully faithful functor is also called a full subcategory-inclusion. We will denote this situation by

$\mathcal{C} \overset{\phantom{A}F\phantom{A}}{\hookrightarrow} \mathcal{D} \,.$
###### Example

(full subcategory on a sub-class of objects)

Let $\mathcal{C}$ be a category (Def. ) and let $S \subset Obj_{\mathcal{C}}$ be a sub-class of its class of objects. The there is a category $\mathcal{C}_S$ whose class of objects is $S$, and whose morphisms are precisely the morphisms of $\mathcal{C}$, between these given objects:

$Hom_{\mathcal{C}_S}(s_1, s_2) \;\coloneqq\; Hom_{\mathcal{C}}(s_1, s_2)$

with identity morphisms and composition defined as in $\mathcal{C}$. Then there is a fully faithful functor (Def. )

$\array{ \mathcal{C}_S &\overset{\phantom{AAAA}}{\hookrightarrow}& \mathcal{C} }$

which is the evident inclsuion on objects, and the identity function on all hom-sets.

This is called the full subcategory of $\mathcal{C}$ on the objects in $S$.

Beware that not every fully faithful functor is, in components, exactly of this form, but, assuming the axiom of choice, every fully faithful functor is so up to equivalence of categories (Def. ).

The concept of faithful functor from Def. allows to make precise the idea of concrete category from Example :

###### Example

(structured sets and faithful functors)

Let $\mathcal{S}$ be a kind of mathematical structure and let $\mathcal{S} Set$ be the category of $\mathcal{S}$-structured sets. Then there is the forgetful functor

$\mathcal{S}Set \longrightarrow Set$

which sends each structured set to the underlying set (“forgetting” the structure that it carries), and which sends functions of sets to themselves. That a homomorphism of structured sets is a function between the underlying sets satisfying a special condition implies that this is a faithful functor (Def. ).

Conversely, it makes sense to define structured sets in general to be the objects of a category $\mathcal{C}$ which is equipped with a faithful functor $\mathcal{C} \overset{faithful}{\longrightarrow} Set$ to the category of sets. See at structure for more on this.

###### Example

(spaces seen via their algebras of functions)

In any given context of geometry, there is typically a functor which sends any space of the given kind to its algebra of functions, and which sends a map (i.e. homomorphism) between the given spaces to the algebra homomorphism given by precomposition with that map (a hom-functor, Def. ). Schematically:

$\array{ \big\{ \text{geometric spaces} \big\} & \overset{ \text{algebra of functions} }{ \longrightarrow } & \big\{ \text{algebras} \big\}^{op} \\ \\ X_1 &\mapsto& FunctionsOn(X_1) \\ {}^{\mathllap{f}}\big\downarrow && \big\uparrow^{ \phi \mapsto \phi \circ f } \\ X_2 &\mapsto& FunctionsOn(X_2) }$

Since the precomposition operation reverses the direction of morphisms, as shown, these are functors from the given category of spaces to the opposite (Example ) of the relevant category of algebras.

In broad generality, there is a duality (“Isbell duality”) between geometry/spaces and algebra/algebras of functions) (“space and quantity”, Lawvere 86).

We now mention some concrete examples of this general pattern:

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topological spaces and C*-algebras

Consider

1. the category Top${}_{cpt}$ of compact topological Hausdorff spaces with continuous functions between them;

2. the category C*Alg of unital C*-algebras over the complex numbers

from Example .

Then there is a functor (Def. )

$C(-) \;\colon\; Top_{H,cpt} \longrightarrow C^\ast Alg^{op}$

from the former to the opposite category of the latter (Example ) which sends any compact topological space $X$ to its C*-algebra $C(X)$ of continuous functions $X \overset{\phi}{\to} \mathbb{C}$ with values in the complex numbers, and which sends every continuous function between compact spaces to the C*-algebra-homomorphism that is given by precomposition:

$C(-) \;\;\;\colon\;\;\; \array{ X &\mapsto & C(X) \\ {}^{\mathllap{ f }}\big\downarrow && \big\uparrow^{\mathrlap{ f^\ast : \phi \mapsto \phi \circ f }} \\ Y &\mapsto& C(Y) }$

Part of the statement of Gelfand duality is that this is a fully faithful functor, hence exhibiting a full subcategory-inclusion (Def. ), namely that of commutative C*-algebras:

$Top_{H,cpt} \overset{\phantom{AAA}}{\hookrightarrow} C^\ast Alg^{op} \,.$

$\,$

affine schemes and commutative algebras

The starting point of algebraic geometry is to consider affine schemes as the formal duals (Example ) of finitely generated commutative algebras over some algebraically closed ground field $\mathbb{K}$:

(2)$Aff_{\mathbb{K}} \;\;\coloneqq\;\; CAlg^{fin}_{\mathbb{K}}^{op} \,.$

Beware that the immediate identification (2) is often obscured by the definition of affine schemes as locally ringed spaces. While the latter is much more complicated, at face value, in the end it yields an equivalent category (Def. below) to the simple formal dualization (Example ) in (2), see here. Already in 1973 Alexander Grothendieck had urged to abandon, as a foundational concept, the more complicated definition in favor of the simpler one in (2), see Lawvere 03.

$\,$

smooth manifolds and real associative algebras

Consider

1. the category SmthMfd of smooth manifolds with smooth functions between them;

2. the category Alg${}_{\mathbb{R}}$ of associative algebras over the real numbers

from Example .

Then there is a functor (Def. )

$C^\infty(-) \;\colon\; SmthMfd \longrightarrow Alg_{\mathbb{R}}^{op}$

from the former to the opposite category of the latter (Def. ), which sends any smooth manifold $X$ to its associative algebra $C^\infty(X)$ of continuous functions $X \overset{\phi}{\to} \mathbb{R}$ to the real numbers, and which sends every smooth function between smooth manifolds to the algebra homomorphism that is given by precomposition:

$C^\infty(-) \;\;\;\colon\;\;\; \array{ X &\mapsto & C^\infty(X) \\ {}^{\mathllap{ f }}\big\downarrow && \big\uparrow^{\mathrlap{ f^\ast : \phi \mapsto \phi \circ f }} \\ Y &\mapsto& C^\infty(Y) }$

The statement of Milnor's exercise is that this this is a fully faithful functor, hence exhibiting a full subcategory-inclusion (Def. ):

$SmthMfd \overset{\phantom{AAAA}}{ \hookrightarrow } Alg_{\mathbb{R}}^{op} \,.$

These two statements, expressing categories of spaces as full subcategories of opposite categories of categories of algebras, are the starting point for many developments in geometry, such as algebraic geometry, supergeometry, noncommutative geometry and noncommutative topology.

$\,$

Since a fully faithful functor/full subcategory-embedding $\mathcal{C} \hookrightarrow \mathcal{D}$ exhibits the objects of $\mathcal{D}$ as a consistent generalization of the objects of $\mathcal{C}$, one may turn these examples around and define more general kinds of spaces as formal duals (Example ) to certain algebras:

$\,$

infinitesimally thickened points and formal Cartesian spaces

The category of infinitesimally thickened points is, by definition, the full subcategory (Example ) of the opposite category (Example ) of that of commutative algebras (Example ) over the real numbers

$\array{ InfThckPoint &\overset{\phantom{AAAA}}{\hookrightarrow}& Alg_{\mathbb{R}}^{op} \\ \mathbb{D} &\mapsto& C^\infty(\mathbb{D}) \\ && \coloneqq \mathbb{R} \oplus V }$

on those with a unique maximal ideal $V$ which is a finite-dimensional as an $\mathbb{R}$-vector space and a nilradical: for each $a \in V$ there exists $n \in \mathbb{N}$ such that $a^n = 0$.

The category of formal Cartesian spaces is, by definition, the full subcategory (Example ) of the opposite category (Example ) of that of commutative algebras (Example ) over the real numbers

$\array{ FormalCartSp &\overset{\phantom{AAAA}}{\hookrightarrow}& Alg_{\mathbb{R}}^{op} \\ \mathbb{R}^n \times \mathbb{D} &\mapsto& C^\infty(\mathbb{R}^n \times \mathbb{D}) \\ && \coloneqq C^\infty(\mathbb{R}^n) \otimes_{\mathbb{R}}(\mathbb{R} \oplus V) }$

on those which are tensor products of algebras, of an algebra of smooth functions on a Cartesian space $\mathbb{R}^n$, for some $n \in \mathbb{Z}$, and the algebra of functions on an infinitesimally thickened point.

Notice that the formal Cartesian spaces $\mathbb{R}^{n\vert q}$ are fully defined by this assignment.

$\,$

super points and super Cartesian spaces

The category of super points is by definition, the full subcategory (Example ) of the opposite category (Example ) of that of supercommutative algebras (Example ) over the real numbers

$\array{ SuperPoint &\overset{\phantom{AAAA}}{\hookrightarrow}& sCAlg_{\mathbb{R}}^{op} \\ \mathbb{R}^{0\vert q} &\mapsto& \Lambda_q }$

on the Grassmann algebras:

$\Lambda_q \;\coloneqq\; \mathbb{R}[ \theta_1, \cdots, \theta_q ]/( \theta_i \theta_j = - \theta_j \theta_i ) \phantom{AAAAA} q \in \mathbb{N} \,.$

More generally, the category of super Cartesian spaces is by definition, the full subcategory

$\array{ SuperCartSp &\overset{\phantom{AAAA}}{\hookrightarrow}& sCAlg_{\mathbb{R}}^{op} \\ \mathbb{R}^{n\vert q} &\mapsto& C^\infty(\mathbb{R}^n) \otimes_{\mathbb{R}} \Lambda_q }$

on the tensor product of algebras, over $\mathbb{R}$ of the algebra of smooth functions on a Cartesian space, and a Grassmann algebra, as above.

Notice that the super Cartesian spaces $\mathbb{R}^{n\vert q}$ are fully defined by this assignment. We discuss this in more detail in the chapter on supergeometry.

$\,$

### Natural transformations and presheaves

Given a system of (homo-)morphisms (“transformations”) in some category (Def. )

$F_X \overset{\phantom{A}\eta_X\phantom{A}}{\longrightarrow} G_X$

between objects that depend on some variable $X$, hence that are values of functors of $X$ (Def. ), one says that this is natural, hence a natural transformation (Def. below) if it is compatible with (homo-)morphisms of the variable itself.

These natural transformations are the evident homomorphisms between functors

$F \overset{\phantom{A}\eta\phantom{A}}{\longrightarrow} G \,,$

and hence there is a category of functors between any two categories (Example below).

A key class of such functor categories are those between an opposite category $\mathcal{C}^{op}$ and the base category of sets, these are also called categories of presheaves (Example below). It makes good sense (Remark below) to think of these as categories of “generalized objects of $\mathcal{C}$”, a perspective which is made precise by the statement of the Yoneda lemma (Prop. below) and the resulting Yoneda embedding (Prop. below). This innocent-looking lemma is the heart that makes category theory tick.

$\,$

###### Definition

(natural transformation and natural isomorphism)

Given two categories $\mathcal{C}$ and $\mathcal{D}$ (Def. ) and given two functors $F$ and $G$ from $\mathcal{C}$ to $\mathcal{D}$ (Def. ), then a natural transformation from $F$ to $G$

$\mathcal{C} \underoverset {\underset{G}{\longrightarrow}} {\overset{F}{\longrightarrow}} {\phantom{AA}\Downarrow \mathrlap{\eta} \phantom{AA}} \mathcal{D}$

is

• for each object $X \in Obj_{\mathcal{C}}$ a morphism

(3)$F(X) \overset{ \eta_X }{\longrightarrow} G(X)$

such that

• for each morphism $X \overset{f}{\longrightarrow} Y$ we have a commuting square (Def. ) of the form

(4)$\eta_Y\circ F(X) \;=\; G(Y)\circ \eta_X \phantom{AAAAAA} \array{ F(X) &\overset{\eta_X}{\longrightarrow}& G(X) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{G(f)}} \\ F(Y) &\underset{\eta_Y}{\longrightarrow}& G(Y) }$

(sometimes called the naturality square of the natural transformation).

If all the component morphisms $\eta_X$ are isomorphisms (Def. ), then the natural transformation $\eta$ is called a natural isomorphism.

For

$\mathcal{C} \underoverset {\underset{G}{\longrightarrow}} {\overset{F}{\longrightarrow}} {\phantom{AA}\Downarrow \mathrlap{\eta} \phantom{AA}} \mathcal{D} \phantom{AA} \text{and} \phantom{AA} \mathcal{C} \underoverset {\underset{H}{\longrightarrow}} {\overset{G}{\longrightarrow}} {\phantom{AA}\Downarrow \mathrlap{\rho} \phantom{AA}} \mathcal{D}$

two natural transformations as shown, their composition is the natural transformation

$\mathcal{C} \underoverset {\underset{H}{\longrightarrow}} {\overset{F}{\longrightarrow}} {\phantom{A}\Downarrow \mathrlap{\rho \circ \eta} \phantom{AAAA}} \mathcal{D}$

whose components (3) are the compositions of the components of $\eta$ and $\rho$:

(5)$(\rho \circ \eta)_X \;\coloneqq\; \rho_{X} \circ \eta_X \phantom{AAAAA} \array{ F(X) &\overset{\eta_X}{\longrightarrow}& G(X) &\overset{\rho_X}{\longrightarrow}& H(X) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{G(f)}} && \downarrow^{\mathrlap{H(f)}} \\ F(Y) &\underset{\eta_Y}{\longrightarrow}& G(Y) &\underset{\rho_Y}{\longrightarrow}& H(Y) }$
###### Example

(reduction of formal Cartesian spaces)

On the category FormalCartSp of formal Cartesian spaces Example , consider the endofunctor

$\array{ FormalCartSp &\overset{ \phantom{AA}\Re \phantom{AA} }{\longrightarrow}& FormalCartSp \\ \mathbb{R}^n \times \mathbb{D} &\mapsto& \mathbb{R}^n }$

which sends each formal Cartesian space to the underlying ordinary Cartesian space, forgetting the infinitesimally thickened point-factor. Moreover, on morphisms this functor is defined via the retraction

$\array{ id \colon & \mathbb{R}^n &\overset{i}{\longrightarrow}& \mathbb{R}^n \times \mathbb{D} &\overset{r}{\longrightarrow}& \mathbb{R}^n \\ & C^\infty(\mathbb{R}^n) &\underoverset{\text{quotient projection}}{i^\ast}{\longleftarrow}& C^\infty(\mathbb{R}^n) \otimes_{\mathbb{R}} (R \oplus V) &\underoverset{f \mapsto f \otimes 1}{r^\ast}{\longleftarrow}& C^\infty(\mathbb{R}^n) }$

as

$\array{ C^\infty(\mathbb{R}^n \times \mathbb{D}) &\phantom{AAA}&& C^\infty(\mathbb{R}^n) &\overset{i^\ast}{\longleftarrow}& C^\infty( \mathbb{R}^n \times \mathbb{D} ) \\ {}^{\mathllap{ f^\ast }}\big\uparrow && & {}^{ \mathllap{\Re( f^\ast ) \coloneqq i^\ast \circ f^\ast \circ r^\ast } }\big\uparrow && \big\uparrow^{ \mathrlap{ f^\ast } } \\ C^\infty(\mathbb{R}^{n'} \times {\mathbb{D}}') &&& C^\infty(\mathbb{R}^{n'}) &\overset{r^\ast}{\longrightarrow}& C^\infty( \mathbb{R}^{n'} \times {\mathbb{D}}') }$

This is indeed functorial due to the fact that any algebra homomorphism $f^\ast$ needs to send nilpotent elements to nilpotent elements, so that the following identity holds:

(6)$i^\ast \circ f^\ast \;=\; i^\ast \circ f^\ast \circ r^\ast \circ i^\ast \,.$

Then there is a natural transformation (Def. ) from this functor to the identity functor

$\Re \overset{ \phantom{A} \eta^{\Re} \phantom{A} }{\longrightarrow} Id$

whose components inject the underlying Cartesian space along the unit point inclusion of the infinitesimally thickened point:

$\array{ \Re\left( \mathbb{R}^n \times \mathbb{D} \right) \coloneqq & \mathbb{R}^n &\overset{ \phantom{A} \eta^\Re_{\mathbb{R}^n \times \mathbb{D}} }{\longrightarrow}& \mathbb{R}^n \times \mathbb{D} \\ & C^\infty(\mathbb{R}^n) &\overset{i^\ast}{\longleftarrow}& C^\infty(\mathbb{R}^n \times \mathbb{D}) \\ & {}^{ \mathllap{ i^\ast \circ f^\ast \circ r^\ast } }\big\uparrow && \big\uparrow^{\mathrlap{ f^\ast }} \\ & C^\infty(\mathbb{R}^{n'}) &\overset{i^\ast}{\longleftarrow}& C^\infty(\mathbb{R}^{n'} \times \mathbb{D}') }$

The commutativity of this naturality square is again the identity (6).

###### Example

(functor category)

Let $\mathcal{C}$ and $\mathcal{D}$ be categories (Def. ). Then the category of functors between them, to be denoted $[\mathcal{C}, \mathcal{D}]$, is the category whose objects are the functors $\mathcal{C} \overset{F}{\to} \mathcal{D}$ (Def. ) and whose morphisms are the natural transformations $F \overset{\eta}{\Rightarrow} G$ between functors (Def. ) and whose composition operation is the composition of natural transformations (5).

###### Example

(category of presheaves)

Given a category $\mathcal{C}$ (Def. ), a functor (Def. ) of the form

$F \;\colon\; \mathcal{C}^{op} \longrightarrow Set \,,$

hence out of the opposite category of $\mathcal{C}$ (Def. ), into the category of sets (Example ) is also called a presheaf (for reasons discussed below) on $\mathcal{C}$ or over $\mathcal{C}$.

The corresponding functor category (Example )

$PSh(\mathcal{C}) \;\coloneqq\; [\mathcal{C}^{op}, Set]$

is hence called the category of presheaves over $\mathcal{C}$.

###### Example

(representable presheaves)

Given a category $\mathcal{C}$ (Def. ), the hom-functor (Example ) induces the following functor (Def. ) from $\mathcal{C}$ to its category of presheaves (Def. ):

(7)$\array{ y & \colon & \mathcal{C} &\longrightarrow& [\mathcal{C}^{op}, Set] \\ \\ && && && c_1 &\overset{g}{\longrightarrow}& c_2 \\ && X &\mapsto& Hom_{\mathcal{C}}(-,X) &\phantom{AA}\colon\phantom{AA}& Hom_{\mathcal{C}}(c_1,X) &\overset{Hom_{\mathcal{C}}( g, X ) }{\longleftarrow}& Hom_{\mathcal{C}}(c_2, X) \\ && {}^{\mathllap{ f }}\big\downarrow && \big\downarrow^{ \mathrlap{ Hom_{\mathcal{C}}(-,f) } } && \big\downarrow^{ \mathrlap{ Hom_{\mathcal{C}}( c_1, f ) } } && \big\downarrow^{ \mathrlap{ Hom_{\mathcal{C}}(c_2,f) } } \\ && Y &\mapsto& Hom_{\mathcal{C}}(-,Y) &\phantom{AA}\colon\phantom{AA}& Hom_{\mathcal{C}}(c_1,Y) &\overset{Hom_{\mathcal{C}}( g, Y ) }{\longleftarrow}& Hom_{\mathcal{C}}(c_2, Y) }$

The presheaves $y(X) \coloneqq Hom_{\mathcal{C}}(-,X)$ in the image of this functor are called the representable presheaves and $X \in Obj_{\mathcal{C}}$ is called their representing object.

The functor (7) is also called the Yoneda embedding, due to Prop. below.

###### Remark

(presheaves as generalized spaces)

If a given category $\mathcal{C}$ (Def. ) is thought of as a category of spaces of sorts, as those in Example , then it will be most useful to think of the corresponding category of presheaves $[\mathcal{C}^{op}, Set]$ (Def. ) as a category of generalized spaces probe-able by the test spaces in $\mathcal{C}$ (Lawvere 86, p. 17).

Namely, imagine a generalized space $\mathbf{X}$ which is at least probe-able by spaces in $\mathcal{C}$. This should mean that for each object $c \in \mathcal{C}$ there is some set of geometric maps “$c \to \mathbf{X}$”. Here the quotation marks are to warn us that, at this point, $\mathbf{X}$ is not defined yet; and even if it were, it is not expected to be an object of $\mathcal{C}$, so that, at this point, an actual morphism from $c$ to $\mathbf{X}$ is not definable. But we may anyway consider some abstract set

(8)$\mathbf{X}(c) \; \text{"=} Hom(c,\mathbf{X})"$

whose elements we do want to think of maps (homomorphisms of spaces) from $c$ to $\mathbf{X}$.

That this is indeed consistent, in that we may actually remove the quotation remarks on the right of (8), is the statement of the Yoneda lemma, which we discuss as Prop. below.

A minimum consistency condition for this to make sense (we will consider further conditions later on when we discuss sheaves) is that we may consistently pre-compose the would-be maps from $c$ to $\mathbf{X}$ with actual morphisms $d \overset{f}{\to} c$ in $\mathcal{C}$. This means that for every such morphism there should be a function between these sets of would-be maps

$\array{ c && \mathbf{X}(d) \\ {}^{\mathllap{ f }}\big\downarrow && \big\uparrow{}^{\mathrlap{ \mathbf{X}(f) \, \text{"=}(-)\circ f\text{"}}} \\ d && \mathbf{X}(c) }$

which respects composition and identity morphisms. But in summary, this says that what we have defined thereby is actually a presheaf on $\mathcal{C}$ (Def. ), namely a functor

$\mathbf{X} \;\colon\; \mathcal{C}^{op} \longrightarrow Set \,.$

For consistency of regarding this presheaf as a presheaf of sets of plots of a generalized space, it ought to be true that every “ordinary space”, hence every object $X \in \mathcal{C}$, is also an example of a “generalized space probe-able by” object of $\mathcal{C}$, since, after all, these are the spaces which may manifestly be probed by objects $c \in \mathcal{C}$, in that morphisms $c \to X$ are already defined.

Hence the incarnation of $X \in \mathcal{C}$ as a generalized space probe-able by objects of $\mathcal{C}$ should be the presheaf $Hom_{\mathcal{C}}(-,X)$, hence the presheaf represented by $X$ (Example ), via the Yoneda functor (7).

At this point, however, a serious consistency condition arises: The “ordinary spaces” now exist as objects of two different categories: on the one hand there is the original $X \in \mathcal{C}$, on the other hand there is its Yoneda image $y(X) \in [\mathcal{C}^{op}, Set]$ in the category of generalized spaces. Hence we need to know that these two perspectives are compatible, notably that maps $X \to Y$ between ordinary spaces are the same whether viewed in $\mathcal{C}$ or in the more general context of $[\mathcal{C}^{op}, Set]$.

That this, too, holds true, is the statement of the Yoneda embedding, which we discuss as Prop. below.

Eventually one will want to impose one more consistency condition, namely that plots are determined by their local behaviour. This is the sheaf condition (Def. below) and is what leads over from category theory to topos theory below.

###### Proposition

(Yoneda lemma)

Let $\mathcal{C}$ be a category (Def. ), $X \in \mathcal{C}$ any object, and $\mathbf{Y} \in [\mathcal{C}^{op}, Set]$ a presheaf over $\mathcal{C}$ (Def. ).

Then there is a bijection

$\array{ Hom_{[\mathcal{C}^{op},Set]}( y(X), \mathbf(Y) ) &\overset{\simeq}{\longrightarrow}& \mathbf{Y}(X) \\ \eta &\mapsto& \eta_X(id_X) }$

between the hom-set of the category of presheaves from the presheaf represented by $X$ (7) to $\mathbf{Y}$, and the set which is assigned by $\mathbf{Y}$ to $X$.

###### Proof

By Example , an element in the set on the left is a natural transformation (Def. ) of the form

$\mathcal{C}^{op} \underoverset {\underset{\mathbf{Y}}{\longrightarrow}} {\overset{y(X)}{\longrightarrow}} {\phantom{AA} \Downarrow \mathrlap{\eta} \phantom{AA}} Set$

hence given by component functions (3)

$Hom_{\mathcal{C}}(c,X) \overset{\eta_c}{\longrightarrow} \mathbf{Y}(X)$

for each $c \in \mathcal{C}$. In particular there is the component at $c = X$

$\array{ Hom_{\mathcal{C}}(X,X) &\overset{\eta_X}{\longrightarrow}& \mathbf{Y}(X) \\ id_X &\mapsto& \eta_X(id_X) }$

and the identity morphism $id_X$ on $X$ is a canonical element in the set on the left. The statement to be proven is hence equivalently that for every element in $\mathbf{Y}(X)$ there is precisely one $\eta$ such that this element equals $\eta_X(id_X)$.

Now the condition to be satisfied by $\eta$ is that it makes its naturality squares (4) commute (Def. ). This includes those of the form

$\array{ id_X \in & Hom_{\mathcal{C}}(X,X) &\overset{\eta_X}{\longrightarrow}& \mathbf{Y}(X) \\ & {}^{\mathllap{ Hom_{\mathcal{C}}(f,X) }} \big\downarrow && \big\downarrow{}^{\mathrlap{\mathbf{Y}(f)}} \\ & Hom_{\mathcal{C}}(Y,X) &\underset{\eta_Y}{\longrightarrow}& \mathbf{Y}(Y) } \phantom{AAAA} \array{ \{id_X\} &\longrightarrow& \{\eta_X(id_X)\} \\ \big\downarrow && \big\downarrow \\ \{f\} &\longrightarrow& \big\{ \eta_Y(F) = \mathbf{Y}(f)( \eta_X(id_X) ) \big\} }$

for any morphism

$(Y \overset{f}{\longrightarrow} X) \;\in\; Hom_{\mathcal{C}}(Y,X) \,.$

As the diagram chase of elements on the right shows, this commutativity (Def. ) fixes $\eta_Y(f)$ for all $Y \in \mathcal{C}$ and all $f \in Hom_{\mathcal{C}}(Y,X)$ uniquely in terms of the element $\eta_{X}(id_X)$.

It remains only to see that there is no condition on the element $\eta_X(id_X)$, hence that with $\eta_Y(f)$ defined this way, the commutativity of all the remaining naturality squares is implies: The general naturality square for a morphism $Y_2 \overset{g}{\longrightarrow} Y_1$ is of the form

$\array{ & Hom_{\mathcal{C}}(Y_1,X) &\overset{\eta_{Y_1}}{\longrightarrow}& \mathbf{Y}(Y_1) \\ & {}^{\mathllap{ Hom_{\mathcal{C}}(g,X) }} \big\downarrow && \big\downarrow{}^{\mathrlap{\mathbf{Y}(g)}} \\ & Hom_{\mathcal{C}}(Y_2,X) &\underset{\eta_{Y_2}}{\longrightarrow}& \mathbf{Y}(Y_2) } \phantom{AAAA} \array{ \{f_1\} &\longrightarrow& \{ \mathbf{Y}(f_1)( \eta_X(id_X) ) \} \\ \big\downarrow && \big\downarrow \\ \{f_2 = f_1\circ g\} &\longrightarrow& \{\mathbf{Y}(f_2)( \eta_X(id_X) ) = \mathbf{Y}(g) \circ \mathbf{Y}(f_1) ( \eta_X(id_X) ) \} }$

As shown on the right, the commutativity of this diagram now follows from the functoriality $\mathbf{Y}(f_2) = \mathbf{Y}(f_1 \circ g)$ of the presheaf $\mathbf{Y}$.

As a direct corollary, we obtain the statement of the Yoneda embedding:

###### Proposition

(Yoneda embedding)

The assignment (7) of represented presheaves (Example ) is a fully faithful functor (Def. ), hence exhibits a full subcategory inclusion

$y \;\;\colon\;\; \array{ \mathcal{C} &\overset{\phantom{AAAA}}{\hookrightarrow}& [\mathcal{C}^{op}, Set] \\ X &\mapsto& Hom_{\mathcal{C}}(-,X) }$

of the given category $\mathcal{C}$ into its category of presheaves.

###### Proof

We need to show that for all $X_1, X_2 \in Obj_{\mathcal{C}}$ the function

(9)$\array{ Hom_{\mathcal{C}}(X_1, X_2) &\overset{ }{\longrightarrow}& Hom_{[\mathcal{C}^{op}, Set]} \big( Hom_{\mathcal{C}}(-,X_1) \;,\; Hom_{\mathcal{C}}(-,X_2) \big) \\ f &\mapsto& Hom_{\mathcal{C}}(-,f) }$

is a bijection. But the Yoneda lemma (Prop. ) states a bijection the other way around

$\array{ Hom_{[\mathcal{C}^{op}, Set]} \big( Hom_{\mathcal{C}}(-,X_1) \;,\; Hom_{\mathcal{C}}(-,X_2) \big) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(-,X_2)(X_1) &=& Hom_{\mathcal{C}}(X_1, X_2) \\ \eta && \mapsto && \eta_{X_1}( id_{X_1} ) \\ Hom_{\mathcal{C}}(-,f) && \mapsto && Hom_{\mathcal{C}}(X_1,f)(id_{X_1}) = f }$

and hence it is sufficient to see that this is a left inverse to (9). This follows by inspection, as shown in the third line above.

As a direct corollary we obtain the following alternative characterization of isomorphisms, to be compared with the definition of epimorphisms/monomorphisms in Def. :

###### Example

(isomorphism via bijection of hom-sets)

Let $\mathcal{C}$ be a category (Def. ), let $X, Y \in Obj_{X}$ be a pair of objects, and let $X \overset{f}{\to} Y \;\; \in Hom_{\mathcal{C}}(X,Y)$ be a morphism between them. Then the following are equivalent:

1. $X \overset{f}{\to} Y$ is an isomorphism (Def. ),

2. the hom-functors into and out of $f$ take values in bijections of hom-sets: i.e. for all objects $A \in Obj_{\mathcal{C}}$, we have

$Hom_{\mathcal{C}}(A,f) \;\colon\; Hom_{\mathcal{C}}(A,X) \overset{\simeq}{\longrightarrow} Hom_{\mathcal{C}}(A,Y)$

and

$Hom_{\mathcal{C}}(f,A) \;\colon\; Hom_{\mathcal{C}}(Y,A) \overset{\simeq}{\longrightarrow} Hom_{\mathcal{C}}(X,A)$

$\,$

The concepts of categories, functors and natural transformations constiture the “language of categories”. This language now allows to formulate the concept of adjoint functors (Def. ) and more generally that of adjunctions (Def. below. This is concept that category theory, as a theory, is all about.

Part of the data involved in an adjunction is its adjunction unit and adjunction counit (Def. below) and depending on their behaviour special cases of adjunctions are identified (Prop. below), which we discuss in detail in following sections:

$\phantom{A}$adjunction$\phantom{A}$
$\phantom{A}$Def. , Def. $\phantom{A}$
$\phantom{A}$unit is iso:$\phantom{A}$
$\phantom{A}$coreflection$\phantom{A}$
$\phantom{A}$Def. $\phantom{A}$
$\phantom{A}$counit is iso:$\phantom{A}$$\phantom{A}$reflection$\phantom{A}$
$\phantom{A}$Def.
$\phantom{A}$adjoint equivalence$\phantom{A}$
$\phantom{A}$Def. $\phantom{A}$

We now discuss four equivalent definitions of adjoint functors:

1. via hom-isomorphism (Def. below);

2. via adjunction unit and -counit satisfying triangle identities (Prop. );

3. via representing objects (Prop. );

4. via universal morphisms (Prop. below).

Then we discuss some key properties:

1. uniqueness of adjoints (Prop. below),

2. epi/mono/iso-characterization of adjunction (co-)units (Prop. below).

$\,$

###### Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be two categories (Def. ), and let

$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{} \mathcal{C}$

be a pair of functors between them (Def. ), as shown. Then this is called a pair of adjoint functors (or an adjoint pair of functors) with $L$ left adjoint and $R$ right adjoint, denoted

$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot} \mathcal{C}$

if there exists a natural isomorphism (Def. ) between the hom-functors (Example ) of the following form:

(10)$Hom_{\mathcal{D}}(L(-),-) \;\simeq\; Hom_{\mathcal{C}}(-,R(-)) \,.$

This means that for all objects $c \in \mathcal{C}$ and $d \in \mathcal{D}$ there is a bijection of hom-sets

$\array{ Hom_{\mathcal{D}}(L(c),d) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(c,R(d)) \\ ( L(c) \overset{f}{\to} d ) &\mapsto& (c \overset{\widetilde f}{\to} R(d)) }$

which is natural in $c$ and $d$. This isomorphism is called the adjunction hom-isomorphism and the image $\widetilde f$ of amorphism $f$ under this bijections is called the adjunct of $f$. Conversely, $f$ is called the adjunct of $\widetilde f$.

Naturality here means that for every morphism $g \colon c_2 \to c_1$ in $\mathcal{C}$ and for every morphisms $h\colon d_1\to d_2$ in $\mathcal{D}$, the resulting square

(11)$\array{ Hom_{\mathcal{D}}(L(c_1), d_1) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_1, R(d_1)) \\ {}^{\mathllap{Hom_{\mathcal{D}}(L(g), h)}}\big\downarrow && \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(g, R(h))}} \\ Hom_{\mathcal{D}}(L(c_2),d_2) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_2,R(d_2)) }$

commutes (Def. ), where the vertical morphisms are given by the hom-functor (Example ).

Explicitly, this commutativity, in turn, means that for every morphism $f \;\colon\; L(c_1) \to d_1$ with adjunct $\widetilde f \;\colon\; c_1 \to R(d_1)$, the adjunct of the composition is

$\widetilde{ \array{ L(c_1) &\overset{f}{\longrightarrow}& d_1 \\ {}^{\mathllap{L(g)}}\big\uparrow && \big\downarrow^{\mathrlap{h}} \\ L(c_2) && d_2 } } \;\;\;=\;\;\; \array{ c_1 &\overset{\widetilde f}{\longrightarrow}& R(d_1) \\ {}^{\mathllap{g}}\big\uparrow && \big\downarrow^{\mathrlap{R(h)}} \\ c_2 && R(d_2) }$
###### Definition

Given a pair of adjoint functors

$\mathcal{C} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot} \mathcal{D}$

according to Def. , one says that

1. for any $c \in \mathcal{C}$ the adjunct of the identity morphism on $L(c)$ is the unit morphism of the adjunction at that object, denoted

$\eta_c \coloneqq \widetilde{id_{L(c)}} \;\colon\; c \longrightarrow R(L(c))$
2. for any $d \in \mathcal{D}$ the adjunct of the identity morphism on $R(d)$ is the counit morphism of the adjunction at that object, denoted

$\epsilon_d \;\colon\; L(R(d)) \longrightarrow d$
###### Remark

It happens that there are subsequence adjoint functors:

If two functors are adjoint to each other as in Def. , we also say that we have an adjoint pair:

$L \;\dashv\; R \,.$

If one of these has yet another adjoint in the other direction, we speak of an adjoint triple

(12)$L \;\dashv\; C \;\dashv\; R \,.$

Notice that in the case of an adjoint triple (12), the adjunction unit of $C \dashv R$ and the adjunction counit of $L \dashv R$ (Def. ) provide, for each object $X$ in the domain of $C$, a diagram

(13)$L(C(X)) \overset{ \phantom{AA} \epsilon_X \phantom{AA} }{\longrightarrow} X \overset{ \phantom{AA} \eta_X \phantom{AA} }{\longrightarrow} R(C(X))$

which is usefully thought of as exhibiting the nature of $X$ as being in between two opposite extreme aspects $L(C(X))$ and $R(C(X))$ of $X$. This is illustrated by the following examples, and formalized by the concept of modalities that we turn to in Def. below.

###### Example

(floor and ceiling as adjoint functors)

Consider the canonical inclusion

$\mathbb{Z}_{\leq} \overset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow} \mathbb{R}_{\leq}$

of the integers into the real numbers, both regarded as preorders in the standard way (“lower or equal”). Regarded as full subcategory-inclusion (Def. ) of the corresponding thin categories, via Example , this inclusion functor has both a left and right adjoint functor (Def. ):

forming an adjoint triple (Def. )

(14)$\lceil(-)\rceil \;\;\dashv\;\; \iota \;\;\dashv\;\; \lfloor (-) \rfloor \,.$

The adjunction unit and adjunction counit express that each real number is in between its “opposite extreme integer aspects” (13) given by floor and ceiling

$\iota \lfloor x \rfloor \;\overset{\epsilon_X}{\leq}\; x \;\overset{\eta_x}{\leq}\; \iota \lceil x \rceil \,.$
###### Proof

First of all, observe that we indeed have functors (Def. )

$\lfloor(-)\rfloor \;,\; \lceil(-)\rceil \;\;\colon\; \mathbb{R} \longrightarrow \mathbb{Z}$

since floor and ceiling preserve the ordering relation.

Now in view of the identification of preorders with thin categories in Example , the hom-isomorphism (10) defining adjoint functors of the form $\iota \dashv \lfloor(-)\rfloor$ says for all $n \in \mathbb{Z}$ and $x \in \mathbb{R}$, that we have

$\underset{ \in \mathbb{Z}}{\underbrace{n \leq \lfloor x \rfloor}} \;\Leftrightarrow\; \underset{ \in \mathbb{R}}{\underbrace{n \leq x }} \,.$

This is clearly already the defining condition on the floor function $\lfloor x \rfloor$.

Similarly, the hom-isomorphism defining adjoint functors of the form $\lceil(-)\rceil \dashv \iota$ says that for all $n \in \mathbb{Z}$ and $x \in \mathbb{R}$, we have

$\underset{ \in \mathbb{Z}}{\underbrace{\lceil x \rceil \leq n}} \;\Leftrightarrow\; \underset{ \in \mathbb{R}}{\underbrace{x \leq n }} \,.$

This is evidently already the defining condition on the floor function $\lfloor x \rfloor$.

Notice that in both cases the condition of a natural isomorphism in both variables, as required for an adjunction, is automatically satisfied: For let $x \leq x'$ and $n' \leq n$, then naturality as in (11) means, again in view of the identifications in Example , that

$\array{ (n \leq \lfloor x \rfloor) &\Leftrightarrow& (n \leq x) \\ \Downarrow && \Downarrow \\ (n' \leq \lfloor x' \rfloor) &\Leftrightarrow& (n' \leq x') \\ \\ \in \mathbb{Z} && \in \mathbb{R} }$

Here the logical implications are equivalently functions between sets that are either empty or singletons. But Functions between such sets are unique, when they exist.

###### Example

(discrete and codiscrete topological spaces)

Consider the “forgetful functor$Top \overset{U}{\longrightarrow} Set$ from the category Top of topological spaces (Example ) to the category of sets (Def. ) which sends every topological space to its underlying set.

This has

These hence form an adjoint triple (Remark )

$Disc \;\dashv\; U \;\dashv\; coDisc \,.$

Hence the adjunction unit of $Disc \dashv U$ and the adjunction counit of $U \dashv coDisc$ exhibit every topology on a given set as “in between the opposite extremes” (13) of the discrete and the co-discrete

$Disc(U(X)) \overset{\epsilon}{\longrightarrow} X \overset{\eta}{\longrightarrow} coDisc(U(X)) \,.$
###### Lemma

If a functor $C$ is a right adjoint

$L \dashv C \;\;\colon\;\; \mathcal{C} \array{ \overset{\phantom{AA} L \phantom{AA} }{\longleftarrow} \\ \underset{\phantom{AA} C \phantom{AA} }{\longrightarrow} } \mathcal{D}$

of a pair of adjoint functors (Def. ), then application of the functor $C$ on any morphism $X \overset{f}{\to} Y \;\;\in \mathcal{C}$ is equal the joint operation of pre-composition with the $(L \dashv C)$-adjunction counit $\epsilon^\flat_{X}$ (Def. ), followed by passing to the $(L \dashv C)$-adjunct:

$\widetilde{\eta^\sharp_{Y} \circ (-)} \;=\; C_{X, Y}$

Dually, if $C$ is a left adjoint

$C \dashv R \;\;\colon\;\; \mathcal{C} \array{ \overset{\phantom{AA} C \phantom{AA} }{\longrightarrow} \\ \underset{\phantom{AA} R \phantom{AA} }{\longleftarrow} } \mathcal{D}$

then its action on any morphism $X \overset{f}{\to} Y \;\;\in \mathcal{C}$ equals the joint operation of post-composition with the $(C \dashv R)$-adjunction unit $\eta^{ \sharp }_{Y}$ (Def. ), followed by passing to the $(C \dashv R)$-adjunct:

$C_{X, Y} \;=\; \widetilde{ (-) \circ \epsilon^\flat_{X} } \,.$

In particular, if $C$ is the middle functor in an adjoint triple (Remark )

$L \dashv C \dashv R \;\;\colon\;\; \mathcal{C} \array{ \overset{\phantom{AA} L \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AAA} C \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} R \phantom{AA} }{\longleftarrow} } \mathcal{D}$

then these two operations coincide:

(15)$\widetilde{\eta^\sharp_{Y} \circ (-)} \;=\; C_{X, Y} \;=\; \widetilde{ (-) \circ \epsilon^\flat_{X} } \,.$
###### Proof

For the first equality, consider the following naturality square (4) for the adjunction hom-isomorphism (10):

$\array{ Hom_{\mathcal{D}}( C Y , C Y ) &\overset{\widetilde {(-)}}{\longrightarrow}& Hom_{\mathcal{C}}( Y, R C Y ) \\ {}^{\mathllap{ Hom_{\mathcal{D}}(C(f), C Y) }} \big\downarrow && \!\!\!\!\! \big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( f, R C Y ) }} \\ Hom_{\mathcal{D}}( C X, C Y ) &\overset{\widetilde{ (-) }}{\longleftarrow}& Hom_{\mathcal{C}}( X, R C Y ) } \phantom{AAAAA} \array{ \{ C Y \overset{id_{C Y}}{\to} C Y\} &\longrightarrow& \{ Y \overset{\eta^\sharp_{Y}}{\to} R C Y \} \\ \big\downarrow && \big\downarrow \\ \{ C X \overset{C(f)}{\to} C Y \} &\longleftarrow& \{ X \overset{\eta^\sharp_{Y} \circ f}{\longrightarrow} R C Y \} }$

Chasing the identity morphism $id_{C Y}$ through this diagram, yields the claimed equality, as shown on the right. Here we use that the right adjunct of the identity morphism is the adjunction unit, as shown.

The second equality is fomally dual:

$\array{ Hom_{\mathcal{D}}( C X, C X) &\overset{\widetilde { (-) }}{\longrightarrow}& Hom_{\mathcal{C}}( L C X , X) \\ {}^{\mathllap{ Hom_{\mathcal{D}}( C X, C(f) ) }} \big\downarrow && \big\downarrow^{ \mathrlap{ Hom_{X}( L C X, f ) } } \\ Hom_{\mathcal{D}}( C X, C Y ) &\overset{ \widetilde{ (-) } }{\longleftarrow}& Hom_{\mathcal{C} }( L C X, Y ) } \phantom{AAAAA} \array{ \{ C X \overset{id_{C X}}{\to} C X \} &\longrightarrow& \{ L C X \overset{\epsilon^{\flat}_X}{\to} X \} \\ \big\downarrow && \big\downarrow \\ \{ C X \overset{C(f)}{\to} C(Y) \} &\longleftarrow& \{ L C X \overset{f\circ \epsilon^\flat_{X} }{\longrightarrow} Y\} }$

$\,$

$\,$

We now consider a sequence of equivalent reformulations of the condition of adjointness.

###### Proposition

(general adjuncts in terms of unit/counit)

Consider a pair of adjoint functors

$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot} \mathcal{C}$

according to Def. , with adjunction units $\eta_c$ and adjunction counits $\epsilon_d$ according to Def. .

Then

1. The adjunct $\widetilde f$ of any morphism $L(c) \overset{f}{\to} d$ is obtained from $R$ and $\eta_c$ as the composite

(16)$\widetilde f \;\colon\; c \overset{\eta_c}{\longrightarrow} R(L(c)) \overset{R(f)}{\longrightarrow} R(d)$

Conversely, the adjunct $f$ of any morphism $c \overset{\widetilde f}{\longrightarrow} R(d)$ is obtained from $L$ and $\epsilon_d$ as

(17)$f \;\colon\; L(c) \overset{L(\widetilde f)}{\longrightarrow} R(L(d)) \overset{\epsilon_d}{\longrightarrow} d$
2. The adjunction units $\eta_c$ and adjunction counits $\epsilon_d$ are components of natural transformations of the form

$\eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L$

and

$\epsilon \;\colon\; L \circ R \Rightarrow Id_{\mathcal{D}}$
3. The adjunction unit and adjunction counit satisfy the triangle identities, saying that

(18)$id_{L(c)} \;\colon\; L(c) \overset{L(\eta_c)}{\longrightarrow} L(R(L(c))) \overset{\epsilon_{L(c)}}{\longrightarrow} L(c)$

and

$id_{R(d)} \;\colon\; R(d) \overset{\eta_{R(d)}}{\longrightarrow} R(L(R(d))) \overset{R(\epsilon_d)}{\longrightarrow} R(d)$
###### Proof

For the first statement, consider the naturality square (11) in the form

$\array{ id_{L(c)} \in & Hom_{\mathcal{D}}(L(c), L(c)) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c, R(L(c))) \\ & {}^{\mathllap{Hom_{\mathcal{D}}(L(id), f)}}\big\downarrow && \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(id, R(f))}} \\ & Hom_{\mathcal{D}}(L(c), d) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}( c, R(d) ) }$

and consider the element $id_{L(c_1)}$ in the top left entry. Its image under going down and then right in the diagram is $\widetilde f$, by Def. . On the other hand, its image under going right and then down is $R(f)\circ \eta_{c}$, by Def. . Commutativity of the diagram means that these two morphisms agree, which is the statement to be shown, for the adjunct of $f$.

The converse formula follows analogously.

The third statement follows directly from this by applying these formulas for the adjuncts twice and using that the result must be the original morphism:

\begin{aligned} id_{L(c)} & = \widetilde \widetilde { id_{L(c)} } \\ & = \widetilde{ c \overset{\eta_c}{\to} R(L(c)) } \\ & = L(c) \overset{L(\eta_c)}{\longrightarrow} L(R(L(c))) \overset{\epsilon_{L(c)}}{\longrightarrow} L(c) \end{aligned}

For the second statement, we have to show that for every moprhism $f \colon c_1 \to c_2$ the following square commutes:

$\array{ c_1 &\overset{f}{\longrightarrow}& c_2 \\ {}^{\mathllap{\eta_{c_1}}}\big\downarrow && \big\downarrow^{\mathrlap{\eta_{c_2}}} \\ R(L(c_1)) &\underset{ R(L(f)) }{\longrightarrow}& R(L(c_2)) }$

To see this, consider the naturality square (11) in the form

$\array{ id_{L(c_2)} \in & Hom_{\mathcal{D}}(L(c_2), L(c_2)) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_2, R(L(c_2))) \\ & {}^{\mathllap{Hom_{\mathcal{D}}(L(f),id_{L(c_2)})}}\big\downarrow && \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(f, R(id_{L(c_2)}))}} \\ & Hom_{\mathcal{D}}(L(c_1),L(c_2)) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_1,R(L(c_1))) }$

The image of the element $id_{L(c_2)}$ in the top left along the right and down is $f \circ \eta_{c_2}$, by Def. , while its image down and then to the right is $\widetilde {L(f)} = R(L(f)) \circ \eta_{c_1}$, by the previous statement. Commutativity of the diagram means that these two morphisms agree, which is the statement to be shown.

The argument for the naturality of $\epsilon$ is directly analogous.

###### Proposition

Two functors

$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{} \mathcal{C}$

are an adjoint pair in the sense that there is a natural isomorphism (10) according to Def. , precisely if they participate in an adjunction in the 2-category Cat, meaning that 1. there exist natural transformations

$\eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L$

and

$\epsilon \;\colon\; L \circ R \Rightarrow Id_{\mathcal{D}}$
2. which satisfy the triangle identities

$id_{L(c)} \;\colon\; L(c) \overset{L(\eta_c)}{\longrightarrow} L(R(L(c))) \overset{\epsilon_{L(c)}}{\longrightarrow} L(c)$

and

$id_{R(d)} \;\colon\; R(d) \overset{\eta_{R(d)}}{\longrightarrow} R(L(R(d))) \overset{R(\epsilon_d)}{\longrightarrow} R(d)$
###### Proof

That a hom-isomorphism (10) implies units/counits satisfying the triangle identities is the statement of the second two items of Prop. .

Hence it remains to show the converse. But the argument is along the same lines as the proof of Prop. : We now define forming of adjuncts by the formula (16). That the resulting assignment $f \mapsto \widetilde f$ is an isomorphism follows from the computation

\begin{aligned} \widetilde {\widetilde f} & = \widetilde{ c \overset{\eta_c}{\to} R(L(c)) \overset{R(f)}{\to} R(d) } \\ & = L(c) \overset{L(\eta_c)}{\to} L(R(L(c))) \overset{L(R(f))}{\to} L(R(d)) \overset{\epsilon_d}{\to} d \\ & = L(c) \overset{L(\eta_c)}{\to} L(R(L(c))) \overset{ \epsilon_{L(c)} }{\to} L(c) \overset{f}{\longrightarrow} d \\ & = L(c) \overset{f}{\longrightarrow} d \end{aligned}

where, after expanding out the definition, we used naturality of $\epsilon$ and then the triangle identity.

Finally, that this construction satisfies the naturality condition (11) follows from the functoriality of the functors involved, and the naturality of the unit/counit:

$\array{ c_2 &\overset{ \eta_{c_2} }{\longrightarrow}& R(L(c_2)) \\ {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{R(L(g))}} & \searrow^{\mathrlap{ R( L(g) \circ f ) }} \\ c_1 &\overset{\eta_{c_1}}{\longrightarrow}& R(L(c_1)) &\overset{R(f)}{\longrightarrow}& R(d_1) \\ && & {}_{R( h\circ f)}\searrow & \downarrow^{\mathrlap{ R(h) }} \\ && && R(d_2) }$

The condition (10) on adjoint functors $L \dashv R$ in Def. implies in particular that for every object $d \in \mathcal{D}$ the functor $Hom_{\mathcal{D}}(L(-),d)$ is a representable functor with representing object $R(d)$. The following Prop. observes that the existence of such representing objects for all $d$ is, in fact, already sufficient to imply that there is a right adjoint functor.

This equivalent perspective on adjoint functors makes manifest that adjoint functors are, if they exist, unique up to natural isomorphism, this is Prop. below.

###### Proposition

(adjoint functor from objectwise representing objects)

A functor $L \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ has a right adjoint $R \;\colon\; \mathcal{D} \to \mathcal{C}$, according to Def. , already if for all objects $d \in \mathcal{D}$ there is an object $R(d) \in \mathcal{C}$ such that there is a natural isomorphism

$Hom_{\mathcal{D}}(L(-),d) \underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow} Hom_{\mathcal{C}}(-,R(d)) \,,$

hence for each object $c \in \mathcal{C}$ a bijection

$Hom_{\mathcal{D}}(L(c),d) \underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow} Hom_{\mathcal{C}}(c,R(d))$

such that for each morphism $g \;\colon\; c_2 \to c_1$, the following diagram commutes

(19)$\array{ Hom_{\mathcal{D}}(L(c_1),d) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_1,R(d)) \\ {}^{\mathllap{ Hom_{\mathcal{C}}(L(g),id_d) }} \big\downarrow && \big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( f, id_{R(d)} ) }} \\ Hom_{\mathcal{D}}(L(c_2),d) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_2,R(d)) }$

(This is as in (11), except that only naturality in the first variable is required.)

In this case there is a unique way to extend $R$ from a function on objects to a function on morphisms such as to make it a functor $R \colon \mathcal{D} \to \mathcal{C}$ which is right adjoint to $L$. , and hence the statement is that with this, naturality in the second variable is already implied.

###### Proof

Notice that

1. in the language of presheaves (Example ) the assumption is that for each $d \in \mathcal{D}$ the presheaf

$Hom_{\mathcal{D}}(L(-),d) \;\in\; [\mathcal{D}^{op}, Set]$

is represented (7) by the object $R(d)$, and naturally so.

2. In terms of the Yoneda embedding (Prop. )

$y \;\colon\; \mathcal{D} \hookrightarrow [\mathcal{D}^{op}, Set]$

we have

(20)$Hom_{\mathcal{C}}(-,R(d)) = y(R(d))$

The condition (11) says equivalently that $R$ has to be such that for all morphisms $h \;\colon\; d_1 \to d_2$ the following diagram in the category of presheaves $[\mathcal{C}^{op}, Set]$ commutes

$\array{ Hom_{\mathcal{D}}(L(-),d_1) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(-,R(d_1)) \\ {}^{\mathllap{ Hom_{\mathcal{C}}( L(-) , h ) }} \big\downarrow && \big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( -, R(h) ) }} \\ Hom_{\mathcal{D}}(L(-),d_2) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(-, R(d_2)) }$

This manifestly has a unique solution

$y(R(h)) \;=\; Hom_{\mathcal{C}}(-,R(h))$

for every morphism $h \colon d_1 \to d_2$ under $y(R(-))$ (20). But the Yoneda embedding $y$ is a fully faithful functor (Prop. ), which means that thereby also $R(h)$ is uniquely fixed.

We consider one more equivalent characterization of adjunctions:

###### Definition

(universal morphism)

Let $\mathcal{C}, \mathcal{D}$ be two categories (Def. ) and let $R \;\colon\; \mathcal{D} \to \mathcal{C}$ be a functor (Def. )

Then for $c\in \mathcal{C}$ an object, a universal morphism from $c$ to $R$ is

1. an object $L(c)\in \mathcal{D}$,

2. a morphism $\eta_c \;\colon\; c \to R(L(c))$, to be called the unit,

such that for any $d\in \mathcal{D}$, any morphism $f \colon c\to R(d)$ factors through this unit $\eta_c$ as

(21)$f \;=\; \eta_c \circ R(\widetilde f) \phantom{AAAA} \array{ && c \\ & {}^{\mathllap{\eta_c}}\swarrow && \searrow^{\mathrlap{f}} \\ R(L(c)) &&\underset{R (\widetilde f)}{\longrightarrow}&& R(d) \\ \\ L(c) &&\underset{ \widetilde f}{\longrightarrow}&& d }$

for a unique morphism $\widetilde f \;\colon\; L(c) \longrightarrow d$, to be called the adjunct of $f$.

###### Proposition

(collection of universal morphisms equivalent to adjoint functor)

Let $R \;\colon\; \mathcal{D} \to \mathcal{C}$ be a functor (Def. ). Then the following are equivalent:

1. $R$ has a left adjoint functor $L \colon \mathcal{C} \to \mathcal{D}$ according to Def. ,

2. For every object $c \in \mathcal{C}$ there is a universal morphism $c \overset{\eta_c}{\longrightarrow} R(L(c))$, according to Def. .

###### Proof

In one direction, assume a left adjoint $L$ is given. Define the would-be universal arrow at $c \in \mathcal{C}$ to be the unit of the adjunction $\eta_c$ via Def. . Then the statement that this really is a universal arrow is implied by Prop. .

In the other direction, assume that universal arrows $\eta_c$ are given. The uniqueness clause in Def. immediately implies bijections

$\array{ Hom_{\mathcal{D}}(L(c),d) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(c,R(d)) \\ \left( L(c) \overset{\widetilde f}{\to} d \right) &\mapsto& \left( c \overset{\eta_c}{\to} R(L(c)) \overset{ R(\widetilde f) }{\to} R(d) \right) }$

Hence to satisfy (10) it remains to show that these are natural in both variables. In fact, by Prop. it is sufficient to show naturality in the variable $d$. But this is immediate from the functoriality of $R$ applied in (21): For $h \colon d_1 \to d_2$ any morphism, we have

$\array{ && c \\ & {}^{\mathllap{\eta_c}}\swarrow && \searrow^{\mathrlap{f}} \\ R (L(c)) &&\underset{R (\widetilde f)}{\longrightarrow}&& R(d_1) \\ && {}_{\mathllap{ R( h\circ \widetilde f ) }}\searrow && \downarrow^{\mathrlap{R(h)}} \\ && && R(d_2) }$

The following equivalent formulation (Prop. ) of universal morphisms is often useful:

###### Example

(comma category)

Let $\mathcal{C}$ be a category, let $c \in \mathcal{C}$ be any object, and let $F \;\colon\; \mathcal{D} \to \mathcal{C}$ be a functor.

1. The comma category $c/F$ is the category whose objects are pairs consisting of an object $d \in \mathcal{D}$ and morphisms $X \overset{f}{\to} F(d)$ in $\mathcal{C}$, and whose morphisms $(d_1,X_1,f_1) \to (d_2,X_2,f_2)$ are the morphisms $X_1 \overset{g}{\longrightarrow} X_2$ in $\mathcal{C}$ that make a commuting triangle (Def. ):

$f_2\circ F(g) \;=\; f_1 \phantom{AAAAAA} \array{ X_1 && \overset{\phantom{AA} g \phantom{AA}}{\longrightarrow} && X_2 \\ F(X_1) && \overset{\phantom{AA} F(g) \phantom{AA}}{\longrightarrow} && F(X_2) \\ & {}_{\mathllap{f_1}}\searrow && \swarrow_{\mathrlap{f_2}} \\ && c }$

There is a canonical functor

$\array{ F/c &\overset{}{\longrightarrow}& \mathcal{D} } \,.$
2. The comma category $F/c$ is the category whose objects are pairs consisting of an object $d \in \mathcal{D}$ and a morphism $F(d) \overset{f}{\to} X$ in $\mathcal{C}$, and whose morphisms $(d_1,X_1,f_1) \to (d_2,X_2,f_2)$ are the morphisms $X_1 \overset{g}{\longrightarrow} X_2$ in $\mathcal{C}$ that make a commuting triangle (Def. ):

$f_2\circ F(g) \;=\; f_1 \phantom{AAAAAA} \array{ && c \\ & {}^{\mathllap{f_1}}\swarrow && \searrow^{\mathrlap{f_2}} \\ F(X_1) && \underset{\phantom{AA} F(g) \phantom{AA}}{\longrightarrow} && F(X_2) \\ X_1 && \underset{ \phantom{AA} g \phantom{AA} }{\longrightarrow} && X_2 }$

Again, there is a canonical functor

(22)$\array{ c/F &\overset{}{\longrightarrow}& \mathcal{D} } ]$

With this definition, the following is evident:

###### Proposition

(universal morphisms are initial objects in the comma category)

Let $\mathcal{C} \overset{R}{\longrightarrow} \mathcal{D}$ be a functor and $d \in \mathcal{D}$ an object. Then the following are equivalent:

1. $d \overset{\eta_d}{\to} R(c)$ is a universal morphism into $R(c)$ (Def. );

2. $(d, \eta_d)$ is the initial object (Def. ) in the comma category $d/R$ (Example ).

$\,$

$\,$

After these equivalent characterizations of adjoint functors, we now consider some of their main properties:

###### Proposition

(adjoint functors are unique up to natural isomorphism)

The left adjoint or right adjoint to a functor (Def. ), if it exists, is unique up to natural isomorphism (Def. ).

###### Proof

Suppose the functor $L \colon \mathcal{D} \to \mathcal{C}$ is given, and we are asking for uniqueness of its right adjoint, if it exists. The other case is directly analogous.

Suppose that $R_1, R_2 \;\colon\; \mathcal{C} \to \mathcal{D}$ are two functors which both are right adjoint to $L$. Then for each $d \in \mathcal{D}$ the corresponding two hom-isomorphisms (10) combine to say that there is a natural isomorphism/

$\Phi_d \;\colon\; Hom_{\mathcal{C}}(-,R_1(d)) \;\simeq\; Hom_{\mathcal{C}}(-,R_2(d))$

As in the proof of Prop. , the Yoneda lemma implies that

$\Phi_d \;=\; y( \phi_d )$

for some isomorphism

$\phi_d \;\colon\; R_1(d) \overset{\simeq}{\to} R_2(d) \,.$

But then the uniqueness statement of Prop. implies that the collection of these isomorphisms for each object constitues a natural isomorphism between the functors (Def. ).

###### Proposition

(characterization of epi/mono/iso (co-)unit of adjunction)

Let

$L \dashv R \;\colon\; \mathcal{D} \underoverset {\underset{\phantom{A}R\phantom{A}}{\longrightarrow}} {\overset{\phantom{A}L\phantom{A}}{\longleftarrow}} {\bot} \mathcal{C}$

be a pair of adjoint functors (Def. ).

Recall the definition of

1. adjunction unit/counit, from Def. )

2. faithful/fully faithful functor from Def.

3. mono/epi/isomorphism from Def. and Def. .

The following holds:

• $R$ is faithful precisely if all components of the counit are epimorphisms $L R(c) \underoverset{\phantom{A}epi\phantom{A}}{\eta_c}{\to} c$;

• $L$ is faithful precisely if all components of the unit are monomorphisms $d \underoverset{mono}{\eta_d}{\to} R L(d)$

• $R$ is full and faithful (exhibits a reflective subcategory, Def. ) precisely if all components of the counit are isomorphisms $L R(c) \underoverset{\phantom{A}iso\phantom{A}}{\eta_c}{\to} c$

• $L$ is full and faithful (exhibits a coreflective subcategory, def. ) precisely if all component of the unit are isomorphisms $d \underoverset{\phantom{A}iso\phantom{A}}{\eta_d}{\to} R L(d)$.

###### Proof

This follows directly by Lemma , using the definition of epi/monomorphism (Def. ) and the characterization of isomorphism from Example .

To complete this pattern, we will see below in Prop. that following are equivalent:

###### Proposition

(right/left adjoint functors preserve monomorphism/epimorphisms and terminal/initial objects)

Every right adjoint functor (Def. ) preserves

1. terminal objects (Def. ),

2. monomorphisms (Def. )

Every left adjoint functor (Def. ) preserves

1. initial objects (Def. ),

2. epimorphisms (Def. ).

###### Proof

This is immediate from the adjunction hom-isomorphism (10), but we spell it out:

We consider the first case, the second is formally dual (Example ). So let $R \;\colon\; \mathcal{C} \to \mathcal{D}$ be a right adjoint functor with left adjoint $L$.

Let $\ast \in \mathcal{C}$ be a terminal object (Def. ). We need to show that for every object $d \in \mathcal{D}$ the hom-set $Hom_{\mathcal{D}}(d,R(\ast)) \simeq \ast$ is a singleton. But by the hom-isomorphism (10) we have a bijection

\begin{aligned} Hom_{\mathcal{d}}(d,R(\ast)) & \simeq Hom_{\mathcal{C}}(L(d), \ast) \\ & \simeq \ast \,, \end{aligned}

where in the last step we used that $\ast$ is a terminal object, by assumption.

Next let $c_1 \overset{f}{\hookrightarrow} c_2$ be a monomorphism. We need to show that for $d \in \mathcal{D}$ any object, the hom-functor out of $d$ yields a monomorphism

$Hom_{\mathcal{D}}(d, R(f)) \;\colon\; Hom_{\mathcal{D}}(d, R(c_1)) \hookrightarrow Hom_{\mathcal{D}}(d, R(c_2)) \,.$

Now consider the following naturality square (11) of the adjunction hom-isomorphism (10):

$\array{ Hom_{\mathcal{D}}(d, R(c_1)) &\simeq& Hom_{\mathcal{C}}(L(d), c_1) \\ {}^{ \mathllap{ Hom_{\mathcal{D}}(d,R(f)) } }\big\downarrow && \big\downarrow^{ \mathrlap{ Hom_{\mathcal{C}}( L(d),f ) } }_{\mathrlap{mono}} \\ Hom_{\mathcal{D}}(d, R(c_2)) &\simeq& Hom_{\mathcal{C}}(L(d), c_2) }$

Here the right vertical function is an injective function, by assumption on $f$ and the definition of monomorphism. Since the two horizontal functions are bijections, this implies that also $Hom_{\mathcal{d}}(d,R(f))$ is an injection.

But the main preservation property of adjoint functors is that adjoints preserve (co-)limits. This we discuss as Prop. below, after introducing limits and colimits in Def. below.

$\,$

Prop. says that adjoint functors are equivalenty “adjunctions in Cat”, as defined there. This is a special case of a general more abstract concept of adjunction, that is useful:

###### Definition

(strict 2-category)

A strict category $\mathcal{C}$ is

1. a class $Obj_{\mathcal{C}}$, called the class of objects;

2. for each pair $X,Y \in Obj_{\mathcal{C}}$ of objects, a small category $Hom_{\mathcal{C}}(X,Y) \in Cat$ (Def. ), called the hom-category from $X$ to $Y$.

We denote the objects of this hom-category by arrows like this:

$X \overset{f}{\longrightarrow} Y \;\;\in Obj_{Hom_{\mathcal{C}}(X,Y)}$

and call them the 1-morphisms of $\mathcal{C}$,

and we denote the morphisms in the hom-category by double arrows, like this:

$X \underoverset {\underset{g}{\longrightarrow}} {\overset{f}{\longrightarrow}} {\Downarrow{}^{\mathrlap{\phi}}} Y$

and call these the 2-morphisms of $\mathcal{C}$;

3. for each object $X \in Obj_{\mathcal{C}}$ a 1-morphism

$X \overset{id_X}{\to} X \;\; \in Hom_{\mathcal{C}}(X,X)$

called the identity morphism on $X$;

4. for each triple $X_1, X_2, X_3 \in Obj$ of objects, a functor (Def. )

$\array{ Hom_{\mathcal{C}}(X_1, X_2) &\times& Hom_{\mathcal{C}}(X_2, X_3) &\overset{\circ_{X_1,X_2,X_3}}{\longrightarrow}& Hom_{\mathcal{C}}(X_1, X_3) \\ X_1 \overset{f}{\to} X_2 &,& X_2 \overset{f}{\to} X_3 &\mapsto& X_1 \overset{ g \circ f }{\longrightarrow} X_3 }$

from the product category (Example ) of hom-categories, called composition;

such that:

1. for all pairs of objects $X,Y \in Obj_{\mathcal{C}}$ unitality holds:

$(-) \circ id_X \;=\; id_{ Hom_{\mathcal{C}}(X,Y) } \phantom{AAAA} id_Y \circ (-) \;=\; id_{ Hom_{\mathcal{C}}(X,Y) }$
2. for all quadruples of objects $X_1, X_2, X_3, X_4 \in Obj_{\mathcal{C}}$ composition satifies associativity, in that the following two composite functors are equal:

$\array{ Hom_{\mathcal{C}}(X_1, X_2) \times Hom_{\mathcal{C}}(X_2, X_3) \times Hom_{\mathcal{C}}(X_3, X_4) &\overset{((-)\circ (-))\circ (-)}{\longrightarrow}& Hom_{\mathcal{C}}(X_1, X_3) \times Hom_{\mathcal{C}}(X_3, X_4) \\ {}^{ \mathllap{ (-) \circ ( (-) \circ (-) ) } }\Big\downarrow && \Big\downarrow{}^{ (-) \circ (-) } \\ Hom_{\mathcal{C}}(X_1, X_2) \times Hom_{\mathcal{C}}(X_2, X_4) &\underset{(-)\circ (-)}{\longrightarrow}& Hom_{\mathcal{C}}(X_1, X)4) }$

The archetypical example of a strict 2-category is the category of categories:

###### Example

(2-category of categories)

There is a strict 2-category (Def. ) Cat whose

with the evident composition operations.

With a concept of 2-category in hand, we may phrase Prop. more abstractly:

###### Definition

Let $\mathcal{C}$ be a strict 2-category (Def. ). Then an adjunction in $\mathcal{C}$ is

1. a pair of objects $\mathcal{C}, \mathcal{D} \in Obj_{\mathcal{C}}$;

2. 1-morphisms

$\mathcal{D} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {} \mathcal{C}$

called the left adjoint $L$ and right adjoint $R$;

3. 2-morphisms

$id_{\mathcal{C}} \overset{\eta}{\Rightarrow} R \circ L$, called the adjunction unit

$L \circ R \overset{\epsilon}{\Rightarrow} id_{\mathcal{D}}$, called the adjunction counit

such that the following triangle identities hold: We denote this situation by

$\mathcal{D} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{C}$

Hence via Example , Prop. says that an adjoint pair of functors is equivalente an adjunction in the general sense of Def. , realized in the very large strict 2-category Cat of categories.

This more abstract perspecive on adjunctions allow us now to understand “duality of dualities” as adjunction in a 2-category of adjunctions:

###### Example

(strict 2-category of categories with adjoint functors between them)

Let $Cat_{Adj}$ be the strict 2-category which is defined just as Cat (Def. ) but with the 1-morphisms being functors that are required to be left adjoints (Def. ).

Since adjoints are unique up to natural isomorphism (Prop. ), this may be thought of as a 2-category whose 1-morphisms are adjoint pairs of functors.

###### Example

An adjunction (Def. ) in the 2-category $Cat_{Adj}$ of categories with adjoint functors between them (Example ) is equivalently an adjoint triple of functors (Remark ):

The adjunction says that two left adjoint functors $L_1$ and $L_2$, which, hence each participate in an adjoint pair

$L_1 \dashv R_1 \phantom{AAAA} L_2 \dashv R_2$

$L_1 \dashv L_2 \,.$

By essentiall uniqueness of adjoints (Prop. ) this implies a natural isomorphism $R_1 \simeq L_2$ and hence an adjoint triple:

$\mathcal{D} \array{ \underoverset{\bot \phantom{\simeq A_a}}{ L_1 \phantom{\simeq A_a} }{\longleftarrow} \\ \underoverset{\phantom{A_a \simeq}\bot}{R_1 \simeq L_2}{\longrightarrow} \\ \overset{ \phantom{A_a \simeq} R_2 }{\longleftarrow} } \mathcal{C}$

Example suggest to consider a slight variant of the concept of strict 2-categories which allows to make the duality between left adjoints and right adjoints explicit:

###### Definition

(double category)

A double category $\mathcal{C}$ is

1. a pair of categories $\mathcal{C}_h$, $\mathcal{C}_v$ (Def. ) which share the same class of objects: $Obj_{\mathcal{C}_1} = Obj_{\mathcal{C}_2}$, to be called the class $Obj_{\mathcal{C}}$ of objects of $\mathcal{C}$

where the morphisms of $\mathcal{C}_h$ are to be called the horizontal morphisms of $\mathcal{C}$,

while the morphisms of $\mathcal{C}_v$ are to be called the vertical morphisms of $\mathcal{C}$,

2. for each quadruple of objects $a,b,c,d,e \in Obj_{\mathcal{C}}$ and pairs of pairs of horizontal/vertical morphisms of the form

$\array{ a &\overset{f \in \mathcal{C}_h}{\longrightarrow}& b \\ {}^{\mathllap{h \in \mathcal{C}_v}}\big\downarrow && \big\downarrow{}^{\mathrlap{k \in \mathcal{C}_v}} \\ c &\underset{g \in \mathcal{C}_h}{\longrightarrow}& }$

a set $2Hom(f,g,h,k)$, to be called the set of 2-morphisms of $\mathcal{C}$ between the given 1-morphisms, whose elements we denote by

$\array{ a &\overset{f \in \mathcal{C}_h}{\longrightarrow}& b \\ {}^{\mathllap{h \in \mathcal{C}_v}}\big\downarrow &\swArrow& \big\downarrow{}^{\mathrlap{k \in \mathcal{C}_v}} \\ c &\underset{g \in \mathcal{C}_h}{\longrightarrow}& d }$
3. a horizontal and a vertical composition operation of 2-morphisms which is unitality and associative in both directions in the evident way, which respects composition in $\mathcal{C}_h$ and $\mathcal{C}_v$, and such that horizontal and vertical composition commute over each other in the evident way.

###### Example

(double category of squares of a strict 2-category)

Let $\mathcal{C}$ be a strict 2-category (Def. ). Then its double category of squares $Sq(\mathcal{C})$ is the double category (Def. ) whose

• objects are those of $\mathcal{C}$;

• horizontal morphisms and vertical morphisms are both the 1-morphisms of $\mathcal{C}$;

• 2-morphisms

$\array{ a &\overset{f \in \mathcal{C}_h}{\longrightarrow}& b \\ {}^{\mathllap{h \in \mathcal{C}_v}}\big\downarrow &{}^{\mathllap{\phi}}\swArrow& \big\downarrow{}^{\mathrlap{k \in \mathcal{C}_v}} \\ c &\underset{g \in \mathcal{C}_h}{\longrightarrow}& d }$

are the 2-morphisms of $\mathcal{C}$ between the evident composites of 1-morphisms:

$k \circ f \overset{\phi}{\Rightarrow} g\circ h$

and composition is given by the evident compositions in $\mathcal{C}$.

###### Remark

(strict and weak 2-functors)

Given two strict 2-categories (Def. ) or double categories (Def. ), $\mathcal{C}, \mathcal{D}$, there is an evident notion of 2-functor or double functor

$\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D}$

between them, namely functions on objects, 1-morphisms and 2-morphisms which respect all the composition operations and identity morphisms.

These are also called strict 2-functors.

This is in contrast to a more flexible concept of weak 2-functors, often called pseudofunctors, which respect composition of 1-morphisms only up to invertible 2-morphisms (which themselves are required to satisfy some coherence condition):

$\array{ && Y \\ & {}^{\mathllap{ F(f) }}\nearrow &\Downarrow{}^{\rho}_{}\simeq& \searrow^{\mathrlap{F(G)}} \\ X && \underset{F(g \circ f)}{\longrightarrow} && Z }$

We will see an important example of a weak double functor in the construction of derived functors of Quillen functors, below in Prop. .

$\,$

### Equivalences

We have seen functors (Def. ) as the homomorphisms between categories (Def. ). But functors themselves are identified only up to natural isomorphism (Def. ), reflective the fact that they are the 1-morphisms in a 2-category of categories (Example ). This means that in identifying two categories, we should not just ask for isomorphisms between them, hence for a functor between them that has a strict inverse morphism, but just for an inverse up to natural isomorphism.

This is called an equivalence of categories (Def. below). A particularly well-behaved equivalence of categories is an equivalence exhibited by an adjoint pair of functors, called an adjoint equivalence of categories (Def. below). In fact every equivalence of categories may be improved to an adjoint equivalence (Prop. ).

$\,$

###### Definition

Let $\mathcal{C}$, $\mathcal{D}$ be two categories (Def. ). Then an adjoint equivalence of categories between them is a pair adjoint functors (Def. )

$\array{ \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\phantom{A} \phantom{{}_{\bot}}\simeq_{\bot} \phantom{A}} \mathcal{D} }$

such that their unit $\eta$ and counit $\epsilon$ (Def. ) are natural isomorphisms (as opposed to just being natural transformations)

$\eta\;\colon\; id_{\mathcal{D}} \overset{\simeq}{\Rightarrow} R \circ L \phantom{AAA} \text{and} \phantom{AAA} \epsilon\;\colon\; L \circ R \overset{\simeq}{\Rightarrow} id_{\mathcal{C}} \,.$

There is also the following, seemingly weaker, notion:

###### Definition

(equivalence of categories)

Let $\mathcal{C}$, $\mathcal{D}$ be two categories (Def. ). Then an equivalence of categories

$\array{ \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\phantom{AA} \simeq \phantom{AA}} \mathcal{D} }$

is a pair of functors back and forth, as shown (Def. ), together with natural isomorphisms (Def. ) between their composition and the identity functors:

$id_{\mathcal{D}} \overset{\simeq}{\Rightarrow} R \circ L \phantom{AAA} \text{and} \phantom{AAA} L \circ R \overset{\simeq}{\Rightarrow} id_{\mathcal{C}} \,.$

If a functor participates in an equivalence of categories, that functor alone is usually already called an equivalence of categories. If there is any equivalence of categories between two categories, these categories are called equivalent.

###### Proposition

(every equivalence of categories comes from an adjoint equivalence of categories)

Let $\mathcal{C}$ and $\mathcal{D}$ be two categories (Def. ). Then the they are equivalent (Def. ) precisely if there exists an adjoint equivalence of categories between them (Def. ).

Moreover, let $R \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a functor (Def. ) which participates in an equivalence of categories (Def. ). Then for every functor $L \;\colon\; \mathcal{D} \to \mathcal{C}$ equipped with a natural isomorphism

$\eta \;\colon\; id_{\mathcal{D}} \overset{\simeq}{\Rightarrow} R \circ L$

there exists a natural isomorphism

$\epsilon \;\colon\; L \circ R \overset{\simeq}{\Rightarrow} id_{\mathcal{C}}$

which completes this to an adjoint equivalence of categories (Def. ).

###### Proposition

(fixed point equivalence of an adjunction)

Let

$\mathcal{D} \underoverset {\underset{ R }{\longrightarrow}} {\overset{ L }{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} \mathcal{C}$

be a pair of adjoint functors (Def. ). Say that

1. an object $c \in \mathcal{C}$ is a fixed point of the adjunction if its adjunction unit (Def. ) is an isomorphism (Def. )

$c \underoverset{\simeq}{\eta_c}{\longrightarrow} R L (c)$

and write

$\mathcal{C}_{fix} \hookrightarrow \mathcal{C}$

for the full subcategory on these fixed objects (Example )

2. an object $d \in \mathcal{D}$ is a fixed point of the adjunction if its adjunction counit (Def. ) is an isomorphism (Def. )

$L R(d) \underoverset{\simeq}{\epsilon_d}{\longrightarrow}$

and write

$\mathcal{D}_{fix} \hookrightarrow \mathcal{D}$

for the full subcategory on these fixed objects (Example )

Then the adjunction (co-)restrics to an adjoint equivalence (Def. ) on these full subcategories of fixed points

$\mathcal{D}_{fix} \underoverset {\underset{ R }{\longrightarrow}} {\overset{ L }{\longleftarrow}} {\phantom{A}\phantom{{}_{\bot}}\simeq_{\bot}\phantom{A}} \mathcal{C}_{fix}$
###### Proof

It is sufficient to see that the functors (co-)restrict as claimed, for then the restricted adjunction unit/counit are isomorphisms by definition, and hence exhibit an adjoint equivalence.

Hence we need to show that

1. for $c \in \mathcal{C}_{fix} \hookrightarrow \mathcal{C}$ we have that $\eta_{R(d)}$ is an isomorphism;

2. for $d \in \mathcal{D}_{fix} \hookrightarrow \mathcal{D}$ we have that $\epsilon_{L(c)}$ is an isomorphism.

For the first case we claim that $R(\eta_{d})$ provides an inverse: by the triangle identity (18) it is a right inverse, but by assumption it is itself an invertible morphism, which implies that $\eta_{R(d)}$ is an isomorphism.

The second claim is formally dual.

$\,$

### Modalities

Generally, a full subcategory-inclusion (Def. ) may be thought of as a consistent proposition about objects in a category: The objects in the full subcategory are those that have the given property.

This basic situation becomes particularly interesting when the inclusion functor has a left adjoint or a right adjoint (Def. ), in which case one speaks of a reflective subcategory, or a coreflective subcategory, respectively (Def. below). The adjunction now implies that each object is reflected or coreflected into the subcategory, and equipped with a comparison morphism to or from its (co-)reflection (the adjunction (co-)unit, Def. ). This comparison morphism turns out to always be an idempotent (co-)projection, in a sense made precise by Prop. below.

This means that, while any object may not fully enjoy the property that defines the subcategory, one may ask for the “aspect” of it that does, which is what is (co-)projected out. Regarding objects only via these aspects of them hence means to regard them only locally (where they exhibit that aspect) or only in the mode of focus on this aspect. Therefore one also calls the (co-)reflection operation into the given subcategory a (co-)localization or (co-)modal operator, or modality, for short (Def. below).

One finds that (co-)modalities are a fully equivalent perspective on the (co-)reflective subcategories of their fully (co-)modal objects (Def. below), this is the statement of Prop. below.

Another alternative perspective on this situation is given by the concept of localization of categories (Def. below), which is about universally forcing a given collection of morphisms (“weak equivalences”, Def. below) to become invertible. A reflective localization is equivalently a reflective subcategory-inclusion (Prop. below), and this exhibits the modal objects (Def. below) as equivalently forming the full subcategory of local objects (Def. below).

Conversely, every reflection onto full subcategories of $S$-local objects (Def. below) satisfies the universal property of a localization at $S$ with respect to left adjoint functors (Prop. below).

In conclusion, we have the following three equivalent perspectives on modalities.

$\phantom{A}$reflective subcategory$\phantom{A}$$\phantom{A}$modal operator$\phantom{A}$$\phantom{A}$reflective localization$\phantom{A}$
$\phantom{A}$object in reflective$\phantom{A}$
$\phantom{A}$full subcategory$\phantom{A}$
$\phantom{A}$modal object$\phantom{A}$$\phantom{A}$local object$\phantom{A}$

$\,$

###### Definition

(reflective subcategory and coreflective subcategory)

Let $\mathcal{D}$ be a category (Def. ) and

$\mathcal{C} \overset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow} \mathcal{D}$

a full subcategory-inclusion (hence a fully faithful functor Def. ). This is called:

1. a reflective subcategory inclusion if the inclusion functor $\iota$ has a left adjoint $L$ def. )

$\mathcal{C} \underoverset {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D} \,,$

then called the reflector;

2. a coreflective subcategory-inclusion if the inclusion functor $\iota$ has a right adjoint $R$ (def. )

$\mathcal{C} \underoverset \underset{R}{\longleftarrow} {\overset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\bot} \mathcal{D} \,,$

then called the coreflector.

###### Example

(reflective subcategory inclusion of sets into small groupoids)

There is a reflective subcategory-inclusion (Def. )

$Set \underoverset {\underset{\phantom{AAAA}}{\hookrightarrow}} {\overset{\pi_0}{\longleftarrow}} {\bot} Grp$

of the category of sets (Example ) into the category Grpd (Example ) of small groupoids (Example ) where

$\,$

We now re-consider the concept of reflective subcategories from the point of view of modalities:

###### Definition

(modality)

Let $\mathcal{D}$ be a category (Def. ). Then

1. a modal operator on $\mathcal{D}$ is

1. $\bigcirc \;\colon\; \mathcal{D} \to \mathcal{D}$

whose full essential image we denote by

$Im(\bigcirc) \overset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow} \mathcal{D} \,,$
2. a natural transformation (Def. )

(23)$X \overset{\eta_X}{\longrightarrow} \bigcirc X$

for all objects $X \in \mathcal{D}$, to be called the unit morphism;

such that:

• for every object $Y \in Im(\bigcirc) \hookrightarrow \mathcal{D}$ in the essential image of $\bigcirc$, every morphism $f$ into $Y$ factors uniquely through the unit (23)

$\array{ && X \\ & {}^{\mathllap{ \eta_X }}\swarrow && \searrow^{\mathrlap{f}} \\ \mathrlap{\bigcirc X\;\;\;\;} && \underset{\exists !}{\longrightarrow} && Y & \in Im(\bigcirc) }$

which equivalently means that if $Y \in Im(\bigcirc)$ the operation of precomposition with the unit $\eta_X$ yields a bijection of hom-sets

(24)$(-)\circ \eta_X \;\colon\; Hom_{\mathcal{D}}(\bigcirc X, Y) \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} Hom_{\mathcal{D}}(X, Y) \,,$
2. a comodal operator on $\mathcal{D}$ is

1. $\Box \;\colon\; \mathcal{D} \to \mathcal{D}$

whose full essential image we denote by

$Im( \Box ) \overset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow} \mathcal{D}$
2. a natural transformation (Def. )

(25)$\Box X \overset{ \epsilon_X }{\longrightarrow} X$

for all objects $X \in \mathcal{D}$, to be called the counit morphism;

such that:

• for every object $Y \in Im( \Box ) \hookrightarrow \mathcal{D}$ in the essential image of $\Box$, every morphism $f$ out of $Y$ factors uniquely through the counit (23)

$\array{ && X \\ & {}^{\mathllap{\epsilon_X}}\nearrow && \nwarrow^{\mathrlap{f}} \\ \mathrlap{\Box X\;\;\;} && \underset{\exists !}{\longleftarrow} && Y \in Im( \Box ) }$

which equivalently means that if $Y \in Im(\bigcirc)$ the operation of postcomposition with the counit $\epsilon_X$ yields a bijection of hom-sets

(26)$\epsilon_X \circ (-) \;\colon\; Hom_{\mathcal{D}}(Y, \Box X) \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} Hom_{\mathcal{D}}(Y , X) \,,$
###### Proposition

(modal operators equivalent to reflective subcategories)

If

$\mathcal{C} \underoverset {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}$

is a reflective subcategory-inclusion (Def. ). Then the composite

$\bigcirc \;\coloneqq\; \iota \circ L \;\colon\; \mathcal{D} \longrightarrow \mathcal{D}$

equipped with the adjunction unit natural transformation (Def. )

$X \overset{\eta_X}{\longrightarrow} \bigcirc X$

is a modal operator on $\mathcal{D}$ (Def. ).

Dually, if

$\mathcal{C} \underoverset {\underset{R}{\longleftarrow}} {\overset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\bot} \mathcal{D}$

is a coreflective subcategory-inclusion (Def. ). Then the composite

$\Box \;\coloneqq\; \iota \circ R \;\colon\; \mathcal{D} \longrightarrow \mathcal{D}$

equipped with the adjunction counit natural transformation (Def. )

$\Box X \overset{ \epsilon_X }{\longrightarrow} X$

is a comodal operator on $\mathcal{D}$ (Def. ).

Conversely:

If an endofunctor $\bigcirc \;\colon\; \mathcal{D} \to \mathcal{D}$ with natural transformation $X \overset{\eta_X}{\to} \bigcirc X$ is a modal operator on a category $\mathcal{D}$ (Def. ), then the inclusion of its full essential image is a reflective subcategory inclusion (Def. ) with reflector given by the corestriction of $\bigcirc$ to its image:

$Im( \bigcirc ) \underoverset {\underset{ \phantom{AA} \iota \phantom{AA} }{\hookrightarrow}} {\overset{ \bigcirc }{\longleftarrow}} {} \mathcal{D} \,.$

Dually, if an endofunctor $\Box \;\colon\; \mathcal{D} \to \mathcal{D}$ with natural transformation $\Box X \overset{\epsilon_X}{\longrightarrow} X$ is a comodal operator (Def. ), then the inclusion of its full essential image is a coreflective subcategory inclusion (Def. ) with coreflector given by the corestriction of $\Box$ to its image

$Im( \Box ) \underoverset {\underset{ \Box }{\longleftarrow}} {\overset{ \phantom{AA} \iota \phantom{AA} }{\hookrightarrow}} {} \mathcal{D} \,.$
###### Proof

The first two statements are immedialy a special case of the characterization of adjunctions via universal morphisms in Prop. : Using that $R = \iota$ is here assumed to be fully faithful, the uniqueness of $\tilde f$ in the universal morphism-factorization condition (21)

$\array{ && c \\ & {}^{\mathllap{\eta_c}}\swarrow && \searrow^{\mathrlap{f}} \\ R(L(c)) &&\underset{R (\widetilde f)}{\longrightarrow}&& R(d) \\ \\ L(c) &&\underset{ \exists ! \, \widetilde f}{\longrightarrow}&& d }$

implies that also $R(\widetilde f) = \iota(\widetilde f)$ is the unique morphism making that triangle commute.

Similarly for the converse: The assumption on a modal operator $\bigcirc$ is just so as to make its unit $\eta$ be a universal morphism (Def. ) into the inclusion functor $\iota$ of its essential image.

###### Proposition

(modal operator is idempotent)

Let $\mathcal{D}$ be a category (Def. ).

For $\bigcirc$ a modal operator on $\mathcal{D}$, with unit $\eta$ (Def. ), it is idempotent, in that it is naturally isomorphic (Def. ) to the composition with itself:

$\bigcirc \;\simeq\; \bigcirc \bigcirc \,.$

In fact, the image under $\bigcirc$ of its unit is such an isomorphism

$\bigcirc\left( X \overset{\eta_X}{\to} \bigcirc X \right) \;\;\colon\;\; \bigcirc X \overset{\simeq}{\longrightarrow} \bigcirc ( \bigcirc X )$

as is its unit on its image

$\eta_{\bigcirc X} \;\;\colon\;\; \bigcirc X \overset{\simeq}{\longrightarrow} \bigcirc ( \bigcirc X ) \,.$

Formally dually, for $\Box$ a comodal operator on $\mathcal{D}$, with counit $\epsilon$ (Def. ), it is idempotent, in that it is naturally isomorphic (Def. ) to the composition with itsef:

$\Box \circ \Box \;\simeq\; \Box \,.$

In fact, the image under $\Box$ of its counit is such an isomorphism

$\Box\left( \Box X \overset{\epsilon_X}{\to} X \right) \;\;\colon\;\; \Box (\Box X) \overset{\simeq}{\longrightarrow} \Box X$

as is its counit on its image

$\epsilon_{\Box X} \;\;\colon\;\; \Box ( \Box X ) \overset{\simeq}{\longrightarrow} \Box X \,.$
###### Proof

We discuss the first case, the second is formally dual (Example ).

By Prop. , the modal operator is equivalent to the composite $\iota \circ L$ obtained from the reflective subcategory-inclusion (Def. ) of its essential image of modal objects:

$Im(\bigcirc) \underoverset {\underset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow}} {\overset{\phantom{AA}L \phantom{AA} }{\longleftarrow}} {\bot} \mathcal{D} \,.$

and its unit is the corresponding adjunction unit (Def. )

$X \overset{\eta_X}{\longrightarrow} \iota \circ L \,.$

Hence it is sufficient to show that the morphisms and $L( \eta_X )$ and $\eta_{\iota Y}$ are isomorphisms.

Now, the triangle identities (18) for the adjunction $L \dashv \iota$, which hold by Prop. , say that their composition with the adjunction counit is the identity morphism

$\epsilon_{L(\eta_X)} \circ L(\eta_X) \;=\; id_{L(X)} \phantom{AA} \text{and} \phantom{AA} \iota( \epsilon_Y )\circ \eta_{\iota(Y)} \;=\; id_{\iota(Y)} \,.$

But by Prop. , the counit $\epsilon$ is a natural isomorphism, since $\iota$ is fully faithful. Hence we may cancel it on both sides of the triangle identities and find that $L(\eta_X)$ and $\eta_{\iota(Y)}$ are indeed isomorphisms.

###### Definition

(modal objects)

Let $\mathcal{D}$ be a category (Def. ).

For $\bigcirc$ a modal operator on $\mathcal{D}$ (Def. ), we say:

1. a $\bigcirc$-modal object is an object $X \in \mathcal{D}$ such that the following conditions hold (which are all equivalent, by Prop. ):

• it is in the $\bigcirc$-essential image: $X \in Im( \bigcirc ) \hookrightarrow \mathcal{D}$,

• it is isomorphic to its own $\bigcirc$-image: $X \simeq \bigcirc X$,

• specifically its $\bigcirc$-unit is an isomorphism $\eta_X \;\colon\; X \overset{\simeq}{\to} \bigcirc X$.

2. a $\bigcirc$-submodal object is an object $X \in \mathcal{D}$, such that

• its $\bigcirc$-unit is a monomorphism (Def. ): $\eta_X \;\colon\; X \hookrightarrow \bigcirc X$.

Dually (Example ):

For $\Box$ a comodal operator on $\mathcal{D}$ (Def. ), we say:

1. a $\Box$-comodal object is an object $X \in \mathcal{D}$ such that the following conditions hold (which are all equivalent, by Prop. ):

• it is in the $\Box$-essential image: $X \in Im( \Box ) \hookrightarrow \mathcal{D}$,

• it is isomorphic to its own $\Box$-image: $\Box X \simeq X$,

• specifically its $\Box$-counit is an isomorphism $\epsilon_X \;\colon\; \Box X \overset{\simeq}{\longrightarrow} X$

2. a $\Box$-supcomodal object is an object $X \in \mathcal{D}$, such that

• its $\Box$-counit is an epimorphism (Def. ): $\epsilon_X \;\colon\; \Box X \overset{epi}{\longrightarrow} X$.
###### Definition

Let

$L \;\dashv\; C \;\dashv\; R \;\colon\; \mathcal{C} \array{ \overset{\phantom{A} L \phantom{A}}{\hookleftarrow} \\ \overset{\phantom{A} C \phantom{A}}{\longrightarrow} \\ \overset{\phantom{A} R \phantom{A}}{\hookleftarrow} } \mathcal{D}$

be an adjoint triple (Remark ) such that $L$ and $R$ are fully faithful functors (necessarily both, by Prop. ). By Prop. , there are induced modal operators

$\Box \;\coloneqq\; L \circ C \phantom{AA} \bigcirc \;\coloneqq\; R \circ C$

which themselves form am adjoint pair

$\Box \;\dashv\; \bigcirc \,,$

hence called an adjoint modality. The adjunction unit and adjunction counit as in (13) may now be read as exhibiting each object $X$ in the domain of $C$ as “in between the opposite extremes of its $\bigcirc$-modal aspect and its $\Box$-modal aspect”

$\Box X \overset{\phantom{AA}\epsilon^\Box_X \phantom{AA}}{\longrightarrow} X \overset{\phantom{AA}\eta^{\bigcirc}_X\phantom{AA}}{\longrightarrow} \bigcirc X \,.$

A formally dual situation (Example ) arises when $C$ is fully faithful.

$L \;\dashv\; C \;\dashv\; R \;\colon\; \mathcal{C} \array{ \overset{\phantom{A} L \phantom{A}}{\longrightarrow} \\ \overset{\phantom{A} C \phantom{A}}{\hookleftarrow} \\ \overset{\phantom{A} R \phantom{A}}{\longrightarrow} } \mathcal{D}$

with

$\left( \bigcirc \;\coloneqq\; C \circ L \right) \;\dashv\; \left( \Box \;\coloneqq\; C \circ R \right)$

and canonical natural transformation between opposite extreme aspects given by

(27)$\Box X \overset{ \phantom{AA} \epsilon^{\Box}_X \phantom{AA} }{\longrightarrow} X \overset{ \phantom{AA} \eta^{\bigcirc}_X \phantom{AA} }{\longrightarrow} \bigcirc X$
###### Proposition

Let $L \dashv C \dashv R$ be an adjoint triple (Remark ). Then the following are equivalent:

1. $L$ is a fully faithful functor;

2. $R$ is a fully faithful functor,

3. $(\Box \;\coloneqq\; L \circ C) \dashv (\bigcirc \;\coloneqq\; R \circ C)$ is an adjoint modality (Def. ).

For proof see this prop..

In order to analyze (in Prop. below) the comparison morphism of opposite extreme aspects (27) induced by an adjoint modality (Def. ), we need the following technical Lemma:

###### Lemma

Let

$\mathcal{C} \array{ \overset{ \phantom{A} L \phantom{A} }{\longrightarrow} \\ \overset{ \phantom{A} C \phantom{A} }{\hookleftarrow} \\ \overset{ \phantom{A} R \phantom{A} }{ \longrightarrow } } \mathcal{D}$

be an adjoint triple with induced adjoint modality (Def. ) to be denoted

$\left( \Box \;\coloneqq\; C \circ L\right) \;\dashv\; \left( \bigcirc \;\coloneqq\; C \circ R \right)$

Denoting the adjunction units/counits (Def. ) as

$\phantom{A}$ adjunction $\phantom{A}$$\phantom{A}$ unit $\phantom{A}$$\phantom{A}$ counit $\phantom{A}$
$\phantom{A}$ $(L \dashv C)$ $\phantom{A}$$\phantom{A}$ $\eta^{\bigcirc}$ $\phantom{A}$$\phantom{A}$ $\epsilon^{\bigcirc}$ $\phantom{A}$
$\phantom{A}$ $(C \dashv R)$ $\phantom{A}$$\phantom{A}$ $\eta^\Box$ $\phantom{A}$$\phantom{A}$ $\epsilon^\Box$ $\phantom{A}$

we have that the following composites of unit/counit components are equal:

(28)$\left( \eta^{\Box}_{L X} \right) \circ \left( L \epsilon^\Box_X \right) \;\;=\;\; \left( R \eta^{\bigcirc}_{X} \right) \circ \left( \epsilon^{\bigcirc}_{R X} \right) \phantom{AAAAAA} \array{ L C R X &\overset{\epsilon^{\bigcirc}_{R X}}{\longrightarrow}& R X \\ {}^{ \mathllap{ L \epsilon^\Box_X } }\big\downarrow && \big\downarrow^{\mathrlap { R \eta^{\bigcirc}_{X} } } \\ L X &\underset{ \eta^\Box_{L X} }{\longrightarrow}& R C L X }$
###### Proof

We claim that the following diagram commutes (Def. ):

$\array{ && && R X \\ && & {}^{ \epsilon^\bigcirc_{R X} }\nearrow && \searrow^{\mathrlap{ R \eta^{\bigcirc}_X }} \\ && L C R X && && R C L X \\ & {}^{ L \epsilon^\Box_X }\swarrow && \searrow^{ \mathrlap{ L C R \eta^{\bigcirc}_X } } && {}^{\mathllap{ \eta^{\bigcirc}_{R C L X} }}\nearrow && \nwarrow^{ \mathrlap{ \eta^{\Box}_{L X} } } \\ L X && && L C R C L X && && L X \\ & {}_{\mathllap{ L \eta^{\bigcirc}_X }}\searrow && {}^{\mathllap{iso}}\swarrow_{\mathrlap{ L \epsilon^{\Box}_{C L X} }} && {}_{\mathllap{ L C \eta^\Box_{L X} }}\nwarrow^{\mathrlap{iso}} && \nearrow_{\mathrlap{ \epsilon^{\bigcirc}_{L X} }} \\ && L C L X && \underset{id_{L C L X}}{\longleftarrow} && L C L X }$

This commutes, because:

1. the left square is the image under $L$ of naturality (4) for $\epsilon^\Box$ on $\eta^{\bigcirc}_X$;

2. the top square is naturality (4) for $\epsilon^{\bigcirc}$ on $R \eta^{\bigcirc}_X$;

3. the right square is naturality (4) for $\epsilon^{\bigcirc}$ on $\eta^{\Box}_{L X}$;

4. the bottom commuting triangle is the image under $L$ of the triangle identity (18) for $(C \dashv R)$ on $L X$.

Moreover, notice that

1. the total bottom composite is the identity morphism $id_{L X}$, due to the triangle identity (18) for $(C \dashv R)$;

2. also the other two morphisms in the bottom triangle are isomorphisms, as shown, due to the idempoency of the $(C-R)$-adjunction (Prop. .)

Therefore the total composite from $L C R X \to R/ C L X$ along the bottom part of the diagram equals the left hand side of (28), while the composite along the top part of the diagram clearly equals the right hand side of (28).

###### Proposition

(comparison transformation between opposite extremes of adjoint modality)

Consider an adjoint triple of the form

$L \dashv C \dashv R \;\;\colon\;\; \mathcal{C} \array{ \overset{\phantom{AA} L \phantom{AA} }{\longrightarrow} \\ \overset{\phantom{AA} C \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} R \phantom{AAA} }{\longrightarrow} } \mathcal{B}$

with induced adjoint modality (Def. ) to be denoted

$\left( \Box \;\coloneqq\; C \circ L \right) \;\dashv\; \left( \bigcirc \;\coloneqq\; C \circ R \right)$

Denoting the adjunction units/counits (Def. ) as

$\phantom{A}$ adjunction $\phantom{A}$$\phantom{A}$ unit $\phantom{A}$$\phantom{A}$ counit $\phantom{A}$
$\phantom{A}$ $(L \dashv C)$ $\phantom{A}$$\phantom{A}$ $\eta^{\bigcirc}$ $\phantom{A}$$\phantom{A}$ $\epsilon^{\bigcirc}$ $\phantom{A}$
$\phantom{A}$ $(C \dashv E)$ $\phantom{A}$$\phantom{A}$ $\eta^\Box$ $\phantom{A}$$\phantom{A}$ $\epsilon^\Box$ $\phantom{A}$

Then for all $X \in \mathcal{C}$ the following two natural transformations, constructed from the adjunction units/counits (Def. ) and their inverse morphisms (using idempotency, Prop. ), are equal:

(29)$comp_{\mathcal{B}} \;\;\coloneqq\;\; \left( L \epsilon^\Box_X \right) \circ \left( \eta^{\bigcirc}_{R X} \right)^{-1} \;\;=\;\; \left( \eta^\Box_{L X} \right)^{-1} \circ \left( \Gamma \eta^{\bigcirc}_X \right) \phantom{AAAAAAA} \array{ R X & \overset{ \Gamma \eta^{\bigcirc}_X }{\longrightarrow} & R C L X \\ {}^{ \mathllap{ \left( \eta^{\bigcirc}_{R X} \right)^{-1} } }\big\downarrow & \searrow^{ { comp_{\mathcal{B}} } } & \big\downarrow^{ \mathrlap{ \left( \eta^\Box_{L X} \right)^{-1} } } \\ L C R X &\underset{ L \epsilon^\Box_X }{\longrightarrow}& L X }$

Moreover, the image of these morphisms under $C$ equals the following composite:

(30)$comp_{\mathcal{C}} \;\colon\; \Box X \overset{ \phantom{A} \epsilon^{\Box}_X \phantom{A} }{\longrightarrow} X \overset{ \phantom{A} \eta^{\bigcirc}_X \phantom{A} }{\longrightarrow} \bigcirc X \,,$

hence

(31)$comp_{\mathcal{C}} \;=\; C(comp_{\mathcal{B}}) \,.$
###### Proof

The first statement follows directly from Lemma .

For the second statement, notice that the $(C \dashv R)$-adjunct (Prop. ) of

$comp_{\mathcal{C}} \;\colon\; C R X \overset{ \phantom{A} \epsilon^{\Box}_X \phantom{A} }{\longrightarrow} X \overset{ \phantom{A} \eta^{\bigcirc}_X \phantom{A} }{\longrightarrow} C L X$

is

(32)$\widetilde{ comp_{\mathcal{C}} } \;\;=\;\; \underset{ = id_{R X} }{ \underbrace{ \Gamma X \underoverset{iso}{ \phantom{A} \eta^{\Box}_{R X} \phantom{A} }{ \longrightarrow } R C R X \underoverset{iso}{ \phantom{A} \Gamma \epsilon^{\Box}_X \phantom{A} }{\longrightarrow} R X }} \overset{ \phantom{A} R \eta^{\bigcirc}_X \phantom{A} }{\longrightarrow} R C L X \,,$

where under the braces we uses the triangle identity (Prop. ).

(As a side remark, for later usage, we observe that the morphisms on the left in (32) are isomorphisms, as shown, by idempotency of the adjunctions.)

From this we obtain the following commuting diagram:

$\array{ C R X &\overset{ \phantom{A} C R \eta^{\bigcirc}_X \phantom{A} }{\longrightarrow}& C R C L X &\underoverset{iso}{ \phantom{A} C \left(\eta^{ \Box }_{L X}\right)^{-1} \phantom{A} }{ \longrightarrow }& C L X \\ &{}_{\mathllap{ comp_{\mathcal{C}} }}\searrow& {}^{ \mathllap{ \epsilon^{\Box}_{C L X} } } \big\downarrow^{\mathrlap{\simeq}} & \nearrow_{\mathrlap{id_{L X}}} \\ && C L X }$

Here:

1. on the left we identified $\widetilde {\widetilde {comp_{\mathcal{C}}}} \;=\; comp_{\mathcal{C}}$ by applying the formula (Prop. ) for $(C \dashv R)$-adjuncts to $\widetilde {comp_{\mathcal{C}}} = R \eta^{\bigcirc}_X$ (32);

2. on the right we used the triangle identity (Prop. ) for $(C \dashv R)$.

This proves the second statement.

###### Definition

(preorder on modalities)

Let $\bigcirc_1$ and $\bigcirc_2$ be two modal operators on a category $\mathcal{C}$. By Prop. these are equivalently characterized by their reflective full subcategories $\mathcal{C}_{\bigcirc_1}, \mathcal{C}_{\bigcirc}_2 \hookrightarrow \mathcal{C}$ of modal objects.

There is an evident preorder on full subcategories of $\mathcal{C}$, given by full inclusions of full subcategories into each other. We write $\mathcal{C}_{\bigcirc_1} \subset \mathcal{C}_{\bigcirc_2}$ if the full subcategory on the left is contained, as a full subcategory of $\mathcal{C}$, in that on the right. Via prop. there is the induced preorder on modal operators, and we write

$\bigcirc_1 \;\lt\; \bigcirc_2 \phantom{AA} iff \phantom{AA} \mathcal{C}_{\bigcirc_1} \;\subset\; \mathcal{C}_{\bigcirc_2} \,.$

There is an analogous preorder on comodal operators (Def. ).

If we have two adjoint modalities (Def. ) of the same type (both modal left adjoint or both comodal left adjoint) such that both the modalities and the comodalities are compatibly ordered in this way, we denote this situation as follows:

$\array{ \bigcirc_2 &\dashv& \Box_2 \\ \vee && \vee \\ \bigcirc_1 &\dashv& \Box_1 } \phantom{AAAA} \text{or} \phantom{AAAA} \array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee && \vee \\ \Box_1 &\dashv& \bigcirc_1 }$

etc.

###### Example

Let $\mathcal{C}$ be a category with both an initial object $\emptyset$ and a terminal object $\ast$ (Def. ). Then, by Example there is an adjoint triple between $\mathcal{C}$ and the terminal category $\ast$ (Example ) of the form

$\mathcal{C} \array{ \overset{ \phantom{A} const_\emptyset \phantom{A} }{\hookleftarrow} \\ \overset{\phantom{AAAA}}{\longrightarrow} \\ \overset{ \phantom{A} const_\ast \phantom{A} }{\hookleftarrow} } \ast \,.$

The induced adjoint modality (Def. ) is

$const_{\emptyset} \;\dashv\; const_\ast \;\;\colon\;\; \mathcal{C} \to \mathcal{C} \,.$

By slight abuse of notation, we will also write this as

(33)$\emptyset \;\dashv\; \ast \;\;\colon\;\; \mathcal{C} \to \mathcal{C} \,.$

On the other extreme, for $\mathcal{C}$ any category whatsoever, the identity functor on it is adjoint functor to itself, and constitutes an adjoint modality (Def. )

(34)$id_{\mathcal{C}} \;\dashv\; id_{\mathcal{C}} \;\;\colon\;\; \mathcal{C} \to \mathcal{C} \,.$

Here

1. (33) is the bottom (or ground)

2. (34) is the top

in the preorder on adjoint modalities according to Def. , in that for every adjoint modality of the form $\bigcirc \dashv \Box$ we have the following:

$\array{ id &\dashv& id \\ \vee && \vee \\ \Box &\dashv& \bigcirc \\ \vee && \vee \\ \emptyset &\dashv& \ast }$
###### Definition

(Aufhebung)

On some category $\mathcal{C}$, consider an inclusion of adjoint modalities, according to Def. :

$\array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee && \vee \\ \Box_1 &\dashv& \bigcirc_1 }$

We say:

1. This provides right Aufhebung of the opposition exhibited by $\Box_1 \dashv \bigcirc_1$ if there is also the diagonal inclusion

$\Box_1 \lt \bigcirc_2 \phantom{AAA} equivalently \phantom{AAA} \mathcal{C}_{\Box_1} \subset \mathcal{C}_{\bigcirc_2}$

We indicate this situation by

$\array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee &/& \vee \\ \Box_1 &\dashv& \bigcirc_1 }$
2. This provides left Aufhebung of the opposition exhibited by $\Box_1 \dashv \bigcirc_1$ if there is also the diagonal inclusion

$\bigcirc_1 \lt \Box_2 \phantom{AAA} equivalently \phantom{AAA} \mathcal{C}_{\bigcirc_1} \subset \mathcal{C}_{\Box_2}$

We indicate this situation by

$\array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee &\backslash& \vee \\ \Box_1 &\dashv& \bigcirc_1 }$
###### Remark

For a progression of adjoint modalities of the form

$\array{ \bigcirc_2 &\dashv& \Box_2 \\ \vee && \vee \\ \bigcirc_1 &\dashv& \Box_1 }$

the analog of Aufhebung (Def. ) is automatic, since, by Prop. , in this situation the full subcategories modal objects at each stage coincide already.

For emphasis we may denote this situation by

$\array{ \bigcirc_2 &\dashv& \Box_2 \\ \vee &\vert& \vee \\ \bigcirc_1 &\dashv& \Box_1 } \,.$
###### Example

(top adjoint modality provides Aufhebung of all oppositions)

For $\mathcal{C}$ any category, the top adjoint modality $id \dashv id$ (Def. ) provides Aufhebung (Def. ) of every other adjoint modality.

But already Aufhebung of the bottom adjoint modality is a non-trivial and interesting condition. We consider this below in Prop. .

$\,$

We now re-consider the concept of reflective subcategories from the point of view of localization of categories:

###### Definition

(category with weak equivalences)

1. a category $\mathcal{C}$ (Def. )

2. a subcategory $W \subset \mathcal{C}$ (i.e. sub-class of objects and morphisms that inherits the structure of a category)

such that the morphisms in $W$

1. include all the isomorphisms of $\mathcal{C}$,

2. satisfy two-out-of-three:

If for $g$, $f$ any two composable morphisms in $\mathcal{C}$, two out of the set $\{g,\, f,\, g \circ f \}$ are in $W$, then so is the third.

$\array{ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ && \underset{ g \circ f }{\longrightarrow} }$
###### Definition

(localization of a category)

Let $W \subset \mathcal{C}$ be a category with weak equivalences (Def. ). Then the localization of $\mathcal{C}$ at $W$ is, if it exsists

1. a category $\mathcal{C}[W^{-1}]$,

2. a functor $\gamma \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}[W^{-1}]$ (Def. )

such that

1. $\gamma$ sends all morphisms in $W \subset \mathcal{C}$ to isomorphisms (Def. ),

2. $\gamma$ is universal with this property: If $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ is any functor with this property, then it factors through $\gamma$, up to natural isomorphism (Def. ):

$F \;\simeq\; D F \circ \gamma \phantom{AAAAAAA} \array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{\gamma}}\searrow &{}^{\rho}\Downarrow_{\simeq}& \nearrow_{\mathrlap{D F}} \\ && \mathcal{C}[W^{-1}] }$

and any two such factorizations $D F$ and $D^' F$ are related by a unique natural isomorphism $\kappa$ compatible with $\rho$ and $\rho^'$:

(35)$\array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{\gamma}}\searrow &{}^{\rho}\Downarrow_{\simeq}& \nearrow_{\mathrlap{D F}} && \searrow^{\mathrlap{id}} \\ && \mathcal{C}[W^{-1}] && {}_{\simeq}\seArrow^{\kappa} && \mathcal{D} \\ && & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{D^' F}} \\ && && \mathcal{C}[W^{-1}] } \phantom{AAAA} = \phantom{AAAA} \array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{\gamma}}\searrow &{}^{\rho^'}\Downarrow_{\simeq}& \nearrow_{\mathrlap{D^' F}} \\ && \mathcal{C}[W^{-1}] }$

Such a localization is called a reflective localization if the localization functor has a fully faithful right adjoint, exhibiting it as the reflection functor of a reflective subcategory-inclusion (Def. )

$\mathcal{C}[W^{-1}] \underoverset {\underset{\phantom{AAAA}}{\hookrightarrow}} {\overset{ \phantom{AA} \gamma \phantom{AA} }{\longleftarrow}} {\bot} \mathcal{C} \,.$
###### Proposition

(reflective subcategories are localizations)

Every reflective subcategory-inclusion (Def. )

$\mathcal{C}_{L} \underoverset {\underset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow}} {\overset{ \phantom{AA} L \phantom{AA} }{\longleftarrow}} {\bot} \mathcal{C}$

is the reflective localization (Def. ) at the class $W \coloneqq L^{-1}(Isos)$ of morphisms that are sent to isomorphisms by the reflector $L$.

###### Proof

Let $F \;\colon\; \mathcal{C} \to \mathcal{D}$ be a functor which inverts morphisms that are inverted by $L$.

First we need to show that it factors through $L$, up to natural isomorphism. But consider the following whiskering of the adjunction unit $\eta$ (Def. ) with $F$:

$\array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{L}}\searrow &\Downarrow& \nearrow_{\mathrlap{D F}} \\ && \mathcal{C}_L } \phantom{AA} \coloneqq \phantom{AA} \array{ \mathcal{C} && \overset{id}{\longrightarrow} && \mathcal{C} & \overset{F}{\longrightarrow}& \mathcal{D} \\ & {}_{\mathllap{L}}\searrow &\Downarrow^{\eta}& \nearrow_{\mathrlap{\iota}} \\ && \mathcal{C}_L }$

By idempotency (Prop. ), the components of the adjunction unit $\eta$ are inverted by $L$, and hence by assumption they are also inverted by $F$, so that on the right the natural transformation $F(\eta)$ is indeed a natural isomorphism.

It remains to show that this factorization is unique up to unique natural isomorphism. So consider any other factorization $D^' F$ via a natural isomorphism $\rho$. Pasting this now with the adjunction counit

$\array{ && \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}^{\mathllap{\iota}}\nearrow & {}^{\epsilon}\Downarrow & {}_{\mathllap{L}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{D^' F}} \\ \mathcal{C}_L && \underset{ id }{\longrightarrow} && \mathcal{C}_L }$

exhibits a natural isomorphism $\epsilon \cdot \rho$ between $D F \simeq D^' F$. Moreover, this is compatible with $F(\eta)$ according to (35), due to the triangle identity (Prop. ):

$\array{ \mathcal{C} && \overset{id}{\longrightarrow} && \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{id}}\searrow & {}^{\mathllap{\eta}}\Downarrow & {}^{\mathllap{\iota}}\nearrow & {}^{\epsilon}\Downarrow & {}_{\mathllap{L}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{D^' F}} \\ && \mathcal{C}_L && \underset{ id }{\longrightarrow} && \mathcal{C}_L } \phantom{AAAA} = \phantom{AAAA} \array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & \searrow &\Downarrow^\rho& \swarrow \\ && \mathcal{C}_L }$

Finally, since $L$ is essentially surjective functor, by idempotency (Prop. ), it is clear that this is the unique such natural isomorphism.

###### Definition

(local object)

Let $\mathcal{C}$ be a category (Def. ) and let $S \subset Mor_{\mathcal{C}}$ be a set of morphisms. Then an object $X \in \mathcal{C}$ is called an $S$-local object if for all $A \overset{s}{\to} B \; \in S$ the hom-functor (Def. ) from $s$ into $X$ yields a bijection

$Hom_{\mathcal{C}}(s,X) \;\colon\; Hom_{\mathcal{C}}(B,X) \overset{ \phantom{AA} \simeq \phantom{AA} }{\longrightarrow} Hom_{\mathcal{C}}(A,X) \,,$

hence if every morphism $A \overset{f}{\longrightarrow} X$ extends uniquely along $w$ to $B$:

$\array{ A &\overset{\phantom{A}f\phantom{A}}{\longrightarrow}& X \\ {}^{\mathllap{w}}\big\downarrow & \nearrow_{\mathrlap{ \exists! }} \\ B }$

We write

(36)$\mathcal{C}_S \overset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow} \mathcal{C}$

for the full subcategory (Example ) of $S$-local objects.

###### Definition

(reflection onto full subcategory of local objects)

Let $\mathcal{C}$ be a category and set $S \subset Mor_{\mathcal{C}}$ be a sub-class of its morphisms. Then the reflection onto local $S$-objects (often just called “localization at the collection $S$” is, if it exists, a left adjoint (Def. ) $L$ to the full subcategory-inclusion of the $S$-local objects (36):

$\mathcal{C}_S \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} \mathcal{C} \,.$

A class of examples is the following, which comes to its full nature (only) after passage to homotopy theory (Example below):

###### Definition

(homotopy localization of 1-categories)

Let $\mathcal{C}$ be a category, let $\mathbb{A} \in \mathcal{C}$ be an object, and consider the class of morphisms given by projection out of the Cartesian product with $\mathbb{A}$, of all objects $X \in \mathcal{C}$:

$X \times \mathbb{A} \overset{p_1}{\longrightarrow} X \,.$

If the corresponding reflection onto the full subcategory of local objects (Def. ) exists, we say this is homotopy localization at that object , and denote the modal operator corresponding to this (via Prop. ) by

$\bigcirc\!\!\!\!\!\!\!\!\mathbb{A} \;\colon\; \mathcal{C} \longrightarrow \mathcal{C} \,.$
###### Proposition

(reflective localization reflects onto full subcategory of local objects)

Let $W \subset \mathcal{C}$ be a category with weak equivalences (Def. ). If its reflective localization (Def. ) exists

$\mathcal{C}[W^{-1}] \underoverset {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\overset{ \phantom{AA} L \phantom{AA} }{\longleftarrow}} {\bot} \mathcal{C}$

then $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$ is equivalently the inclusion of the full subcategory (Example ) on the $W$-local objects (Def. ), and hence $L$ is equivalently reflection onto the $W$-local objects, according to Def. .

###### Proof

We need to show that

1. every $X \in \mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$ is $W$-local,

2. every $Y \in \mathcal{C}$ is $W$-local precisely if it is isomorphic to an object in $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$.

The first statement follows directly with the adjunction isomorphism (10):

$Hom_{\mathcal{C}}(w, \iota(X)) \simeq Hom_{\mathcal{C}[W^{-1}]}(L(w), X)$

and the fact that the hom-functor takes isomorphisms to bijections (Example ).

For the second statement, consider the case that $Y$ is $W$-local. Observe that then $Y$ is also local with respect to the class

$W_{sat} \;\coloneqq\; L^{-1}(Isos)$

of all morphisms that are inverted by $L$ (the “saturated class of morphisms”): For consider the hom-functor $\mathcal{C} \overset{Hom_{\mathcal{C}}(-,Y)}{\longrightarrow} Set^{op}$ to the opposite of the category of sets. By assumption on $Y$ this takes elements in $W$ to isomorphisms. Hence, by the defining universal property of the localization-functor $L$, it factors through $L$, up to natural isomorphism.

Since, by idempotency (Prop. ), the adjunction unit $\eta_Y$ is in $W_{sat}$, this implies that we have a bijection of the form

$Hom_{\mathcal{C}}( \eta_Y, Y ) \;\colon\; Hom_{\mathcal{C}}( \iota L(Y), Y ) \overset{\simeq}{\longrightarrow} Hom_{\mathcal{C}}(Y, Y) \,.$

In particular the identity morphism $id_Y$ has a preimage $\eta_Y^{-1}$ under this function, hence a left inverse to $\eta$:

$\eta_Y^{-1} \circ \eta_Y \;=\; id_Y \,.$

But by 2-out-of-3 this implies that $\eta_Y^{-1} \in W_{sat}$. Since the first item above shows that $\iota L(Y)$ is $W_{sat}$-local, this allows to apply this same kind of argument again,

$Hom_{\mathcal{C}}( \eta^{-1}_Y, \iota L(Y) ) \;\colon\; Hom_{\mathcal{C}}( Y, \iota L(Y) ) \overset{\simeq}{\longrightarrow} Hom_{\mathcal{C}}( \iota L(Y) , \iota L(Y)) \,,$

to deduce that also $\eta_Y^{-1}$ has a left inverse $(\eta_Y^{-1})^{-1} \circ \eta_Y^{-1}$. But since a left inverse that itself has a left inverse is in fact an inverse morphisms (this Lemma), this means that $\eta^{-1}_Y$ is an inverse morphism to $\eta_Y$, hence that $\eta_Y \;\colon\; Y \to \iota L (Y)$ is an isomorphism and hence that $Y$ is isomorphic to an object in $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$.

Conversely, if there is an isomorphism from $Y$ to a morphism in the image of $\iota$ hence, by the first item, to a $W$-local object, it follows immediatly that also $Y$ is $W$-local, since the hom-functor takes isomorphisms to bijections and since bijections satisfy 2-out-of-3.

###### Proposition

(reflection onto local objects is localization with respect to left adjoints)

Let $\mathcal{C}$ be a category (Def. ) and let $S \subset Mor_{\mathcal{C}}$ be a class of morphisms in $\mathcal{C}$. Then the reflection onto the $S$-local objects (Def. ) satisfies, if it exists, the universal property of a localization of categories (Def. ) with respect to left adjoint functors inverting $S$.

###### Proof

Write

$\mathcal{C}_S \underoverset {\underset{ \phantom{AA}\iota\phantom{AA} }{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} \mathcal{C}$

for the reflective subcategory-inclusion of the $S$-local objects.

Say that a morphism $f$ in $\mathcal{C}$ is an $S$-local morphism if for every $S$-local object $A \in \mathcal{C}$ the hom-functor (Example ) from $f$ to $A$ yields a bijection $Hom_{\mathcal{C}}(f,A)$. Notice that, by the Yoneda embedding for $\mathcal{C}_S$ (Prop. ), the $S$-local morphisms are precisely the morphisms that are taken to isomorphisms by the reflector $L$ (via Example ).

Now let

$(F \dashv G) \;\colon\; \mathcal{C} \underoverset {\underset{G}{\longleftarrow}} {\overset{ \phantom{AA} F \phantom{AA} }{\longrightarrow}} {\bot} \mathcal{D}$

be a pair of adjoint functors, such that the left adjoint $F$ inverts the morphisms in $S$. By the adjunction hom-isomorphism (10) it follows that $G$ takes values in $S$-local objects. This in turn implies, now via the Yoneda embedding for $\mathcal{D}$, that $F$ inverts all $S$-local morphisms, and hence all morphisms that are inverted by $L$.

Thus the essentially unique factorization of $F$ through $L$ now follows by Prop. .

$\,$

## Basic notions of Categorical algebra

We have seen that the existence of Cartesian products in a category $\mathcal{C}$ equips is with a functor of the form

$\mathcal{C} \times \mathcal{C} \overset{ (-) \times (-) }{\longrightarrow} \mathcal{C}$

which is directly analogous to the operation of multiplication in an associative algebra or even just in a semigroup (or monoid), just “categorified” (Example below). This is made precise by the concept of a monoidal category (Def. below).

This relation between category theory and algebra leads to the fields of categorical algebra and of universal algebra.

Here we are mainly interested in monoidal categories as a foundations for enriched category theory, to which we turn below.

$\,$

### Monoidal categories

###### Definition

(monoidal category)

An_monoidal category_ is a category $\mathcal{C}$ (Def. ) equipped with

1. a functor (Def. )

$\otimes \;\colon\; \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}$

out of the product category of $\mathcal{C}$ with itself (Example ), called the tensor product,

2. an object

$1 \in Obj_{\mathcal{C}}$

called the unit object or tensor unit,

3. a natural isomorphism (Def. )

$a \;\colon\; ((-)\otimes (-)) \otimes (-) \overset{\simeq}{\longrightarrow} (-) \otimes ((-)\otimes(-))$

called the associator,

4. $\ell \;\colon\; (1 \otimes (-)) \overset{\simeq}{\longrightarrow} (-)$

called the left unitor, and a natural isomorphism

$r \;\colon\; (-) \otimes 1 \overset{\simeq}{\longrightarrow} (-)$

called the right unitor,

such that the following two kinds of diagrams commute, for all objects involved:

1. triangle identity:

$\array{ & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y) \\ & {}_{\rho_x \otimes 1_y}\searrow && \swarrow_{1_x \otimes \lambda_y} & \\ && x \otimes y && }$
2. $\array{ && (w \otimes x) \otimes (y \otimes z) \\ & {}^{\mathllap{\alpha_{w \otimes x, y, z}}}\nearrow && \searrow^{\mathrlap{\alpha_{w,x,y \otimes z}}} \\ ((w \otimes x ) \otimes y) \otimes z && && (w \otimes (x \otimes (y \otimes z))) \\ {}^{\mathllap{\alpha_{w,x,y}} \otimes id_z }\downarrow && && \uparrow^{\mathrlap{ id_w \otimes \alpha_{x,y,z} }} \\ (w \otimes (x \otimes y)) \otimes z && \underset{\alpha_{w,x \otimes y, z}}{\longrightarrow} && w \otimes ( (x \otimes y) \otimes z ) }$
###### Example

(cartesian monoidal category)

Let $\mathcal{C}$ be a category in which all finite products exist. Then $\mathcal{C}$ becomes a monoidal category (Def. ) by

1. taking the tensor product to be the Cartesian product

$X \otimes Y \;\coloneqq\; X \times Y$
2. taking the unit object to be the terminal object (Def. )

$I \;\coloneqq\; \ast$

Monoidal categories of this form are called cartesian monoidal categories.

###### Lemma

(Kelly 64)

Let $(\mathcal{C}, \otimes, 1)$ be a monoidal category, def. . Then the left and right unitors $\ell$ and $r$ satisfy the following conditions:

1. $\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1$;

2. for all objects $x,y \in \mathcal{C}$ the following diagrams commutes:

$\array{ (1 \otimes x) \otimes y & & \\ {}^\mathllap{\alpha_{1, x, y}} \downarrow & \searrow^\mathrlap{\ell_x \otimes id_y} & \\ 1 \otimes (x \otimes y) & \underset{\ell_{x \otimes y}}{\longrightarrow} & x \otimes y } \,;$

and

$\array{ x \otimes (y \otimes 1) & & \\ {}^\mathllap{\alpha^{-1}_{1, x, y}} \downarrow & \searrow^\mathrlap{id_x \otimes r_y} & \\ (x \otimes y) \otimes 1 & \underset{r_{x \otimes y}}{\longrightarrow} & x \otimes y } \,;$

For proof see at monoidal category this lemma and this lemma.

###### Remark

Just as for an associative algebra it is sufficient to demand $1 a = a$ and $a 1 = a$ and $(a b) c = a (b c)$ in order to have that expressions of arbitrary length may be re-bracketed at will, so there is a coherence theorem for monoidal categories which states that all ways of freely composing the unitors and associators in a monoidal category (def. ) to go from one expression to another will coincide. Accordingly, much as one may drop the notation for the bracketing in an associative algebra altogether, so one may, with due care, reason about monoidal categories without always making all unitors and associators explicit.

(Here the qualifier “freely” means informally that we must not use any non-formal identification between objects, and formally it means that the diagram in question must be in the image of a strong monoidal functor from a free monoidal category. For example if in a particular monoidal category it so happens that the object $X \otimes (Y \otimes Z)$ is actually equal to $(X \otimes Y)\otimes Z$, then the various ways of going from one expression to another using only associators and this equality no longer need to coincide.)

###### Definition

(braided monoidal category)

A braided monoidal category, is a monoidal category $\mathcal{C}$ (def. ) equipped with a natural isomorphism (Def. )

(37)$\tau_{x,y} \colon x \otimes y \to y \otimes x$

called the braiding, such that the following two kinds of diagrams commute for all objects involved (“hexagon identities”):

$\array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{\tau_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{\tau_{x,y}\otimes Id} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{Id \otimes \tau_{x,z}}{\to}& y \otimes (z \otimes x) }$

and

$\array{ x \otimes (y \otimes z) &\stackrel{a^{-1}_{x,y,z}}{\to}& (x \otimes y) \otimes z &\stackrel{\tau_{x \otimes y, z}}{\to}& z \otimes (x \otimes y) \\ \downarrow^{Id \otimes \tau_{y,z}} &&&& \downarrow^{a^{-1}_{z,x,y}} \\ x \otimes (z \otimes y) &\stackrel{a^{-1}_{x,z,y}}{\to}& (x \otimes z) \otimes y &\stackrel{\tau_{x,z} \otimes Id}{\to}& (z \otimes x) \otimes y } \,,$

where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the associator of $\mathcal{C}^\otimes$.

###### Definition

A symmetric monoidal category is a braided monoidal category (def. ) for which the braiding

$\tau_{x,y} \colon x \otimes y \to y \otimes x$

satisfies the condition:

$\tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}$

for all objects $x, y$

###### Remark

In analogy to the coherence theorem for monoidal categories (remark ) there is a coherence theorem for symmetric monoidal categories (def. ), saying that every diagram built freely (see remark ) from associators, unitors and braidings such that both sides of the diagram correspond to the same permutation of objects, coincide.

###### Definition

(symmetric closed monoidal category)

Given a symmetric monoidal category $\mathcal{C}$ with tensor product $\otimes$ (def. ) it is called a closed monoidal category if for each $Y \in \mathcal{C}$ the functor $Y \otimes(-)\simeq (-)\otimes Y$ has a right adjoint, denoted $hom(Y,-)$

(38)$\mathcal{C} \underoverset {\underset{ [Y,-]}{\longrightarrow}} {\overset{(-) \otimes Y}{\longleftarrow}} {\bot} \mathcal{C} \,,$

hence if there are natural bijections

$Hom_{\mathcal{C}}(X \otimes Y, Z) \;\simeq\; Hom_{\mathcal{C}}{C}(X, [Y,Z])$

for all objects $X,Z \in \mathcal{C}$.

Since for the case that $X = 1$ is the tensor unit of $\mathcal{C}$ this means that

$Hom_{\mathcal{C}}(1, [Y,Z]) \simeq Hom_{\mathcal{C}}(Y,Z) \,,$

the object $[Y,Z] \in \mathcal{C}$ is an enhancement of the ordinary hom-set $Hom_{\mathcal{C}}(Y,Z)$ to an object in $\mathcal{C}$. Accordingly, it is also called the internal hom between $Y$ and $Z$.

The adjunction counit (Def. ) in this case is called the evaluation morphism

(39)$X \otimes [X,Y] \overset{ev}{\longrightarrow} Y$
###### Example

(Set is a cartesian closed category)

The category Set of all sets (Example ) equipped with its cartesian monoidal category-structure (Example ) is a closed monoidal category (Def. ), hence a cartesian closed category. The Cartesian product is the original Cartesian product of sets, and the internal hom is the function set $[X,Y]$ of functions from $X$ to $Y$

###### Example

(tensor product of abelian groups is closed monoidal category symmetric monoidal category-structure)

The category Ab of abelian groups (as in Example ) becomes a symmetric monoidal category (Def. ) with tensor product the actual tensor product of abelian groups $\otimes_{\mathbb{Z}}$ and with tensor unit the additive group $\mathbb{Z}$ of integers. Again the associator, unitor and braiding isomorphism are the evident ones coming from the underlying sets.

This is a closed monoidal category with internal hom $hom(A,B)$ being the set of homomorphisms $Hom_{Ab}(A,B)$ equipped with the pointwise group structure for $\phi_1, \phi_2 \in Hom_{Ab}(A,B)$ then $(\phi_1 + \phi_2)(a) \coloneqq \phi_1(a) + \phi_2(b) \; \in B$.

This is the archetypical case that motivates the notation “$\otimes$” for the pairing operation in a monoidal category.

###### Example

(Cat and Grpd are cartesian closed categories)

The category Cat (Example ) of all small categories (Example ) is a cartesian monoidal category-structure (Example ) with Cartesian product given by forming product categories (Example ).

Inside this, the full subcategory (Example ) Grpd (Example ) of all small groupoids (Example ) is itself a cartesian monoidal category-structure (Example ) with Cartesian product given by forming product categories (Example ).

In both cases this yields a closed monoidal category (Def. ), hence a cartesian closed category: the internal hom is given by the functor category construction (Example ).

###### Example

(categories of presheaves are cartesian closed)

Let $\mathcal{C}$ be a category and write $[\mathcal{C}^{op}, Set]$ for its category of presheaves (Example ).

This is

1. a cartesian monoidal category (Example ), whose Cartesian product is given objectwise in $\mathcal{C}$ by the Cartesian product in Set:

for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$, their Cartesian product $\mathbf{X} \times \mathbf{Y}$ exists and is given by

$\mathbf{X} \times \mathbf{Y} \;\;\colon\;\;\phantom{A} \array{ c_1 &\mapsto& \mathbf{X}(c_1) \times \mathbf{Y}(c_1) \\ {}^{\mathllap{f}}\big\downarrow && \big\uparrow^{ \mathrlap{ \mathbf{X}(f) \times \mathbf{Y}(f) } } \\ c_2 &\mapsto& \mathbf{X}(c_2) \times \mathbf{Y}(c_2) }$
2. a cartesian closed category (Def. ), whose internal hom is given for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ by

$[\mathbf{X}, \mathbf{Y}] \;\;\colon\;\;\phantom{A} \array{ c_1 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_1) \times \mathbf{X}, \mathbf{y} ) \\ {}^{ \mathllap{ f } }\big\downarrow && \big\uparrow^{ \mathrlap{ Hom_{[\mathcal{C}^{op}, Set]}( y(f) \times \mathbf{X}, \mathbf{y} ) } } \\ c_2 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_2) \times \mathbf{X}, \mathbf{y} ) }$

Here $y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set]$ denotes the Yoneda embedding and $Hom_{[\mathcal{C}^{op}, Set]}(-,-)$ is the hom-functor on the category of presheaves.

###### Proof

The first statement is a special case of the general fact that limits of presheaves are computed objectwise (Example ).

For the second statement, first assume that $[\mathbf{X}, \mathbf{Y}]$ does exist. Then by the adjunction hom-isomorphism (10) we have for any other presheaf $\mathbf{Z}$ a natural isomorphism of the form

(40)$Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z}, [\mathbf{X},\mathbf{Y}]) \;\simeq\; Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z} \times \mathbf{X}, \mathbf{Y}) \,.$

This holds in particular for $\mathbf{Z} = y(c)$ a representable presheaf (Example ) and so the Yoneda lemma (Prop. ) implies that if it exists, then $[\mathbf{X}, \mathbf{Y}]$ must have the claimed form:

\begin{aligned} [\mathbf{X}, \mathbf{Y}](c) & \simeq Hom_{[\mathcal{C}^{op}, Set]}( y(c), [\mathbf{X}, \mathbf{Y}] ) \\ & \simeq Hom_{ [\mathcal{C}^{op}, Set] }( y(c) \times \mathbf{X}, \mathbf{Y} ) \,. \end{aligned}

Hence it remains to show that this formula does make (40) hold generally.

For this we use the equivalent characterization of adjoint functors from Prop. , in terms of the adjunction counit providing a system of universal arrows (Def. ).

Define a would-be adjunction counit, hence a would-be evaluation morphism (39), by

$\array{ \mathbf{X} \times [\mathbf{X} , \mathbf{Y}] &\overset{ev}{\longrightarrow}& \mathbf{Y} \\ \mathbf{X}(c) \times Hom_{[\mathcal{C}^{op}, Set]}(y(c) \times \mathbf{X}, \mathbf{Y}) &\overset{ev_c}{\longrightarrow}& \mathbf{Y}(c) \\ (x, \phi) &\mapsto& \phi_c( id_c, x ) }$

Then it remains to show that for every morphism of presheaves of the form $\mathbf{X} \times \mathbf{A} \overset{\phantom{A}f\phantom{A}}{\longrightarrow} \mathbf{Y}$ there is a unique morphism $\widetilde f \;\colon\; \mathbf{A} \longrightarrow [\mathbf{X}, \mathbf{Y}]$ such that

(41)$\array{ \mathbf{X} \times \mathbf{A} && \overset{ \mathbf{X} \times \widetilde f }{\longrightarrow} && \mathbf{X} \times [\mathbf{X}, \mathbf{Y}] \\ & {}_{\mathllap{ \mathrlap{f} }}\searrow && \swarrow_{ \mathrlap{ ev } } \\ && \mathbf{Y} }$

The commutativity of this diagram means in components at $c \in \mathcal{C}$ that, that for all $x \in \mathbf{X}(c)$ and $a \in \mathbf{A}(c)$ we have

\begin{aligned} ev_c( x, \widetilde f_c(a) ) & \coloneqq (\widetilde f_c(a))_c( id_c, x ) \\ & = f_c( x, a ) \end{aligned}

Hence this fixes the component $\widetilde f_c(a)_c$ when its first argument is the identity morphism $id_c$. But let $g \;\colon\; d \to c$ be any morphism and chase $(id_c, x )$ through the naturality diagram for $\widetilde f_c(a)$:

$\array{ Hom_{\mathcal{C}}(c,c) \times \mathbf{X}(c) &\overset{ (\widetilde f_c(a))_c }{\longrightarrow}& \mathbf{Y}(c) \\ {}^{\mathllap{ g^\ast }}\big\downarrow && \big\downarrow^{\mathrlap{ g^\ast }} \\ Hom_{\mathcal{C}}(d,c) \times \mathbf{X}(d) &\overset{ (\widetilde f_c(a))_d }{\longrightarrow}& \mathbf{Y}(d) } \phantom{AAAA} \array{ \{ (id_c, x ) \} &\longrightarrow& \{ f_c( x, a ) \} \\ \big\downarrow && \big\downarrow \\ \{ (g, g^\ast(x)) \} &\longrightarrow& \{ f_d( g^\ast(x), g^\ast(a) ) \} }$

This shows that $(\widetilde f_c(a))_d$ is fixed to be given by

(42)$(\widetilde f_c(a))_d( g, x' ) \;=\; f_d( x', g^\ast(a) )$

at least on those pairs $(g,x')$ such that $x'$ is in the image of $g^\ast$.

But, finally, $(\widetilde f_c(a))_d$ is also natural in $c$

$\array{ \mathbf{A}(c) &\overset{ \widetilde f_c }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](c) \\ {}^{\mathllap{g^\ast}}\big\downarrow && \big\downarrow^{\mathrlap{g^\ast}} \\ \mathbf{A}(d) &\overset{ \widetilde f_d }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](d) }$

which implies that (42) must hold generally. Hence naturality implies that (41) indeed has a unique solution.

$\,$

The internal hom (Def. ) turns out to share all the abstract properties of the ordinary (external) hom-functor (Def. ), even though this is not completely manifest from its definition. We make this explicit by the following three propositions.

###### Proposition

(internal hom bifunctor)

For $\mathcal{C}$ a closed monoidal category (Def. ), there is a unique functor (Def. ) out of the product category (Def. ) of $\mathcal{C}$ with its opposite category (Def. )

$[-,-] \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{C}$

such that for each $X \in \mathcal{C}$ it coincides with the internal hom $[X,-]$ (38) as a functor in the second variable, and such that there is a natural isomorphism

$Hom(X, [Y,Z]) \;\simeq\; Hom(X \otimes Y, Z)$

which is natural not only in $X$ and $Z$, but also in $Y$.

###### Proof

We have a natural isomorphism for each fixed $Y$, and hence in particular for fixed $Y$ and fixed $Z$ by (38). With this the statement follows by Prop. .

In fact the 3-variable adjunction from Prop. even holds internally:

###### Proposition

In a symmetric closed monoidal category (def. ) there are natural isomorphisms

$[X \otimes Y, Z] \;\simeq\; [X, [Y,Z]]$

whose image under $Hom_{\mathcal{C}}(1,-)$ (see also Example below) are the defining natural bijections of Prop. .

###### Proof

Let $A \in \mathcal{C}$ be any object. By applying the natural bijections from Prop. , there are composite natural bijections

\begin{aligned} Hom_{\mathcal{C}}(A , [X \otimes Y, Z]) & \simeq Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z) \\ & \simeq Hom_{\mathcal{C}}((A \otimes X)\otimes Y, Z) \\ & \simeq Hom_{\mathcal{C}}(A \otimes X, [Y,Z]) \\ & \simeq Hom_{\mathcal{C}}(A, [X, [Y,Z]]) \end{aligned}

Since this holds for all $A$, the fully faithfulness of the Yoneda embedding (Prop. ) says that there is an isomorphism $[ X\otimes Y, Z ] \simeq [X, [Y,Z]]$. Moreover, by taking $A = 1$ in the above and using the left unitor isomorphisms $A \otimes (X \otimes Y) \simeq X \otimes Y$ and $A\otimes X \simeq X$ we get a commuting diagram

$\array{ Hom_{\mathcal{C}}(1, [X\otimes Y, Z )) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(1, [X, [Y,Z]]) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ Hom_{\mathcal{C}}(X \otimes Y, Z) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(X, [Y,Z]) } \,.$

Also the key respect of the hom-functor for limits is inherited by internal hom-functors

###### Proposition

(internal hom preserves limits)

Let $\mathcal{C}$ be a symmetric closed monoidal category with internal hom-bifunctor $[-,-]$ (Prop. ). Then this bifunctor preserves limits in the second variable, and sends colimits in the first variable to limits:

$[X, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Y(j)] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [X, Y(j)]$

and

$[\underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j),X] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j),X]$
###### Proof

For $X \in \mathcal{X}$ any object, $[X,-]$ is a right adjoint by definition, and hence preserves limits by Prop. .

For the other case, let $Y \;\colon\; \mathcal{L} \to \mathcal{C}$ be a diagram in $\mathcal{C}$, and let $C \in \mathcal{C}$ be any object. Then there are isomorphisms

\begin{aligned} Hom_{\mathcal{C}}(C, \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ) & \simeq Hom_{\mathcal{C}}( C \otimes \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ) \\ & \simeq Hom_{\mathcal{C}}( \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( C, [Y(j), X] ) \\ & \simeq Hom_{\mathcal{C}}( C, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] ) \end{aligned}

which are natural in $C \in \mathcal{C}$, where we used that the ordinary hom-functor preserves limits (Prop. ), and that the left adjoint $C \otimes (-)$ preserves colimits, since left adjoints preserve colimits (Prop. ).

Hence by the fully faithfulness of the Yoneda embedding, there is an isomorphism

$\left[ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X \right] \overset{\simeq}{\longrightarrow} \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] \,.$

$\,$

Now that we have seen monoidal categories with various extra properties, we next look at functors which preserve these:

###### Definition

(monoidal functors)

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two monoidal categories (def. ). A lax monoidal functor between them is

1. a functor (Def. )

$F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,$
2. (43)$\epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})$
3. a natural transformation (Def. )

(44)$\mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)$

for all $x,y \in \mathcal{C}$

satisfying the following conditions:

1. (associativity) For all objects $x,y,z \in \mathcal{C}$ the following diagram commutes

$\array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) ) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } \,,$

where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the associators of the monoidal categories;

2. (unitality) For all $x \in \mathcal{C}$ the following diagrams commutes

$\array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} x ) }$

and

$\array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) } \,,$

where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right unitors of the two monoidal categories, respectively.

If $\epsilon$ and alll $\mu_{x,y}$ are isomorphisms, then $F$ is called a strong monoidal functor.

If moreover $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are equipped with the structure of braided monoidal categories (def. ) with braidings $\tau^{\mathcal{C}}$ and $\tau^{\mathcal{D}}$, respectively, then the lax monoidal functor $F$ is called a braided monoidal functor if in addition the following diagram commutes for all objects $x,y \in \mathcal{C}$

$\array{ F(x) \otimes_{\mathcal{C}} F(y) &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}& F(y) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\mu_{x,y}}}\downarrow && \downarrow^{\mathrlap{\mu_{y,x}}} \\ F(x \otimes_{\mathcal{C}} y ) &\underset{F(\tau^{\mathcal{C}}_{x,y} )}{\longrightarrow}& F( y \otimes_{\mathcal{C}} x ) } \,.$

A homomorphism $f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2)$ between two (braided) lax monoidal functors is a monoidal natural transformation, in that it is a natural transformation $f_x \;\colon\; F_1(x) \longrightarrow F_2(x)$ of the underlying functors

compatible with the product and the unit in that the following diagrams commute for all objects $x,y \in \mathcal{C}$:

$\array{ F_1(x) \otimes_{\mathcal{D}} F_1(y) &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F_2(x) \otimes_{\mathcal{D}} F_2(y) \\ {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\mu_2)_{x,y}}} \\ F_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& F_2(x \otimes_{\mathcal{C}} y) }$

and

$\array{ && 1_{\mathcal{D}} \\ & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}} \\ F_1(1_{\mathcal{C}}) &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&& F_2(1_{\mathcal{C}}) } \,.$

We write $MonFun(\mathcal{C},\mathcal{D})$ for the resulting category of lax monoidal functors between monoidal categories $\mathcal{C}$ and $\mathcal{D}$, similarly $BraidMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between braided monoidal categories, and $SymMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between symmetric monoidal categories.

###### Remark

In the literature the term “monoidal functor” often refers by default to what in def. is called a strong monoidal functor. But for the purpose of the discussion of functors with smash product below, it is crucial to admit the generality of lax monoidal functors.

If $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are symmetric monoidal categories (def. ) then a braided monoidal functor (def. ) between them is often called a symmetric monoidal functor.

###### Proposition

For $\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D} \overset{G}{\longrightarrow} \mathcal{E}$ two composable lax monoidal functors (def. ) between monoidal categories, then their composite $F \circ G$ becomes a lax monoidal functor with structure morphisms

$\epsilon^{G\circ F} \;\colon\; 1_{\mathcal{E}} \overset{\epsilon^G}{\longrightarrow} G(1_{\mathcal{D}}) \overset{G(\epsilon^F)}{\longrightarrow} G(F(1_{\mathcal{C}}))$

and

$\mu^{G \circ F}_{c_1,c_2} \;\colon\; G(F(c_1)) \otimes_{\mathcal{E}} G(F(c_2)) \overset{\mu^{G}_{F(c_1), F(c_2)}}{\longrightarrow} G( F(c_1) \otimes_{\mathcal{D}} F(c_2) ) \overset{G(\mu^F_{c_1,c_2})}{\longrightarrow} G(F( c_1 \otimes_{\mathcal{C}} c_2 )) \,.$

### Algebras and modules

###### Definition

Given a monoidal category $(\mathcal{C}, \otimes, 1)$ (Def. ), then a monoid internal to $(\mathcal{C}, \otimes, 1)$ is

1. an object $A \in \mathcal{C}$;

2. a morphism $e \;\colon\; 1 \longrightarrow A$ (called the unit)

3. a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the product);

such that

1. (associativity) the following diagram commutes

$\array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{A \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes A}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,,$

where $a$ is the associator isomorphism of $\mathcal{C}$;

2. (unitality) the following diagram commutes:

$\array{ 1 \otimes A &\overset{e \otimes id}{\longrightarrow}& A \otimes A &\overset{id \otimes e}{\longleftarrow}& A \otimes 1 \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\mu}} & & \swarrow_{\mathrlap{r}} \\ && A } \,,$

where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.

Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a symmetric monoidal category (def. ) $(\mathcal{C}, \otimes, 1, B)$ with symmetric braiding $\tau$, then a monoid $(A,\mu, e)$ as above is called a commutative monoid in $(\mathcal{C}, \otimes, 1, B)$ if in addition

• (commutativity) the following diagram commutes

$\array{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,.$

A homomorphism of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism

$f \;\colon\; A_1 \longrightarrow A_2$

in $\mathcal{C}$, such that the following two diagrams commute

$\array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 }$

and

$\array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,.$

Write $Mon(\mathcal{C}, \otimes,1)$ for the category of monoids in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its full subcategory of commutative monoids.

###### Example

Given a monoidal category $(\mathcal{C}, \otimes, 1)$ (Def. ), the tensor unit $1$ is a monoid in $\mathcal{C}$ (def. ) with product given by either the left or right unitor

$\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1 \,.$

By lemma , these two morphisms coincide and define an associative product with unit the identity $id \colon 1 \to 1$.

If $(\mathcal{C}, \otimes , 1)$ is a symmetric monoidal category (def. ), then this monoid is a commutative monoid.

###### Example

Given a symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. ), and given two commutative monoids $(E_i, \mu_i, e_i)$ $i \in \{1,2\}$ (def. ), then the tensor product $E_1 \otimes E_2$ becomes itself a commutative monoid with unit morphism

$e \;\colon\; 1 \overset{\simeq}{\longrightarrow} 1 \otimes 1 \overset{e_1 \otimes e_2}{\longrightarrow} E_1 \otimes E_2$

(where the first isomorphism is, $\ell_1^{-1} = r_1^{-1}$ (lemma )) and with product morphism given by

$E_1 \otimes E_2 \otimes E_1 \otimes E_2 \overset{id \otimes \tau_{E_2, E_1} \otimes id}{\longrightarrow} E_1 \otimes E_1 \otimes E_2 \otimes E_2 \overset{\mu_1 \otimes \mu_2}{\longrightarrow} E_1 \otimes E_2$

(where we are notationally suppressing the associators and where $\tau$ denotes the braiding of $\mathcal{C}$).

That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of $(E_i,\mu_i, e_i)$, and from the hexagon identities for the braiding (def. ) and from symmetry of the braiding.

Similarly one checks that for $E_1 = E_2 = E$ then the unit maps

$E \simeq E \otimes 1 \overset{id \otimes e}{\longrightarrow} E \otimes E$
$E \simeq 1 \otimes E \overset{e \otimes 1}{\longrightarrow} E \otimes E$

and the product map

$\mu \;\colon\; E \otimes E \longrightarrow E$

and the braiding

$\tau_{E,E} \;\colon\; E \otimes E \longrightarrow E \otimes E$

are monoid homomorphisms, with $E \otimes E$ equipped with the above monoid structure.

###### Definition

Given a monoidal category $(\mathcal{C}, \otimes, 1)$ (def. ), and given $(A,\mu,e)$ a monoid in $(\mathcal{C}, \otimes, 1)$ (def. ), then a left module object in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is

1. an object $N \in \mathcal{C}$;

2. a morphism $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the action);

such that

1. (unitality) the following diagram commutes:

$\array{ 1 \otimes N &\overset{e \otimes id}{\longrightarrow}& A \otimes N \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && N } \,,$

where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.

2. (action property) the following diagram commutes

$\array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,,$

A homomorphism of left $A$-module objects

$(N_1, \rho_1) \longrightarrow (N_2, \rho_2)$

is a morphism

$f\;\colon\; N_1 \longrightarrow N_2$

in $\mathcal{C}$, such that the following diagram commutes:

$\array{ A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2 \\ {}^{\mathllap{\rho_1}}\downarrow && \downarrow^{\mathrlap{\rho_2}} \\ N_1 &\underset{f}{\longrightarrow}& N_2 } \,.$

For the resulting category of modules of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write

$A Mod(\mathcal{C}) \,.$
###### Example

Given a monoidal category $(\mathcal{C},\otimes, 1)$ (def. ) with the tensor unit $1$ regarded as a monoid in a monoidal category via example , then the left unitor

$\ell_C \;\colon\; 1\otimes C \longrightarrow C$

makes every object $C \in \mathcal{C}$ into a left module, according to def. , over $C$. The action property holds due to lemma . This gives an equivalence of categories

$\mathcal{C} \simeq 1 Mod(\mathcal{C})$

of $\mathcal{C}$ with the category of modules over its tensor unit.

###### Example

The archetypical case in which all these abstract concepts reduce to the basic familiar ones is the symmetric monoidal category Ab of abelian groups from example .

1. A monoid in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently a ring.

2. A commutative monoid in in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently a commutative ring $R$.

3. An $R$-module object in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently an $R$-module;

4. The tensor product of $R$-module objects (def. ) is the standard tensor product of modules.

5. The category of module objects $R Mod(Ab)$ (def. ) is the standard category of modules $R Mod$.

###### Example

Closely related to the example , but closer to the structure we will see below for spectra, are monoids in the category of chain complexes $(Ch_\bullet, \otimes, \mathbb{Z})$ from example . These monoids are equivalently differential graded algebras.

###### Proposition

In the situation of def. , the monoid $(A,\mu, e)$ canonically becomes a left module over itself by setting $\rho \coloneqq \mu$. More generally, for $C \in \mathcal{C}$ any object, then $A \otimes C$ naturally becomes a left $A$-module by setting:

$\rho \;\colon\; A \otimes (A \otimes C) \underoverset{\simeq}{a^{-1}_{A,A,C}}{\longrightarrow} (A \otimes A) \otimes C \overset{\mu \otimes id}{\longrightarrow} A \otimes C \,.$

The $A$-modules of this form are called free modules.

The free functor $F$ constructing free $A$-modules is left adjoint to the forgetful functor $U$ which sends a module $(N,\rho)$ to the underlying object $U(N,\rho) \coloneqq N$.

$A Mod(\mathcal{C}) \underoverset {\underset{U}{\longrightarrow}} {\overset{F}{\longleftarrow}} {\bot} \mathcal{C} \,.$
###### Proof

A homomorphism out of a free $A$-module is a morphism in $\mathcal{C}$ of the form

$f \;\colon\; A\otimes C \longrightarrow N$

fitting into the diagram (where we are notationally suppressing the associator)

$\array{ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,.$

Consider the composite

$\tilde f \;\colon\; C \underoverset{\simeq}{\ell_C}{\longrightarrow} 1 \otimes C \overset{e\otimes id}{\longrightarrow} A \otimes C \overset{f}{\longrightarrow} N \,,$

i.e. the restriction of $f$ to the unit “in” $A$. By definition, this fits into a commuting square of the form (where we are now notationally suppressing the associator and the unitor)

$\array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\underset{id \otimes f}{\longrightarrow}& A \otimes N } \,.$

Pasting this square onto the top of the previous one yields

$\array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,,$

where now the left vertical composite is the identity, by the unit law in $A$. This shows that $f$ is uniquely determined by $\tilde f$ via the relation

$f = \rho \circ (id_A \otimes \tilde f) \,.$

This natural bijection between $f$ and $\tilde f$ establishes the adjunction.

###### Definition

Given a closed symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. , def. ), given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. ), and given $(N_1, \rho_1)$ and $(N_2, \rho_2)$ two left $A$-module objects (def.), then

1. the tensor product of modules $N_1 \otimes_A N_2$ is, if it exists, the coequalizer

$N_1 \otimes A \otimes N_2 \underoverset {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}} {\overset{N_1 \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} N_1 \otimes N_1 \overset{coeq}{\longrightarrow} N_1 \otimes_A N_2$

and if $A \otimes (-)$ preserves these coequalizers, then this is equipped with the left $A$-action induced from the left $A$-action on $N_1$

2. the function module $hom_A(N_1,N_2)$ is, if it exists, the equalizer

$hom_A(N_1, N_2) \overset{equ}{\longrightarrow} hom(N_1, N_2) \underoverset {\underset{hom(A \otimes N_1, \rho_2)\circ (A \otimes(-))}{\longrightarrow}} {\overset{hom(\rho_1,N_2)}{\longrightarrow}} {\phantom{AAAAAA}} hom(A \otimes N_1, N_2) \,.$

equipped with the left $A$-action that is induced by the left $A$-action on $N_2$ via

$\frac{ A \otimes hom(X,N_2) \longrightarrow hom(X,N_2) }{ A \otimes hom(X,N_2) \otimes X \overset{id \otimes ev}{\longrightarrow} A \otimes N_2 \overset{\rho_2}{\longrightarrow} N_2 } \,.$
###### Proposition

Given a closed symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. , def. ), and given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. ). If all coequalizers exist in $\mathcal{C}$, then the tensor product of modules $\otimes_A$ from def. makes the category of modules $A Mod(\mathcal{C})$ into a symmetric monoidal category, $(A Mod, \otimes_A, A)$ with tensor unit the object