category theory

# Contents

## Definition

The slice category or over category $\mathbf{C}/c$ of a category $\mathbf{C}$ over an object $c \in \mathbf{C}$ has

• objects that are all arrows $f \in \mathbf{C}$ such that $cod(f) = c$, and

• morphisms $g: X \to X' \in \mathbf{C}$ from $f:X \to c$ to $f': X' \to c$ such that $f' \circ g = f$.

$C/c = \left\lbrace \array{ X &&\stackrel{g}{\to}&& X' \\ & {}_f \searrow && \swarrow_{f'} \\ && c } \right\rbrace$

The slice category is a special case of a comma category.

There is a forgetful functor $U_c: \mathbf{C}/c \to \mathbf{C}$ which maps an object $f:X \to c$ to its domain $X$ and a morphism $g: X \to X' \in \mathbf{C}/c$ (from $f:X \to c$ to $f': X' \to c$ such that $f' \circ g = f$) to the morphism $g: X \to X'$.

The dual notion is an under category.

## Examples

• If $\mathbf{C} = \mathbf{P}$ is a poset and $p \in \mathbf{P}$, then the slice category $\mathbf{P}/p$ is the down set $\downarrow (p)$ of elements $q \in \mathbf{P}$ with $q \leq p$.

• If $1$ is a terminal object in $\mathbf{C}$, then $\mathbf{C}/1$ is isomorphic to $\mathbf{C}$.

## Properties

### Relation to codomain fibration

The assignment of overcategories $C/c$ to objects $c \in C$ extends to a functor

$C/(-) : C \to Cat$

Under the Grothendieck construction this functor corresponds to the codomain fibration

$cod : [I,C] \to C$

from the arrow category of $C$. (Note that unless $C$ has pullbacks, this functor is not actually a fibration, though it is always an opfibration.)

###### Proposition

Let

$(L \dashv R) : D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} C$

be a pair of adjoint functors, where the category $C$ has all pullbacks.

Then for every object $X \in C$ there is induced a pair of adjoint functors between the slice categories

$(L/X \dashv R/X) : D/(L X) \stackrel{\overset{L/X}{\leftarrow}}{\underset{R/X}{\to}} C/X$

where

• $L/X$ is the evident induced functor;

• $R/X$ is the composite

$R/X : D/{L X} \stackrel{R}{\to} C/{(R L X)} \stackrel{i_{X}^*}{\to} C/X$

of the evident functor induced by $R$ with the pullback along the $(L \dashv R)$-unit at $X$.

### Presheaves on over-categories and over-categories of presheaves

Let $C$ be a category, $c$ an object of $C$ and let $C/c$ be the over category of $C$ over $c$. Write $PSh(C/c) = [(C/c)^{op}, Set]$ for the category of presheaves on $C/c$ and write $PSh(C)/Y(c)$ for the over category of presheaves on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(c)$ is the Yoneda embedding.

###### Proposition

There is an equivalence of categories

$e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c) \,.$
###### Proof

The functor $e$ takes $F \in PSh(C/c)$ to the presheaf $F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map $\eta_d \sqcup_{f \in C(d,c)} F(f) \to C(d,c)$.

A weak inverse of $e$ is given by the functor

$\bar e : PSh(C)/Y(c) \to PSh(C/c)$

which sends $\eta : F' \to Y(C))$ to $F \in PSh(C/c)$ given by

$F : (f : d \to c) \mapsto F'(d)|_c \,,$

where $F'(d)|_c$ is the pullback

$\array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.$
###### Example

Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphsims to $C$, i.e. suppose that it factors through the forgetful functor from the over category to $C$:

$F : (C/c)^{op} \to C^{op} \to Set \,.$

Then $F'(d) = \sqcup_{f \in C(d,c)} F(f) = \sqcup_{f \in C(d,c)} F(d) \simeq C(d,c) \times F(d)$ and hence $F ' = Y(c) \times F$ with respect to the closed monoidal structure on presheaves.

For the analog statement in (∞,1)-category theory see (∞,1)-category of (∞,1)-presheaves – Interaction with overcategories

### Limits and colimits

###### Proposition

A limit in an under category is computed as a limit in the underlying category.

Precisely: let $C$ be a category, $t \in C$ an object, and $t/C$ the corresponding under category, and $p : t/C \to C$ the obvious projection.

Let $F : D \to t/C$ be any functor. Then, if it exists, the limit over $p \circ F$ in $C$ is the image under $p$ of the limit over $F$:

$p(\lim F) \simeq \lim (p F)$

and $\lim F$ is uniquely characterized by $\lim (p F)$.

###### Proof

Over a morphism $\gamma : d \to d'$ in $D$ the limiting cone over $p F$ (which exists by assumption) looks like

$\array{ && \lim p F \\ & \swarrow && \searrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }$

By the universal property of the limit this has a unique lift to a cone in the under category $t/C$ over $F$:

$\array{ && t \\ & \swarrow &\downarrow & \searrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }$

It therefore remains to show that this is indeed a limiting cone over $F$. Again, this is immediate from the universal property of the limit in $C$. For let $t \to Q$ be another cone over $F$ in $t/C$, then $Q$ is another cone over $p F$ in $C$ and we get in $C$ a universal morphism $Q \to \lim p F$

$\array{ && t \\ & \swarrow & \downarrow & \searrow \\ && Q \\ \downarrow & \swarrow &\downarrow & \searrow & \downarrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }$

A glance at the diagram above shows that the composite $t \to Q \to \lim p F$ constitutes a morphism of cones in $C$ into the limiting cone over $p F$. Hence it must equal our morphism $t \to \lim p F$, by the universal property of $\lim p F$, and hence the above diagram does commute as indicated.

This shows that the morphism $Q \to \lim p F$ which was the unique one giving a cone morphism on $C$ does lift to a cone morphism in $t/C$, which is then necessarily unique, too. This demonstrates the required universal property of $t \to \lim p F$ and thus identifies it with $\lim F$.

###### Remark

One often says “$p$ reflects limits” to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if $U: A \to C$ is monadic (i.e., has a left adjoint $F$ such that the canonical comparison functor $A \to (U F)-Alg$ is an equivalence), then $U$ both reflects and preserves limits. In the present case, the projection $p: A = t/C \to C$ is monadic, is essentially the category of algebras for the monad $T(-) = t + (-)$, at least if $C$ admits binary coproducts. (Added later: the proof is even simpler: if $U: A \to C$ is the underlying functor for the category of algebras of an endofunctor on $C$ (as opposed to algebras of a monad), then $U$ reflects and preserves limits; then apply this to the endofunctor $T$ above.)

###### Proposition

For $\mathcal{C}$ a category, $X \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ a diagram, $\mathcal{C}_{/X}$ the comma category (the over-category if $\mathcal{D}$ is the point) and $F \;\colon\; K \to \mathcal{C}_{/X}$ a diagram in the comma category, then the limit $\underset{\leftarrow}{\lim} F$ in $\mathcal{C}_{/X}$ coincides with the limit $\underset{\leftarrow}{\lim} F/X$ in $\mathcal{C}$.

For a proof see at (∞,1)-limit here.

### Initial and terminal objects

As a special case of the above discussion of limits and colimits in a slice $\mathcal{C}_{/X}$ we obtain the following statement, which of course is also immediately checked explicitly.

###### Corollary
• If $\mathcal{C}$ has an initial object $\emptyset$, then $\mathcal{C}_{/X}$ has an initial object, given by $\langle \emptyset \to X\rangle$.

• The terminal object of $\mathcal{C}_{/X}$ is $\mathrm{id}_X$.

Revised on September 2, 2015 13:07:43 by Urs Schreiber (195.82.63.198)