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\begin{document}

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\section*{nerve and realization}

[[!redirects realization and nerve]]

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

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

[[!include category theory - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - 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{TopologicalRealizationOfSimplicialSets}{Topological realization of simplicial sets}\dotfill \pageref*{TopologicalRealizationOfSimplicialSets} \linebreak
\noindent\hyperlink{NervesOfCategories}{Nerve and realization of categories}\dotfill \pageref*{NervesOfCategories} \linebreak
\noindent\hyperlink{doldkan_correspondence}{Dold--Kan correspondence}\dotfill \pageref*{doldkan_correspondence} \linebreak
\noindent\hyperlink{higher_lie_integration__smooth_sullivan_construction}{Higher Lie integration / smooth Sullivan construction}\dotfill \pageref*{higher_lie_integration__smooth_sullivan_construction} \linebreak
\noindent\hyperlink{simplicial_models_for_categories}{Simplicial models for $(\infty,1)$-categories}\dotfill \pageref*{simplicial_models_for_categories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{full_and_faithfulness}{Full and faithfulness}\dotfill \pageref*{full_and_faithfulness} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Under rather general conditions a [[functor]]

\begin{displaymath}
S_C : S \to C
\end{displaymath}
into a [[cocomplete category]] $C$ (possibly a $V$-[[enriched category]] with $V$ some [[complete category|complete]] [[symmetric monoidal category]]) induces a pair of [[adjoint functors]]

\begin{displaymath}
C \stackrel{\xleftarrow{|-|}}{\underset{N}{\rightarrow}} [S^{op}, V],
\end{displaymath}
where $|-| \dashv N$, between $C$ and the [[category of presheaves]] $PSh(S) = [S^{op}, V]$ on $S$ (here $V$ = [[Set]] for the unenriched case)

where

\begin{itemize}%
\item $N$ behaves like a [[nerve]] operation;


\item $|-|$ behaves like [[geometric realization]].



\end{itemize}
\begin{remark}
\label{}\hypertarget{}{}
Here ``$S$'' is supposed to be suggestive of a category of certain ``[[geometric shapes for higher structures|geometric Shapes]]''. The canonical example is $S = \Delta$, the [[simplex category]], and the reader may find it helpful to keep that example in mind.

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

We place ourselves in the context of $V$-[[enriched category theory]]. The reader wishing to stick to the ordinary notions in [[locally small category|locally small categories]] takes $V$= [[Set]].

The \emph{realization} operation is the left [[Kan extension]] of $S_C : S \to C$ along the [[Yoneda embedding]] $S \hookrightarrow [S^{op},V]$ (i.e. the [[Yoneda extension]]):

\begin{displaymath}
\itexarray{
    S &\stackrel{S_C}{\to}&& C
    \\
    \downarrow^{Y} &\Downarrow& \nearrow_{|-|}
    \\
    [S^{op},V]
  }
  \,.
\end{displaymath}
If we assume that $C$ is [[copower|tensored]] over $V$, then by the general [[coend]] formula for left [[Kan extension]] we find that for $X \in [S^{op}, V]$ we have

\begin{displaymath}
|X| \simeq 
   \int^{s \in S}
   S_C(s) \cdot X_s
  \,.
\end{displaymath}
For instance when $S = \Delta$ is the [[simplex category]] this reads more recognizably

\begin{displaymath}
|X| \simeq 
   \int^{[n] \in \Delta}
   \Delta_C[n] \cdot X_n
  \,.
\end{displaymath}
The corresponding [[nerve]] operation

\begin{displaymath}
N :
  C \stackrel{}{\to} [S^{op},V]
\end{displaymath}
is given by

\begin{displaymath}
N(c) : S^{op} \stackrel{S_C^{op}}{\to} C^{op}
  \stackrel{C(-,c)}{\to}  
  V
  \,.
\end{displaymath}
\begin{theorem}
\label{}\hypertarget{}{}
Nerve and realization are a pair of [[adjoint functors]]

\begin{displaymath}
(|-| \dashv N)
\end{displaymath}
with $N$ [[right adjoint]].

\end{theorem}
\begin{proof}
Using the fact that the Hom in its first argument sends [[coends]] to [[ends]] and then using the definition of [[copower|tensoring]] over $V$, we check the hom-isomorphism

\begin{displaymath}
\begin{aligned}
    Hom_C(|X|, c)
    &:=
    Hom_C( \int^{s} S_C(s) \cdot X_s, c)
    \\
    & \simeq
    \int_{s} Hom_C( S_C(s) \cdot X_s, c)
    \\
    & \simeq
    \int_{s} Hom_V( X_s , C(S_C(s), c))
    \\
    & =
    \int_{s} Hom_V( X_s , N(c)_s)
    \\
    & \simeq
    Hom_{[S^{op},V]}(X, N(c))
    \,,
  \end{aligned}
\end{displaymath}
where in the last step we used the definition of the [[enriched functor category]] in terms of an [[end]].

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
In many cases we have $V =$ [[Set]] and the [[copower|tensoring]] of an object $c$ over a set $I$ is given by coproducts as

\begin{displaymath}
c \cdot I = \coprod_{i \in I} c
    \,.
\end{displaymath}
This is the case for instance for the below examples of realization of [[simplicial sets]], nerves of categories and the [[Dold-Kan correspondence]].

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

\hypertarget{TopologicalRealizationOfSimplicialSets}{}\subsubsection*{{Topological realization of simplicial sets}}\label{TopologicalRealizationOfSimplicialSets}

A classical example is given by the [[simplicial topological space|cosimplicial topological space]]

\begin{displaymath}
\Delta_{Top} : \Delta \to Top
\end{displaymath}
that sends the abstract $n$-[[simplex]] $[n]$ to the standard topological $n$-simplex $\Delta_{Top}[n] \subset \mathbb{R}^n$.

\begin{itemize}%
\item The corresponding realization is what is traditionally called \emph{[[geometric realization]]} of simplicial sets.

By restricting this to simplicial sets which are themselves simplicial nerves of [[categories]] (see \href{NervesOfCategories}{below}) or more generally are [[quasi-categories]], this also induces the notion of geometric realization of categorical structures.

The construction generalizes also to a notion of \emph{[[geometric realization of simplicial topological spaces]]}.


\item The corresponding [[nerve]] is the [[singular simplicial complex]] functor, producing the [[fundamental ∞-groupoid]] of a topological space.



\end{itemize}
This topological nerve and realization adjunction plays a central role as a presentation of the [[Quillen equivalence]] between the [[model structure on simplicial sets]] and the [[model structure on topological spaces]]. This is discussed in detail at \emph{[[homotopy hypothesis]]}.

\hypertarget{NervesOfCategories}{}\subsubsection*{{Nerve and realization of categories}}\label{NervesOfCategories}

Pretty much every notion of [[category]] and [[higher category theory|higher category]] comes, or should come, with its canonical notion of simplicial nerve, induced from a functor

\begin{displaymath}
\Delta_C : \Delta \to n Cat
\end{displaymath}
that sends the standard $n$-[[simplex]] to something like the free $n$-category on the $n$-[[directed n-graph|directed graph]] underlying that simplex.

For ordinary categories see the discussion at \emph{[[nerve]]} and at \emph{[[geometric realization of categories]]}.

One formalization of this for $n = \infty$ in the context of [[strict ∞-categories]] is the cosimplicial $\omega$-category called the [[orientals]]

\begin{displaymath}
\Delta_{\omega} : \Delta \to \omega Cat
    \,.
\end{displaymath}
\begin{itemize}%
\item The induced [[nerve]] is the [[∞-nerve]].


\item The induced realization operation is the operation of forming the free $\omega$-category on a simplicial set. See [[oriental]] for more details.



\end{itemize}
\hypertarget{doldkan_correspondence}{}\subsubsection*{{Dold--Kan correspondence}}\label{doldkan_correspondence}

The [[Dold-Kan correspondence]] is the nerve/realization adjunction for the [[homology]] functor

\begin{displaymath}
\Delta_{C_\bullet} : \Delta \to Ch_+
\end{displaymath}
to the [[category of chain complexes]] of abelian groups, which sends the standard $n$-[[simplex]] to its [[homology]] chain complex, more precisely to its normalized [[Moore complex]].

\begin{itemize}%
\item The induced realization is the normalized [[Moore complex]] functor extended from $\Delta$ to all [[simplicial sets]].

\end{itemize}
\hypertarget{higher_lie_integration__smooth_sullivan_construction}{}\subsubsection*{{Higher Lie integration / smooth Sullivan construction}}\label{higher_lie_integration__smooth_sullivan_construction}

see at \emph{[[Lie integration]]} \href{Lie+integration#LieIntegrationIsRightQuillenFunctor}{this Prop.}

\hypertarget{simplicial_models_for_categories}{}\subsubsection*{{Simplicial models for $(\infty,1)$-categories}}\label{simplicial_models_for_categories}

The canonical cosimplicial [[simplicially enriched category]]

\begin{displaymath}
\Delta \to SSet\text{-}Cat
\end{displaymath}
induces the [[homotopy coherent nerve]] of [[SSet]]-[[enriched category|enriched categories]] and establishes the relation between the [[quasi-category]] and the simplicially enriched model for [[(infinity,1)-category|(infinity,1)-categories]]. See

\begin{itemize}%
\item [[relation between quasi-categories and simplicial categories]].

\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{full_and_faithfulness}{}\subsubsection*{{Full and faithfulness}}\label{full_and_faithfulness}

Under some conditions one can characterize when and where the nerve construction is a [[full and faithful functor]]. For the moment see for instance \emph{[[monad with arities]]}.

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

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


\item [[geometric realization]]


\item [[totalization]]



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

The notion of nerve and realization (not with these names yet) was introduced and proven to be an [[adjunction]] in section 3 of

\begin{itemize}%
\item [[Daniel 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}
In fact, in that very article apparently what is now called [[Kan extension]] is first discussed.

Also, in that article, as an example of the general mechanism, also the [[Dold?Kan correspondence]] was found and discussed, independently of the work by Dold and Puppe shortly before, who used a much less general-nonsense approach.

In an article in 1984, Dwyer and Kan look at this `nerve-realization' context from a different viewpoint, using the term `singular functor' where the above has used `nerve'. Their motivation example is that in which $S$ is the [[orbit category]] of a group $G$, and the realisation starts with a functor on that category with values in spaces and returns a $G$-space:

\begin{itemize}%
\item [[W. G. Dwyer]] and [[D. M. Kan]], \emph{Singular functors and realization functors}, Nederl. Akad. Wetensch. Indag. Math., 87, (1984), 147 -- 153.

\end{itemize}
We should also mention the treatment in Leinster's book and the relation to the notions of [[dense subcategory]] or adequate subcategory in the sense of Isbell.

In a \href{https://golem.ph.utexas.edu/category/2010/02/sheaves_do_not_belong_to_algeb.html}{blog post} on the n-Category Caf\'e{}, Tom Leinster illustrates that ``sections of a bundle'' is a nerve operation, and its corresponding geometric realization is the construction of the \'e{}tal\'e{} space of a presheaf.

[[!redirects realization]] [[!redirects realizations]]



\end{document}