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\begin{document}

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\section*{simplicial localization}

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

\hypertarget{locality_and_descent}{}\paragraph*{{Locality and descent}}\label{locality_and_descent}

[[!include descent and locality - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\begin{quote}%
See also [[derived hom space]]


\end{quote}
\hypertarget{simplicial_localisation}{}\section*{{Simplicial localisation}}\label{simplicial_localisation}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{construction}{Construction}\dotfill \pageref*{construction} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{standard_simplicial_localization}{``Standard'' simplicial localization}\dotfill \pageref*{standard_simplicial_localization} \linebreak
\noindent\hyperlink{hammock_localization}{Hammock localization}\dotfill \pageref*{hammock_localization} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{basic_properties}{Basic properties}\dotfill \pageref*{basic_properties} \linebreak
\noindent\hyperlink{simplical_localization_gives_all_categories}{Simplical localization gives all $(\infty,1)$-categories}\dotfill \pageref*{simplical_localization_gives_all_categories} \linebreak
\noindent\hyperlink{equivalences_between_simplicial_localizations}{Equivalences between simplicial localizations}\dotfill \pageref*{equivalences_between_simplicial_localizations} \linebreak
\noindent\hyperlink{simplicial_localization_of_model_categories}{Simplicial localization of model categories}\dotfill \pageref*{simplicial_localization_of_model_categories} \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}

A [[category with weak equivalences]] or [[homotopical category]] is a [[category]] $C$ equipped with the information that some of its [[morphisms]], specifically, a subcategory $W \supset Core(C)$, are to be regarded as ``weakly invertible''. One way to make this notion precise is through the concept of \textbf{simplicial localization}:

The \emph{simplicial localization} $L C$ of a category $C$ is an [[(∞,1)-category]] realized concretely as a [[simplicially enriched category]] which is such that the original category injects into it, $C \hookrightarrow L C$, such that every morphism in $C$ that is labeled as a weak equivalence becomes an actual equivalence in the sense of morphisms in [[(∞,1)-categories]] in $L C$. And $L C$ is in some sense universal with this property.

Passing to the [[homotopy category of an (∞,1)-category]] of $L C$ then reproduces the [[homotopy category]] that can also directly be obtained from $C$:

\begin{displaymath}
Ho_C(a,b) \simeq \Pi_0 (L C(a,b))
\end{displaymath}
(where $\Pi_0$ gives the [[simplicial homotopy group|0th simplicial homotopy groupoid]]).

If the homotopical structure on $C$ extends to that of a (combinatorial) [[model category]], then there is another procedure to obtain a simplicially enriched category from $C$, the [[presentable (infinity,1)-category|(∞,1)-category presented by a combinatorial model category]]. This $(\infty,1)$-category is equivalent to the one obtained by simplicial localization but typically more explicit and more tractable.

See also [[localization of a simplicial model category]].

\hypertarget{construction}{}\subsection*{{Construction}}\label{construction}

See [[simplicial localization of a homotopical category]].

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

\hypertarget{standard_simplicial_localization}{}\subsubsection*{{``Standard'' simplicial localization}}\label{standard_simplicial_localization}

Let $U : \mathbf{Cat} \to \mathbf{Grph}$ be the [[forgetful functor]] that sends a (small) category to its underlying \emph{[[reflexive graph|reflexive]]} [[graph]], and let $F : \mathbf{Grph} \to \mathbf{Cat}$ be its [[left adjoint]]. We then get a [[comonad]] $\mathbb{G} = (G, \epsilon, \delta)$ on $\mathbf{Cat}$, and as usual this defines a functor $G_\bullet : \mathbf{Cat} \to [\mathbf{\Delta}^{op}, \mathbf{Cat}]$ from [[Cat]] to [[simplicial objects in Cat]] equipped with a canonical augmentation, where $G_n C = G^{n+1} C$.

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{standard resolution} of a small category $C$ is defined to be the simplicial category $G_\bullet C$.

\end{defn}
Note that this is also a [[simplicial category]] in the strong sense, i.e. $ob G_\bullet C$ is discrete! Thus we may also regard $G_\bullet C$ as an [[sSet]]-[[simplicially enriched category|category]]. This is a [[resolution]] in the sense that the augmentation $\epsilon : G_\bullet C \to C$ is a [[Dwyer-Kan equivalence]]. (In fact, for objects $X$ and $Y$ in $C$, the morphism $G_\bullet C (X, Y) \to C (X, Y)$ admits an extra degeneracy and hence a contracting homotopy.)

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{standard simplicial localization} of a [[relative category]] $(C, W)$ is the simplicial category $L_\bullet (C, W)$ where $L_n (C, W) = G_n C [{G_n W}^{-1}]$.

\end{defn}
This appears as (\hyperlink{DwyerKanLocalizations}{DwyerKanLocalizations, def. 4.1}). Again, $L_\bullet C$ is a simplicial category in the strong sense, because $G_\bullet C$ is.

\hypertarget{hammock_localization}{}\subsubsection*{{Hammock localization}}\label{hammock_localization}

\begin{defn}
\label{}\hypertarget{}{}
Let $(C,W)$ be a [[category with weak equivalences]]. For $X,Y \in C$ any two objects, write

\begin{displaymath}
L^H C(X,Y) \in sSet
\end{displaymath}
for the simplicial set defined as follows. For each natural number $n$ there is a [[category]] defined as follows:

\begin{itemize}%
\item its [[objects]] are length-$n$ zig-zags of morphisms in $C$

\begin{displaymath}
X \stackrel{\simeq}{\leftarrow}
  K_1 \to K_2 \stackrel{\simeq}{\leftarrow} K_3 \to
  \cdots \to Y
  \,,
\end{displaymath}
where the left-pointing morphisms are to be in $W$;


\item its [[morphisms]] are ``natural transformations'' between such objects, fixing the endpoints:

\begin{displaymath}
\itexarray{
     && K_1 &\to& K_2 &\stackrel{\simeq}{\leftarrow}&
     \cdots 
     \\
     & {}^{\mathllap{\simeq}}\swarrow
     &&&&& && \searrow^{}
     \\
    X 
    && \downarrow^{\simeq}&& \downarrow^{\simeq}& \cdots&&  && Y
    \\
     & {}_{\mathllap{\simeq}}\nwarrow
     &&&&& && \nearrow^{}
    \\
     && L_1 &\to& L_2 &\stackrel{\simeq}{\leftarrow}&
     \cdots 
  }
  \;
\end{displaymath}


\end{itemize}
$L^H C(X,Y)$ is obtained by

\begin{itemize}%
\item taking the [[coproduct]] of the [[nerves]] of these categories over all $n$, and


\item quotienting by the [[equivalence relation]] generated by inserting or removing identity morphisms and composing composable morphisms.



\end{itemize}
For $X,Y,Z$ three objects, there is an evident compositing morphism

\begin{displaymath}
L^H C(X,Y) \times L^H C(Y,Z) \to L^H C(X,Z)
\end{displaymath}
given by horizontally concatenating hammock diagrams as above.

The [[simplicially enriched category]] $L^H C$ obtained this way is the \textbf{hammock localization} of $(C,W)$.

\end{defn}
This appears as (\hyperlink{DwyerKanCalculating}{DwyerKanCalculating, def. 2.1}).

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

\hypertarget{basic_properties}{}\subsubsection*{{Basic properties}}\label{basic_properties}

\begin{prop}
\label{}\hypertarget{}{}
For $(C,W)$ a [[category with weak equivalences]], write $L^H C \in sSet Cat$ for its hammock localization and $C[W^{-1}] \in Cat$ for its ordinary [[localization]]. Write $Ho(L^H C) \in Cat$ for the category with the same objects as $C$ and morphisms between $X$ and $Y$ being $\pi_0 L^H C(X,Y)$.

There is an [[equivalence of categories]]

\begin{displaymath}
Ho L^H C
   \simeq
  C[W^{-1}]
  \,.
\end{displaymath}
\end{prop}
This appears as (\hyperlink{DwyerKanCalculating}{DwyerKanCalculating, prop. 3.1}).

\begin{prop}
\label{PrePostCompositionWithWeakEquivalences}\hypertarget{PrePostCompositionWithWeakEquivalences}{}
Let $(C,W)$ be a [[category with weak equivalences]], and let

\begin{displaymath}
(f : X \to Y) \in W \subset Mor(C)
\end{displaymath}
be a weak equivalence. Then for all objects $U \in C$ we have that the to concatenation operations on hammocks induce [[weak homotopy equivalence]]s

\begin{displaymath}
f_* : L^H C(U,X) \stackrel{\simeq}{\to}
     L^H C(U,Y)
\end{displaymath}
and

\begin{displaymath}
f^* : L^H C(Y,U) \stackrel{\simeq}{\to} L^H C(X, U)
  \,.
\end{displaymath}
\end{prop}
This appears as (\hyperlink{DwyerKanCalculating}{DwyerKanCalculating, prop. 3.3}).

\hypertarget{simplical_localization_gives_all_categories}{}\subsubsection*{{Simplical localization gives all $(\infty,1)$-categories}}\label{simplical_localization_gives_all_categories}

\begin{prop}
\label{}\hypertarget{}{}
Up to Dwyer-Kan equivalence --the [[weak equivalences]] in the [[model structure on sSet-categories]] -- every [[simplicial category]] is the simplicial localization of a [[category with weak equivalences]].

This is (\hyperlink{DwyerKanEquivalences}{DwyerKan 87, 2.5}).

\end{prop}
If the [[category with weak equivalences]] in question happens to carry even the structure of a [[model category]] there exist more refined tools for computing the [[SSet]]-[[hom object]] of the simplicial localization. These are described at [[(∞,1)-categorical hom-space]].

\hypertarget{equivalences_between_simplicial_localizations}{}\subsubsection*{{Equivalences between simplicial localizations}}\label{equivalences_between_simplicial_localizations}

\begin{prop}
\label{SimplicialLocalizationOfNaturalTransformation}\hypertarget{SimplicialLocalizationOfNaturalTransformation}{}
Let $(C,W)$ and $(C', W')$ be [[categories with weak equivalences]]. Write $L^H C, L^H C' \in sSet Cat$ for the corresponding [[hammock localizations]].

Then for $F_1, F_2 :  C \to C'$ two [[homotopical functors]] (functors respecting the weak equivalences, i.e. $F_i(W) \subset W'$) with

\begin{displaymath}
\eta : F_1 \Rightarrow F_2
\end{displaymath}
a [[natural transformation]] with components in the $W'$, we have that for all objects $X,Y \in C$, there is induced a [[natural transformation|natural]] [[homotopy]] between morphisms of [[simplicial sets]]

\begin{displaymath}
\itexarray{
    && L^H C'(F_1(X), F_1(Y) )     
    \\
    & 
    {}^{\mathllap{L^H F_1}}\nearrow 
    && 
    \searrow^{\mathrlap{\eta(Y)_*}}
    \\
    L^H C(X,Y) 
     && \Downarrow && 
    L^H C'(F_1(X), F_2(Y))
    \\
    & 
     {}_{\mathllap{L^H F_2}}\searrow 
    &&  
     \nearrow_{\mathrlap{\eta(X)^*}}
    \\
    && L^H C' (F_2(X), F_2(Y))
  }
  \,.
\end{displaymath}
\end{prop}
This is (\hyperlink{DwyerKanComputations}{DwyerKanComputations, prop. 3.5}).

\begin{cor}
\label{}\hypertarget{}{}
Let $i : (C_1, W_1) \hookrightarrow (C_2, W_2)$ be a [[full subcategory]] such that

\begin{enumerate}%
\item $i$ is homotopy-essentially surjective: for every object $c_2 \in C_2$ there is an object $c_1 \in C_1$ and a weak equivalence $c_2 \stackrel{\simeq}{\to} i(c_1)$;


\item there is a [[functor]] $Q : (C_2,W_2) \to (C_1, W_1)$ and a [[natural transformation]]

\begin{displaymath}
i \circ Q  \Rightarrow Id_{C_2}
  \,.
\end{displaymath}


\end{enumerate}
Then we have an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
L^H C_1 \simeq L^H C_2
  \,.
\end{displaymath}
\end{cor}
\begin{proof}
We have to check that $i$ is an [[essentially surjective (∞,1)-functor]] and a [[full and faithful (∞,1)-functor]].

The first condition is immediate from the first assumption. The second follows with prop. \ref{SimplicialLocalizationOfNaturalTransformation} (using prop. \ref{PrePostCompositionWithWeakEquivalences}) from the second assumption.

\end{proof}
\hypertarget{simplicial_localization_of_model_categories}{}\subsubsection*{{Simplicial localization of model categories}}\label{simplicial_localization_of_model_categories}

\begin{prop}
\label{}\hypertarget{}{}
Let $C$ be a [[simplicial model category]]. Write $C^\circ$ for the full $Set$-subcategory on the fibrant and cofibrant objects.

Then $C^\circ$ and $L^H C$ are connected by an [[equivalence of (∞,1)-categories]].

\end{prop}
This is one of the central statements in (\hyperlink{DwyerKanFunctionComplexes}{DwyerKanFunctionComplexes}). The weak homotopy equivalence between $C^\circ(X,Y)$ and $L^H C(X,Y)$ is in corollary 4.7. The equivalence of $\infty$-categories stated above follows with this and the diagram of morphisms of simplicial categories in prop. 4.8.

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

\begin{itemize}%
\item [[localizer]]


\item [[localizing subcategory]]



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

The original articles are

\begin{itemize}%
\item [[William Dwyer]], [[Daniel Kan]], \emph{Simplicial localizations of categories} , J. Pure Appl. Algebra 17 (1980), 267--284. (\href{http://www.nd.edu/~wgd/Dvi/SimplicialLocalizations.pdf}{pdf})


\item [[William Dwyer]], [[Daniel Kan]], \emph{Calculating simplicial localizations} , J. Pure Appl. Algebra 18 (1980), 17--35. (\href{http://www.nd.edu/~wgd/Dvi/CalculatingSimplicialLocalizations.pdf}{pdf})


\item [[William Dwyer]], [[Daniel Kan]], \emph{Function complexes in homotopical algebra} , Topology 19 (1980), 427--440.


\item [[William Dwyer]], [[Daniel Kan]], \emph{Equivalences between homotopy theories of diagrams} , Algebraic topologx and algebraic K-theory, (Princeton, N.J. 1983) , Ann. of Math. Stud. 113, Princeton University Press, Princeton, N.J. 1987 .


\item [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], \emph{[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]} , volume 113 of Mathematical Surveys and Monographs



\end{itemize}
A survey of the general topic involved here can be found in the following paper:

\begin{itemize}%
\item [[Tim Porter]], \emph{$S$-Categories, $S$-groupoids, Segal categories and quasicategories} (\href{http://arxiv.org/abs/math/0401274}{arXiv})

\end{itemize}
Hammock localization is described in Section 4.1 there.

A useful quick collection of facts can be found at the beginning of Section 2 in the following paper:

\begin{itemize}%
\item [[Clark Barwick]], \emph{On (enriched) Bousfield localization of model categories} (\href{http://arxiv.org/abs/0708.2067}{arXiv})

\end{itemize}
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[[!redirects simplicial localisation]] [[!redirects simplicial localisations]]

[[!redirects Dwyer-Kan localization]] [[!redirects Dwyer-Kan localisation]] [[!redirects Dwyer?Kan localization]] [[!redirects Dwyer?Kan localisation]] [[!redirects Dwyer--Kan localization]] [[!redirects Dwyer--Kan localisation]] [[!redirects Hammock-Localization]] [[!redirects Hammock localization]] [[!redirects Hammock localisation]]

[[!redirects hammock localization]] [[!redirects hammock localizations]]

[[!redirects hammock localisation]] [[!redirects hammock localisations]]

[[!redirects (infinity,1)-categorical localization]] [[!redirects (infinity,1)-categorical localizations]] [[!redirects (∞,1)-categorical localization]] [[!redirects (∞,1)-categorical localizations]]



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