Contents

category theory

# Contents

The nerve is the right adjoint of a pair of adjoint functors that exists in many situations. For the general abstract theory behind this see

## Idea

As soon as any locally small category $C$ comes equipped with a cosimplicial object

$\Delta_C : \Delta \to C$

that we may think of as determining a realization of the standard $n$-simplex in $C$, we make every object of $C$ probeable by simplices in that there is now a way to find the set

$N(A)_n := Hom_C(\Delta_C[n],A)$

of ways to map the $n$-simplex into a given object $A$.

These collections of sets evidently organize into a simplicial set

$N(A) : \Delta^{op} \to Set \,.$

This simplicial set is called the nerve of $A$ (with respect to the chosen realization of the standard simplices in $C$). Typically the nerve defines a functor $N \colon C \to Set^{\Delta^op}$ that has a left adjoint $|\cdot| \colon Set^{\Delta^op} \to C$ called realization.

There are many generalizations of this procedure, some of which are described below.

## Definition

(notice that for the moment the following gives just one particular case of the more general notion of nerve)

Let $S$ be one of the categories of geometric shapes for higher structures, such as the globe category $G$, the simplex category $\Delta$, the cube category $\Box$, the cycle category $\Lambda$ of Connes, or certain category $\Omega$ related to trees whose corresponding presheaves are dendroidal sets.

If $C$ is any locally small category or, more generally, a $V$-enriched category equipped with a functor

$i : S \to C$

we obtain a functor

$N : C \to V^{S^{op}}$

from $C$ to globular sets or simplicial sets or cubical sets, respectively, (or the corresponding $V$-objects) given on an object $c \in C$ by

$N_i(c) : S^{op} \stackrel{i}\to C^{op} \stackrel{C(-,c)}{\to} V \,.$

This $N_i(c)$ is the nerve of $c$ with respect to the chosen $i : S \to C$. In other words, $N = i^* \circ Y$ where $Y: C \to [C^{op}, V]$ is the curried Hom functor; if $V=\mathsf{Sets}$ then $Y$ is the Yoneda embedding.

Typically, one wants that $i$ is dense functor, i.e. that every object $c$ of $C$ is canonically a colimit of a diagram of objects in $M$, more precisely,

$\mathrm{colim}((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C) = c,$

which is equivalent to the requirement that the corresponding nerve functor is fully faithful (in other words, if $i$ is inclusion then $S$ is a left adequate subcategory of $C$ in terminology of [Isbell 1960]). The nerve functor may be viewed as a singular functor? of the functor $i$.

## Examples

### Nerve of a 1-category

For fixing notation, recall that the source and target maps of a small category form a span in the category $Span(Set)$ where composition is given by a pullback (fiber product). The pairs of composable morphisms of a category are then obtained composing its source/target span with itself.

###### Definition

A small category $\mathcal{C}_\bullet$ is

• a pair of sets $\mathcal{C}_0 \in Set$ (the set of objects) and $\mathcal{C}_1 \in Set$ (the set of morphisms)

• equipped with functions

$\array{ \mathcal{C}_1 \times_{\mathcal{C}_0} \mathcal{C}_1 &\stackrel{\circ}{\to}& \mathcal{C}_1 & \stackrel{\overset{t}{\to}}{\stackrel{\overset{i}{\leftarrow}}{\underset{s}{\to}}}& \mathcal{C}_0 }\,,$

where the fiber product on the left is that over $\mathcal{C}_1 \stackrel{t}{\to} \mathcal{C}_0 \stackrel{s}{\leftarrow} \mathcal{C}_1$,

such that

• $i$ takes values in endomorphisms;

$t \circ i = s \circ i = id_{\mathcal{G}_0}, \;\;\;$
• $\circ$ defines a partial composition operation which is associative and unital for $i(\mathcal{C}_0)$ the identities; in particular

$s (g \circ f) = s(f)$ and $t (g \circ f) = t(g)$.

#### Definition

###### Definition

For $\mathcal{C}_\bullet$ a small category, def. , its simplicial nerve $N(\mathcal{C}_\bullet)_\bullet$ is the simplicial set with

$N(\mathcal{C}_\bullet)_n \coloneqq \mathcal{C}_1^{\times_{\mathcal{C}_0}^n}$

the set of sequences of composable morphisms of length $n$, for $n \in \mathbb{N}$;

with face maps

$d_k \colon N(\mathcal{C}_\bullet)_{n+1} \to N(\mathcal{C}_\bullet)_{n}$

being

• for $n = 0$, $d_0= target:arr(\mathcal{C})\to ob(\mathcal{C})$, whilst $d_1$ is similarly the domain / source function;

• for $n \geq 1$

• the two outer face maps $d_0$ and $d_n$ are given by forgetting the first and the last morphism in such a sequence, respectively;

• the $n-1$ inner face maps $d_{0 \lt k \lt n}$ are given by composing the $k$th morphism with the $k+1$st in the sequence.

The degeneracy maps

$s_k \colon N(\mathcal{C}_\bullet)n \to N(\mathcal{C}_\bullet)_{n+1} \,.$

are given by inserting an identity morphism on $x_k$.

###### Remark

Spelling this out in more detail: write

$\mathcal{C}_n = \left\{ x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_2 \stackrel{f_{2,3}}{\to} \cdots \stackrel{f_{n-1,n}}{\to} x_n \right\}$

for the set of sequences of $n$ composable morphisms. Given any element of this set and $0 \lt k \lt n$, write

$f_{i-1,i+1} \coloneqq f_{i,i+1} \circ f_{i-1,i}$

for the composition of the two morphism that share the $i$th vertex.

With this, face map $d_k$ acts simply by “removing the index $k$”:

$d_0 \colon (x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto (x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n )$
$d_{0\lt k \lt n} \colon ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n ) \mapsto ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n )$
$d_n \colon ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} ) \,.$

Similarly, writing

$f_{k,k} \coloneqq id_{x_k}$

for the identity morphism on the object $x_k$, then the degeneracy map acts by “repeating the $k$th index”

$s_k \colon ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \mapsto ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \,.$

This makes it manifest that these functions organise into a simplicial set.

More abstractly, this construction is described as follows. Recall that

###### Definition

The simplex category $\Delta$ is equivalent to the full subcategory

$i \colon \Delta \hookrightarrow Cat$

of Cat on non-empty finite linear orders regarded as categories, meaning that the object $[n] \in Obj(\Delta)$ may be identified with the category $[n] = \{0 \to 1 \to 2 \to \cdots \to n\}$. The morphisms of $\Delta$ are all functors between these total linear categories.

###### Definition

For $\mathcal{C}$ a small category its nerve $N(\mathcal{C})$ is the simplicial set given by

$N(\mathcal{C}) \colon \Delta^{op} \hookrightarrow Cat^{op} \stackrel{Cat(-,\mathcal{C})}{\to} Set \,,$

where Cat is regarded as a 1-category with objects locally small categories, and morphisms being functors between these.

So the set $N(\mathcal{C})_n$ of $n$-simplices of the nerve is the set of functors $\{0 \to 1 \to \cdots \to n\} \to \mathcal{C}$. This is clearly the same as the set of sequences of composable morphisms in $\mathcal{C}$ of length $n$ obtained by iterated fiber product (as above for pairs of composables):

$N(\mathcal{C})_n = \underbrace{ Mor(\mathcal{C}) \times_{Obj(\mathcal{C})} Mor(\mathcal{C}) \times_{Obj(\mathcal{C})} \cdots \times_{Obj(\mathcal{C})} Mor(\mathcal{C}) }_{n \medspace factors}$

The collection of all functors between linear orders

$\{ 0 \to 1 \to \cdots \to n \} \to \{ 0 \to 1 \to \cdots \to m \}$

is generated from those that map almost all generating morphisms $k \to k+1$ to another generating morphism, except at one position, where they

• map a single generating morphism to the composite of two generating morphisms

$\delta^n_i : [n-1] \to [n]$
$\delta^n_i : ((i-1) \to i) \mapsto ((i-1) \to i \to (i+1))$
• map one generating morphism to an identity morphism

$\sigma^n_i : [n+1] \to [n]$
$\sigma^n_i : (i \to i+1) \mapsto Id_i$

It follows that, for instance

• for $(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{f_2}{\to} d_2, d_2 \stackrel{f_3}{\to} d_3) \in N(D)_3$ the image under $d_1 := N(\mathcal{C})(\delta_1) : N(\mathcal{C})_3 \to N(\mathcal{C})_2$ is obtained by composing the first two morphisms in this sequence: $(d_0 \stackrel{f_2 \circ f_1}{\to} d_2, d_2 \stackrel{f_3}{\to} d_3) \in N(\mathcal{C})_2$

• for $(d_0 \stackrel{f_1}{\to} d_1) \in N(\mathcal{C})_1$ the image under $s_1 := N(\mathcal{C})(\sigma_1) : N(\mathcal{C})_1 \to N(\mathcal{C})_2$ is obtained by inserting an identity morphism: $(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{Id_{d_1}}{\to} d_1) \in N(\mathcal{C})_2$.

In this way, generally the face and degeneracy maps of the nerve of a category come from composition of morphisms and from inserting identity morphisms.

In particular in light of their generalization to nerves of higher categories, discussed below, the cells in the nerve $N(\mathcal{C})$ have the following interpretation:

• $N(\mathcal{C})_0 = \{d | d \in Obj(\mathcal{C})\}$ is the collection of objects of $\mathcal{C}$;

• $N(\mathcal{C})_1 = Mor(\mathcal{C}) = \{d \stackrel{f}{\to} d' | f \in Mor(D)\}$ is the collection of morphisms of $D$;

• $N(\mathcal{C})_2 = \left\{ \left. \array{ && d_1 \\ & {}^{f_1}\nearrow &\Downarrow^{\exists !}& \searrow^{f_2} \\ d_0 &&\stackrel{f_2 \circ f_1}{\to}&& d_2 } \right| (f_1, f_2) \in Mor(D) {}_t \times_s Mor(D) \right\}$ is the collection of composable morphisms in $\mathcal{C}$: the 2-cell itself is to be read as the composition operation, which is unique for an ordinary category (there is just one way to compose two morphisms);

• $N(\mathcal{C})_3 = \left\{ \left. \array{ d_1 &\stackrel{f_2}{\to}& d_2 \\ {}^{f_1}\uparrow & {}^{f_2 \circ f_1}\nearrow & \downarrow^{f_3} \\ d_0 &\stackrel{f_3\circ (f_2\circ f_1)}{\to}& d_3 } \;\;\;\;\;\stackrel{\exists !}{\Rightarrow} \;\;\;\;\; \array{ d_1 &\stackrel{f_2}{\to}& d_2 \\ {}^{f_1}\uparrow & \searrow^{f_3\circ f_2} & \downarrow^{f_3} \\ d_0 &\stackrel{(f_3\circ f_2) \circ f_1}{\to}& d_3 } \right| (f_3,f_2, f_1) \in Mor(D) {}_t \times_s Mor(D) {}_t \times_s Mor(D) \right\}$ is the collection of triples of composable morphisms, to be read as the unique associators that relate one way to compose three morphisms using the above 2-cells to the other way.

#### Examples

###### Example

(bar construction)

Let $A$ be a monoid (for instance a group) and write $\mathbf{B} A$ for the corresponding one-object category with $Mor(\mathbf{B} A) = A$. Then the nerve $N(\mathbf{B} A)$ of $\mathbf{B}A$ is the simplicial set which is the usual bar construction of $A$

$N(\mathbf{B}A) = \left( \cdots A \times A \times A \stackrel{\to}{\stackrel{\to}{\to}} A \times A \stackrel{\to}{\to} A \to {*} \right)$

In particular, when $A = G$ is a discrete group, then the geometric realization $|N(\mathbf{B} G)|$ of the nerve of $\mathbf{B}G$ is the classifying topological space $\cdots \simeq B G$ for $G$-principal bundles.

#### Properties

The following lists some characteristic properties of simplicial sets that are nerves of categories.

###### Proposition

A simplicial set is the nerve of a category precisely if it satisfies the Segal condition.

See at Segal condition for more on this.

###### Proposition

A simplicial set is the nerve of a small category precisely if all inner horns have unique fillers.

See inner fibration for details on this.

###### Proposition

A simplicial set is the nerve of a groupoid precisely if all horns of dimension $\gt 1$ have unique fillers.

###### Proposition

The nerve $N(C)$ of a category is 2-coskeletal.

Hence all horn inclusions $\Lambda[n]_i \hookrightarrow \Delta[n]$ have unique fillers for $n \gt 3$, and all boundary inclusions $\partial \Delta[n] \hookrightarrow \Delta[n]$ have unique fillers for $n \geq 3$.

Here the point as compared to the previous statements is that in particular all the outer horns have fillers for $n \gt 3$.

###### Proposition

The nerve $N(C)$ of a small category is a Kan complex precisely if $C$ is a groupoid.

The existence of inverse morphisms in $C$ corresponds to the fact that in the Kan complex $N(C)$ the “outer” horns

$\array{ && d_0 \\ & && \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_1 }$

have fillers

$\array{ && d_0 \\ & {}^{f^{-1}}\nearrow&& \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \searrow^{f^{-1}} \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_0 }$

(even unique fillers, due to the above).

It suggests the sense that a Kan complex models an ∞-groupoid. The possible lack of uniqueness of fillers in general gives the ‘weakness’ needed, whilst the lack of a coskeletal property requirement means that the homotopy type it represents has enough generality, not being constrained to be a 1-type.

###### Proposition

The nerve functor

$N : Cat \to SSet$

So functors between locally small categories are in bijection with morphisms of simplicial sets between their nerves.

###### Proposition

A simplicial set $S$ is the nerve of a locally small category $C$ precisely if it satisfies the Segal conditions: precisely if all the commuting squares

$\array{ S_{n+m} &\stackrel{\cdots \circ d_0 \circ d_0}{\to}& S_m \\ {}^{\cdots d_{n+m-1}\circ d_{n+m}}\downarrow && \downarrow \\ S_n &\stackrel{d_0 \circ \cdots d_0}{\to}& S_0 }$

are pullback diagrams.

Unwrapping this definition inductively in $(n+m)$, this says that a simplicial set is the nerve of a category if and only if all its cells in degree $\geq 2$ are unique compositors, associators, pentagonators, etc of composition of 1-morphisms. No non-trivial such structure cells appear and no further higher cells appear.

This characterization of categories in terms of nerves directly leads to the model of (∞,1)-category in terms of complete Segal spaces by replacing in the above discussion sets by topological spaces (or something similar, like Kan complexes) and pullbacks by homotopy pullbacks.

### Nerve of a 2-category

For 2-categories modeled as bicategories the nerve operation is called the Duskin nerve.

###### Proposition

A simplicial set is the Duskin nerve of a bigroupoid precisely if it is a 2-hypergroupoid: a Kan complex such that the horn fillers in dimension $\geq 3$ are unique .

This is theorem 8.6 in (Duskin)

For a 2-category, regarded as a Cat-internal category one can apply the nerve operation for categories in stages, to obtain the double nerve.

### Nerve of chain complexes

Let $Ch_+$ be the category of chain complexes of abelian groups, then there is a cosimplicial chain complex

$C_\bullet : \Delta \to Ch_+$

given by sending the standard $n$-simplex $\Delta[n]$ first to the free simplicial group $F(\Delta[n])$ over it and then that to the normalized Moore complex. This is a small version of the ordinary homology chain complex of the standard $n$-simplex.

The nerve induced by this cosimplicial object was first considered in

• D. Kan, Functors involving c.s.s complexes, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar., 1958), pp. 330–346 (jstor)

The nerve/realization adjunction induced from this is the Dold?Kan correspondence. See there for more details.

## Remarks

### Geometric realization

Often the operation of taking the nerve of a (higher) category is followed by forming the geometric realization of the corresponding cellular set.

### Nerves and higher categories

For many purposes it is convenient to conceive categories and especially ∞-categories entirely in terms of their nerves: those simplicial sets that arise as certain nerves are usually characterized by certain properties. So one can turn this around and define an ∞-category as a simplicial set with certain properties. This is the strategy of a geometric definition of higher category. Examples for this are complicial sets, Kan complexes, quasi-categories, simplicial T-complexes,…

### Internal nerve

A variant of the nerve construction can also be applied internally within a category, to any internal category, see the discussion at internal category.

For an explanation of how the category $\Delta$ and the nerve construction arise canonically from the free category monad on the category of quivers, see:

### Historical note

The notion of the nerve of a category seems to be due to Grothendieck, which is in turn based on the nerve of a covering from 1926 work of Pavel Sergeevič Aleksandrov. One of the first papers to consider the properties of the nerve and to apply it to problems in algebraic topology was

• Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. No. 34 (1968) 105-112.

Many of the later developments can already be seen there in ‘embryonic’ form.