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nerve
- Provides a functor to simplicial set
\begin{align*}
{ \mathcal{N}({{-}}) }: \mathsf{Cat}&\to {\mathsf{sSet}}\\
\mathsf{C} &\mapsto { \mathcal{N}({\mathsf{C}}) }
\end{align*}
- After application:
\begin{align*}
{ \mathcal{N}({\mathsf{C}}) }: \Delta^{\operatorname{op}}&\to {\mathsf{Set}}\\ \quad
[n] &\mapsto {\mathsf{Fun}}([n], \mathsf{C})
\end{align*}
- So \({ \mathcal{N}({\mathsf{C}}) }({-}) = {\mathsf{Fun}}({-}, \mathsf{C})\) - A simplicial set whose skeleton is - \({ \mathcal{N}({\mathsf{C}}) }_0\): The objects of \(x,y,z,\cdots \in \mathsf{C}\) - \({ \mathcal{N}({\mathsf{C}}) }_1\): Morphisms \(\mathsf{C}(x, y), \mathsf{C}(y, z), \cdots\) - \({ \mathcal{N}({\mathsf{C}}) }_2\): Composable morphisms:
- The nerve has sufficient data to reconstruct \(\mathsf{C}\) up to isomorphism of categories.
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\({ \mathcal{N}({{-}}) }: \mathsf{Cat}\to {\mathsf{sSet}}\) is fully faithful.
- Actual statement: \({ \mathcal{N}({\mathsf{C}}) }\) is a Kan complex (with a unique filler for every horn) iff \(\mathsf{C}\) is a groupoid.
- \({ \mathcal{N}({\mathsf{C}}) }_n\): tuples \(f_0, f_1, \cdots, f_{n-1}\) of composable morphisms
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Alternative functor definition:
- Define a functor \begin{align*} \mathcal{P}: \mathsf{Poset}\to \mathsf{Cat}^{{\mathrm{small}}} \end{align*} which takes a poset to its poset category, where there is a unique morphism \(p\to q \iff p\leq q\).
- Using the definition of a simplicial set as a functor \(\Delta^{\operatorname{op}}\to {\mathsf{Set}}\), define \begin{align*} { \mathcal{N}({\mathsf{C}}) }({-}) := {\mathsf{Fun}}({-}, \mathsf{C}) \circ \mathcal{P}({-}) = {\mathsf{Fun}}( \mathcal{P}({-}), \mathsf{C}) \end{align*} Thus \({ \mathcal{N}({\mathsf{C}}) }([n]) = {\mathsf{Fun}}([n], \mathsf{C})\) where \([n]\) is the poset category on \((\left\{{0, 1, \cdots, n}\right\}, \leq)\).
Actual Definition
Given an ordinary category \(\mathsf{C}\), define the nerve of \(\mathsf{C}\) to be the simplicial set given by \begin{align*} N(\mathsf{C})_n \coloneqq\left\{{\text{Functors } F: [n] \to \mathsf{C}}\right\} \end{align*}
where \([n]\) is the poset category on \(\left\{{1, 2, \cdots, n}\right\}\). So an \(n{\hbox{-}}\)simplex is a diagram of objects \(X_0, \cdots, X_n \in {\operatorname{Ob}}(\mathsf{C})\) and a sequence of maps.
This defines an \(\infty{\hbox{-}}\)category, and there is a correspondence \begin{align*} {\mathsf{Fun}}_{\mathsf{Cat}}(\mathsf{C}, \mathsf{D}) &\rightleftharpoons [{ \mathcal{N}({\mathsf{C}}) }, { \mathcal{N}({\mathsf{D}}) }]_{{ \underset{\infty}{ \mathsf{Cat}} }} .\end{align*} Note that taking the nerve of a category preserves the usual categorical structure, since the objects are the 0-simplices and the morphisms are the 1-simplices.
Notes
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If \(\mathsf{C}\) has any initial or terminal objects, \({ \mathcal{N}({\mathsf{C}}) }\) is contractible..?
- What does this mean? Define homotopy direct on \({\mathsf{sSet}}\), or take geometric realization to \({\mathsf{Top}}\)..?
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\(\operatorname{im}{ \mathcal{N}({{-}}) } \hookrightarrow{\mathsf{sSet}}\) are precisely Segal spaces
- I.e. \({ \mathcal{N}({\mathsf{C}}) }\) is a Segal space, regarding \({\mathsf{Set}}\hookrightarrow{ \underset{\infty}{ {\mathsf{Grpd}}} }\) as the discrete objects.
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There is an adjunction:
\begin{align*}
\adjunction{{ {\left\lvert {{-}} \right\rvert} }}{ \mathcal{N}({{-}}) }{{\mathsf{sSet}}}{\mathsf{Cat}}
.\end{align*}
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Here the geometric realization is left-adjoint to the nerve.
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Note that the nerve doesn’t have a right adjoint? Seemingly because it doesn’t preserve colimits.
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Relative nerve
Adjunction
Coherent nerve
