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\begin{document}

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\section*{nerve}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{NerveOfACategory}{Nerve of a 1-category}\dotfill \pageref*{NerveOfACategory} \linebreak
\noindent\hyperlink{definition_3}{Definition}\dotfill \pageref*{definition_3} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{PropNerveCat}{Properties}\dotfill \pageref*{PropNerveCat} \linebreak
\noindent\hyperlink{nerve_of_a_2category}{Nerve of a 2-category}\dotfill \pageref*{nerve_of_a_2category} \linebreak
\noindent\hyperlink{nerve_of_an_category}{Nerve of an $\omega$-category}\dotfill \pageref*{nerve_of_an_category} \linebreak
\noindent\hyperlink{nerve_of_chain_complexes}{Nerve of chain complexes}\dotfill \pageref*{nerve_of_chain_complexes} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{geometric_realization}{Geometric realization}\dotfill \pageref*{geometric_realization} \linebreak
\noindent\hyperlink{nerves_and_higher_categories}{Nerves and higher categories}\dotfill \pageref*{nerves_and_higher_categories} \linebreak
\noindent\hyperlink{internal_nerve}{Internal nerve}\dotfill \pageref*{internal_nerve} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{historical_note}{Historical note}\dotfill \pageref*{historical_note} \linebreak


The \emph{nerve} is the right adjoint of a pair of [[adjoint functors]] that exists in many situations. For the general abstract theory behind this see

\begin{itemize}%
\item [[nerve and realization]].

\end{itemize}
\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

As soon as any [[locally small category]] $C$ comes equipped with a [[simplicial object|cosimplicial object]]

\begin{displaymath}
\Delta_C : \Delta \to C
\end{displaymath}
that we may think of as determining a [[geometric realization|realization]] of the standard $n$-[[simplex]] in $C$, we make every [[object]] of $C$ [[space and quantity|probeable]] by [[simplex|simplices]] in that there is now a way to find the set

\begin{displaymath}
N(A)_n := Hom_C(\Delta_C[n],A)
\end{displaymath}
of ways to map the $n$-[[simplex]] into a given object $A$.

These collections of sets evidently organize into a [[simplicial set]]

\begin{displaymath}
N(A) : \Delta^{op} \to Set
  \,.
\end{displaymath}
This [[simplicial set]] is called the \emph{nerve} of $A$ (with respect to the chosen [[geometric realization|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.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{quote}%
(notice that for the moment the following gives just one particular case of the more general notion of nerve)


\end{quote}
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 set|dendroidal sets]].

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

\begin{displaymath}
i : S \to C
\end{displaymath}
we obtain a functor

\begin{displaymath}
N : C \to V^{S^{op}}
\end{displaymath}
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

\begin{displaymath}
N_i(c) : S^{op} \stackrel{i}\to C^{op} 
    \stackrel{C(-,c)}{\to}
    V
  \,.
\end{displaymath}
This $N_i(c)$ is the \textbf{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,

\begin{displaymath}
\mathrm{colim}((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C) = c,
\end{displaymath}
which is equivalent to the requirement that the corresponding nerve functor is [[full and faithful functor|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 $i$.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{NerveOfACategory}{}\subsubsection*{{Nerve of a 1-category}}\label{NerveOfACategory}

For fixing notation, recall that the source and target maps of a small [[category\#OneCollectionOfMorphisms|category]] form a [[span]] in the category $Span(Set)$ where composition [[span\#categories\_of\_spans|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.

\begin{defn}
\label{SmallCategory}\hypertarget{SmallCategory}{}
A \emph{[[small category]]} $\mathcal{C}_\bullet$ is

\begin{itemize}%
\item a pair of [[sets]] $\mathcal{C}_0 \in Set$ (the set of [[objects]]) and $\mathcal{C}_1 \in Set$ (the set of [[morphisms]])


\item equipped with [[functions]]

\begin{displaymath}
\itexarray{
    \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
  }\,,
\end{displaymath}
where the [[fiber product]] on the left is that over $\mathcal{C}_1 \stackrel{t}{\to} \mathcal{C}_0 \stackrel{s}{\leftarrow} \mathcal{C}_1$,



\end{itemize}
such that

\begin{itemize}%
\item $i$ takes values in [[endomorphisms]];

\begin{displaymath}
t \circ i = s \circ i =   id_{\mathcal{G}_0}, \;\;\;
\end{displaymath}

\item $\circ$ defines a partial [[composition]] operation which is [[associativity|associative]] and [[unitality|unital]] for $i(\mathcal{C}_0)$ the [[identities]]; in particular

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



\end{itemize}
\end{defn}
\hypertarget{definition_3}{}\paragraph*{{Definition}}\label{definition_3}

\begin{defn}
\label{NerveOfSmallCategory}\hypertarget{NerveOfSmallCategory}{}
For $\mathcal{C}_\bullet$ a [[small category]], def. \ref{SmallCategory}, its \emph{simplicial nerve} $N(\mathcal{C}_\bullet)_\bullet$ is the [[simplicial set]] with

\begin{displaymath}
N(\mathcal{C}_\bullet)_n \coloneqq \mathcal{C}_1^{\times_{\mathcal{C}_0}^n}
\end{displaymath}
the set of sequences of composable morphisms of length $n$, for $n \in \mathbb{N}$;

with face maps

\begin{displaymath}
d_k \colon N(\mathcal{C}_\bullet)_{n+1} \to N(\mathcal{C}_\bullet)_{n}
\end{displaymath}
being

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


\item for $n \geq 1$

\begin{itemize}%
\item 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;


\item 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.



\end{itemize}


\end{itemize}
The degeneracy maps

\begin{displaymath}
s_k \colon N(\mathcal{C}_\bullet)n \to N(\mathcal{C}_\bullet)_{n+1}
  \,.
\end{displaymath}
are given by inserting an [[identity]] morphism on $x_k$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Spelling this out in more detail: write

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

\begin{displaymath}
f_{i-1,i+1} \coloneqq f_{i,i+1} \circ f_{i-1,i}
\end{displaymath}
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 $k$'':

\begin{displaymath}
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 )
\end{displaymath}
\begin{displaymath}
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
  )
\end{displaymath}
\begin{displaymath}
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}
  )
  \,.
\end{displaymath}
Similarly, writing

\begin{displaymath}
f_{k,k} \coloneqq id_{x_k}
\end{displaymath}
for the identity morphism on the object $x_k$, then the degeneracy map acts by ``repeating the $k$th index''

\begin{displaymath}
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
  )
  \,.
\end{displaymath}
This makes it manifest that these functions organise into a [[simplicial set]].

\end{remark}
More abstractly, this construction is described as follows. Recall that

\begin{defn}
\label{}\hypertarget{}{}
The [[nLab:simplex category|simplex category]] $\Delta$ is equivalent to the [[full subcategory]]

\begin{displaymath}
i \colon \Delta \hookrightarrow Cat
\end{displaymath}
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.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{C}$ a [[small category]] its \emph{nerve} $N(\mathcal{C})$ is the [[simplicial set]] given by

\begin{displaymath}
N(\mathcal{C}) 
  \colon 
  \Delta^{op} 
     \hookrightarrow
     Cat^{op}
    \stackrel{Cat(-,\mathcal{C})}{\to}
    Set
  \,,
\end{displaymath}
where [[Cat]] is regarded as a [[1-category]] with objects locally small categories, and morphisms being [[functors]] between these.

\end{defn}
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 \hyperlink{NerveOfACategory}{above} for pairs of composables):

\begin{displaymath}
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}
\end{displaymath}
The collection of all functors between linear orders

\begin{displaymath}
\{
    0 \to 1 \to \cdots \to n
  \}
  \to
  \{
    0 \to 1 \to \cdots \to m
  \}
\end{displaymath}
is generated from those that map almost all generating morphisms $k \to k+1$ to another generating morphism, except at one position, where they

\begin{itemize}%
\item map a single generating morphism to the composite of two generating morphisms

\begin{displaymath}
\delta^n_i : [n-1] \to [n]
\end{displaymath}
\begin{displaymath}
\delta^n_i : ((i-1) \to i) \mapsto ((i-1) \to i \to (i+1))
\end{displaymath}

\item map one generating morphism to an identity morphism

\begin{displaymath}
\sigma^n_i : [n+1] \to [n]
\end{displaymath}
\begin{displaymath}
\sigma^n_i : (i \to i+1) \mapsto Id_i
\end{displaymath}


\end{itemize}
It follows that, for instance

\begin{itemize}%
\item 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: $(d_0 \stackrel{f_2 \circ f_1}{\to} d_2, d_2 \stackrel{f_3}{\to} d_3) \in N(\mathcal{C})_2$


\item 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: $(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{Id_{d_1}}{\to} d_1) \in N(\mathcal{C})_2$.



\end{itemize}
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:

\begin{itemize}%
\item $N(\mathcal{C})_0 = \{d | d \in Obj(\mathcal{C})\}$ is the collection of [[objects]] of $\mathcal{C}$;


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


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


\item $N(\mathcal{C})_3 = \left\{
    \left.
      \itexarray{
    d_1 &\stackrel{f_2}{\to}& d_2
    \\
    {}^{f_1}\uparrow &
    {}^{f_2 \circ f_1}\nearrow
    & \downarrow^{f_3}
    \\
    d_0 &\stackrel{f_3\circ (f_2\circ f_1)}{\to}&
    d_3
      }
      \;\;\;\;\;\stackrel{\exists !}{\Rightarrow}
      \;\;\;\;\;
      \itexarray{
    d_1 &\stackrel{f_2}{\to}& d_2
    \\
    {}^{f_1}\uparrow &
    \searrow^{f_3\circ f_2}
    & \downarrow^{f_3}
    \\
    d_0 &\stackrel{(f_3\circ f_2) \circ f_1}{\to}&
    d_3
      }
    \right|
    (f_3,f_2, f_1) \in
    Mor(D) {}_t \times_s Mor(D) {}_t \times_s Mor(D)
  \right\}$ 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.



\end{itemize}
\hypertarget{examples_2}{}\paragraph*{{Examples}}\label{examples_2}

\begin{example}
\label{}\hypertarget{}{}
\textbf{(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 $A$

\begin{displaymath}
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)
\end{displaymath}
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 space|classifying]] [[topological space]] $\cdots \simeq B G$ for $G$-[[principal bundles]].

\end{example}
\hypertarget{PropNerveCat}{}\paragraph*{{Properties}}\label{PropNerveCat}

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

\begin{prop}
\label{}\hypertarget{}{}
A simplicial set is the nerve of a category precisely if it satisfies the [[Segal condition]].

\end{prop}
See at \emph{[[Segal condition]]} for more on this.

\begin{prop}
\label{}\hypertarget{}{}
A [[simplicial set]] is the nerve of a [[small category]] precisely if all \emph{inner} [[horns]] have \emph{unique} fillers.

\end{prop}
See [[inner fibration]] for details on this.

\begin{prop}
\label{}\hypertarget{}{}
A [[simplicial set]] is the nerve of a [[groupoid]] precisely if \emph{all} [[horns]] of dimension $\gt 1$ have \emph{unique} fillers.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
The nerve $N(C)$ of a category is [[coskeleton|2-coskeletal]].

\end{prop}
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 $n \gt 3$.

\begin{prop}
\label{}\hypertarget{}{}
The nerve $N(C)$ of a [[small category]] is a [[Kan complex]] precisely if $C$ is a [[groupoid]].

\end{prop}
The existence of [[inverse]] morphisms in $C$ corresponds to the fact that in the [[Kan complex]] $N(C)$ the ``outer'' [[horns]]

\begin{displaymath}
\itexarray{
    && d_0
    \\
    & && \searrow^{f}
    \\
    d_1 &&\stackrel{Id_{d_1}}{\to} && d_1
  }
  \;\;\; 
  \;\;\; 
  and
  \;\;\; 
  \;\;\; 
  \itexarray{
    && d_1
    \\
    & {}^f\nearrow &&     
    \\
    d_0 &&\stackrel{Id_{d_0}}{\to} && d_1
  }
\end{displaymath}
have fillers

\begin{displaymath}
\itexarray{
    && d_0
    \\
    & {}^{f^{-1}}\nearrow&& \searrow^{f}
    \\
    d_1 &&\stackrel{Id_{d_1}}{\to} && d_1
  }
  \;\;\; 
  \;\;\; 
  and
  \;\;\; 
  \;\;\; 
  \itexarray{
    && d_1
    \\
    & {}^f\nearrow && \searrow^{f^{-1}}
    \\
    d_0 &&\stackrel{Id_{d_0}}{\to} && d_0
  }
\end{displaymath}
(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.

\begin{prop}
\label{}\hypertarget{}{}
The nerve functor

\begin{displaymath}
N : Cat \to SSet
\end{displaymath}
is a [[full and faithful functor]].

\end{prop}
So [[functors]] between [[locally small category|locally small categories]] are in [[bijection]] with morphisms of [[simplicial sets]] between their nerves.

\begin{prop}
\label{}\hypertarget{}{}
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

\begin{displaymath}
\itexarray{
    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
  }
\end{displaymath}
are [[pullback]] diagrams.

\end{prop}
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 pullback|homotopy pullbacks]].

\hypertarget{nerve_of_a_2category}{}\subsubsection*{{Nerve of a 2-category}}\label{nerve_of_a_2category}

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

\begin{prop}
\label{}\hypertarget{}{}
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 \emph{unique} .

\end{prop}
This is theorem 8.6 in (\href{http://www.tac.mta.ca/tac/volumes/9/n10/9-10abs.html}{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]].

\hypertarget{nerve_of_an_category}{}\subsubsection*{{Nerve of an $\omega$-category}}\label{nerve_of_an_category}

\begin{itemize}%
\item For [[strict omega-category|strict omega-categories]] there is a nerve induced by the [[orientals]]; see [[omega-nerve]].

\end{itemize}
\hypertarget{nerve_of_chain_complexes}{}\subsubsection*{{Nerve of chain complexes}}\label{nerve_of_chain_complexes}

Let $Ch_+$ be the [[category of chain complexes]] of abelian groups, then there is a [[simplicial object|cosimplicial chain complex]]

\begin{displaymath}
C_\bullet : \Delta \to Ch_+
\end{displaymath}
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

\begin{itemize}%
\item D. Kan, \emph{Functors involving c.s.s complexes}, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar., 1958), pp. 330--346 (\href{http://www.jstor.org/stable/1993103}{jstor})

\end{itemize}
The nerve/[[geometric realization|realization]] adjunction induced from this is the [[Dold?Kan correspondence]]. See there for more details.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\hypertarget{geometric_realization}{}\subsubsection*{{Geometric realization}}\label{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.

\hypertarget{nerves_and_higher_categories}{}\subsubsection*{{Nerves and higher categories}}\label{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 \emph{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 set|complicial sets]], [[Kan complex|Kan complexes]], [[quasi-category|quasi-categories]], [[simplicial T-complex|simplicial T-complexes]],\ldots{}

\hypertarget{internal_nerve}{}\subsubsection*{{Internal nerve}}\label{internal_nerve}

A variant of the nerve construction can also be applied \emph{internally} within a category, to any internal category, see the discussion at [[internal category]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[monad with arities]]


\item [[Duskin nerve]]


\item [[∞-nerve]]


\item [[homotopy coherent nerve]]


\item [[dg-nerve]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[W. G. Dwyer]], [[D. M. Kan]], Singular functors and realization functors, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), no. 2, 147--153. \href{http://www.nd.edu/~wgd/Dvi/SingularAndRealization.pdf}{pdf}


\item [[John Isbell]], Adequate subcategories, Illinois J. Math. 4, 541--552 (1960)


\item [[Tom Leinster]], \emph{Higher operads, higher categories} , London Mathematical Society Lecture Note Series, 298. Cambridge Univ. Press 2004. xiv+433 pp. ISBN: 0-521-53215-9, \href{http://front.math.ucdavis.edu/0305.5049}{arXiv:math.CT/0305049}


\item [[Ross Street]], The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987), no. 3, 283--335.


\item [[Paul Bressler]], Alexander Gorokhovsky, [[Ryszard Nest]], [[Boris Tsygan]], \emph{Formality for algebroids I: Nerves of two-groupoids}, \href{http://arxiv.org/abs/1211.6603}{arxiv/1211.6603}



\end{itemize}
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:

\begin{itemize}%
\item [[Tom Leinster]], \href{https://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html}{How I learned to love the nerve construction}, $n$-Category Caf\'e{}, January 6, 2008.

\end{itemize}
\hypertarget{historical_note}{}\subsubsection*{{Historical note}}\label{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

\begin{itemize}%
\item [[Graeme Segal]], \emph{Classifying spaces and spectral sequences,} Inst. Hautes \'E{}tudes Sci. Publ. Math. No. 34 (1968) 105-112.

\end{itemize}
Many of the later developments can already be seen there in `embryonic' form.

[[!redirects nerves]] [[!redirects nerve functor]] [[!redirects nerve functors]] [[!redirects simplicial nerve]] [[!redirects simplicial nerves]] [[!redirects simplicial nerve functor]] [[!redirects simplicial nerve functors]]

[[!redirects nerve of a category]] [[!redirects nerves of a categories]]



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