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\section*{localization at geometric homotopies}

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

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

[[!include descent and locality - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In general, given a [[category]] $\mathcal{C}$ and a [[class]] $W$ of [[morphisms]], one may ask for the \emph{[[localization]]} $\mathcal{C}[W^{-1}]$, or more specifically for the [[reflective subcategory]] of $W$-[[local objects]] (the [[reflective localization]]). Similarly for variants in [[higher category]], such as \emph{[[localization of an (∞,1)-category]]}.

If $\mathcal{C}$ has [[finite products]], then for a given [[object]] $\mathbb{A} \in \mathcal{C}$, one may take $W \coloneqq W_{\mathbb{A}}$ to be the class of morphisms of the form

\begin{displaymath}
X \times (\mathbb{A} \overset{\exists!}{\to} \ast)
  \;\;\colon\;\;
  X \times \mathbb{A}
    \overset{p_1}{\longrightarrow}
  X
  \,,
\end{displaymath}
where $X$ is any [[object]], and where $\ast$ is the [[terminal object]], and where $(-) \times (-) \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ denotes the [[Cartesian product]] [[functor]].

The [[reflective localization]] at such a class of morphisms $W_{\mathbb{A}}$ is often referred to as \emph{homotopy localization at the object $\mathbb{A}$} or similar.

The idea is that if $\mathbb{A}$ is, or is regarded as, an [[interval object]], then ``geometric'' [[left homotopies]] between morphisms $X \to Y$ are, or would be, given by morphisms out of $X \times \mathbb{A}$, and hence forcing the projections $X \times \mathbb{A} \to X$ to be equivalences means forcing all morphisms to be \emph{[[homotopy invariance|homotopy invariant]]} with respect to $\mathbb{A}$.

Typically this is considered in the case that $\mathcal{C}$ is a [[locally presentable category]] with a [[small set]] of [[generators and relations|generating objects]] $G_i$ such that it becomes sufficient to enforce the localization only on the resulting [[small set]] of [[morphisms]] of the form $G_i \times (\mathbb{A} \to \ast)$.

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

\begin{itemize}%
\item The localization of the [[(∞,1)-category of (∞,1)-sheaves]] on the [[Nisnevich site]] at the [[affine line]] $\mathbb{A}^1$ is known as \emph{[[A1-homotopy theory]]}.


\item The localization of [[smooth ∞-groupoids]] at the [[real line]] $\mathbb{R}^1$ is equivalently ([[geometrically discrete ∞-groupoids|geometrically discrete]]) [[∞-groupoids]].

After realizing [[smooth ∞-groupoids]] as the [[(∞,1)-sheaves]] over [[CartSp]], $Smooth\infty Grpd \simeq Sh_{\infty}(CartSp)$, this follows from the following Prop. \ref{HomotopyLocalizationOverSiteOfAns}.



\end{itemize}
\begin{prop}
\label{HomotopyLocalizationOverSiteOfAns}\hypertarget{HomotopyLocalizationOverSiteOfAns}{}
\textbf{([[homotopy localization]] at $\mathbb{A}^1$ over the [[site]] of $\mathbb{A}^n$s)}

Let $\mathcal{C}$ be any [[site]] (\href{geometry+of+physics+--+categories+and+toposes#Coverage}{this Def.}), and write $[\mathcal{C}^{op}, sSet_{Qu}]_{proj, loc}$ for its local projective [[model category of simplicial presheaves]] (\href{geometry+of+physics+--+categories+and+toposes#TopologicalLocalization}{this Prop.}).

Assume that $\mathcal{C}$ contains an [[object]] $\mathbb{A} \in \mathcal{C}$, such that every other object is a [[finite product]] $\mathbb{A}^n \coloneqq \underset{n \; \text{factors}}{\underbrace{\mathbb{A} \times \cdots \times \mathbb{A}}}$, for some $n \in \mathbb{N}$. (In other words, assume that $\mathcal{C}$ is also the [[syntactic category]] of [[Lawvere theory]].)

Consider the $\mathbb{A}^1$-[[homotopy localization]] (\href{geometry+of+physics+--+categories+and+toposes#HomotopyLocalizationOfCombinatorialModelCategories}{this Def.}) of the [[(∞,1)-sheaf (∞,1)-topos]] over $\mathcal{C}$ (\href{geometry+of+physics+--+categories+and+toposes#TopologicalLocalization}{this Prop.})

\begin{displaymath}
Sh_\infty(\mathcal{C})_{\mathbb{A}}
    \underoverset
      {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}}
      {\overset{L_{\mathbb{A}}}{\longleftarrow}}
      {\bot}
  Sh_\infty(\mathcal{C})
  \;\;
  \in
  Ho(CombModCat)
\end{displaymath}
hence the [[left Bousfield localization]] of [[model categories]]

\begin{displaymath}
[\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc,\mathbb{A}}
    \underoverset
      {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}}
      {\overset{id}{\longleftarrow}}
      {\phantom{{}_{Qu}}\bot_{Qu}}
  [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc}
  \;\;
  \in 
  CombModCat
\end{displaymath}
at the set of morphisms

\begin{displaymath}
S
    \;\coloneqq\;
  \big\{
    \mathbb{A}^n \times \mathbb{A}
      \overset{p_1}{\longrightarrow}
    \mathbb{A}^n
  \big\}
\end{displaymath}
(according to \href{geometry+of+physics+--+categories+and+toposes#ExistenceOfLeftBousfieldLocalization}{this Prop.}).

Then this is [[equivalence of (∞,1)-categories|equivalent]] (\href{geometry+of+physics+--+categories+and+toposes#HoCombModCat}{this Def.}) to [[∞Grpd]] (\href{geometry+of+physics+--+categories+and+toposes#InfinityGroupoid}{this Example}),

\begin{displaymath}
\infty Grpd
  \;\simeq\;
  Sh_\infty(\mathcal{C})_{\mathbb{A}}
    \underoverset
      {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}}
      {\overset{L_{\mathbb{A}}}{\longleftarrow}}
      {\bot}
  Sh_\infty(\mathcal{C})
  \;\;
  \in
  Ho(CombModCat)
\end{displaymath}
in that the ([[constant functor]] $\dashv$ [[limit]])-[[adjunction]] (\href{geometry+of+physics+--+categories+and+toposes#Limits}{this Def.})

\begin{equation}
[\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}}
    \underoverset
      {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}}
      {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}}
      {\bot}
  sSet_{Qu}    
  \;\;\;\;
  \in 
  CombModCat
\label{QuillenEquivalenceInABousfLocalization}\end{equation}
is a [[Quillen equivalence]] (\href{geometry+of+physics+--+homotopy+types#QuillenEquivalence}{this Def.}).

\end{prop}
\begin{proof}
First to see that \eqref{QuillenEquivalenceInABousfLocalization} is a [[Quillen adjunction]]: Since we have a [[simplicial Quillen adjunction]] before localization

\begin{displaymath}
[\mathcal{C}^{op}, sSet_{Qu}]_{inj}
    \underoverset
      {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}}
      {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}}
      {\bot}
  sSet_{Qu}
\end{displaymath}
(by \href{geometry+of+physics+--+categories+and+toposes#HomotopyLimitOfSimplicialSets}{this Example}) and since both [[model categories]] here are [[left proper model category|left proper]] [[simplicial model categories]] (by \href{geometry+of+physics+--+categories+and+toposes#SimplicialPresheavesIsProperCombinatorialSimplicial}{this Prop.} and \href{geometry+of+physics+--+categories+and+toposes#ExistenceOfLeftBousfieldLocalization}{this Prop.}), and since [[left Bousfield localization]] does not change the class of [[cofibrations]] (\href{geometry+of+physics+--+categories+and+toposes#BousfieldLocalizationOfModelCategories}{this Def.}) it is sufficient to show that $\underset{\longleftarrow}{\lim}$ preserves [[fibrant objects]] (by \href{geometry+of+physics+--+categories+and+toposes#RecognitionOfSimplicialQuillenAdjunction}{this Prop.}).

But by assumption $\mathcal{C}$ has a [[terminal object]] $\ast = \mathbb{A}^0$, which is hence the [[initial object]] of $\mathcal{C}^{op}$, so that the [[limit]] operation is given just by evaluation on that object:

\begin{displaymath}
\underset{\longleftarrow}{\lim}
  \mathbf{X}
  \;=\;
  \mathbf{X}(\mathbb{A}^0)
  \,.
\end{displaymath}
Hence it is sufficient to see that an injectively fibrant simplicial presheaf $\mathbf{X}$ is objectwise a [[Kan complex]]. This is indeed the case, by \href{geometry+of+physics+--+categories+and+toposes#ModelCategoriesOfSimplicialPresheaves}{this Prop.}.

To check that \eqref{QuillenEquivalenceInABousfLocalization} is actually a [[Quillen equivalence]], we check that the [[derived adjunction unit]] and [[derived adjunction counit]] are [[weak equivalences]]:

For $X \in sSet$ any simplicial set (necessarily cofibrant), the [[derived adjunction unit]] is

\begin{displaymath}
X 
    \overset{id_X}{\longrightarrow} 
  const(X)(\mathbb{A}^0)
    \overset{ const(j_X)(\mathbb{A}^0) }{\longrightarrow}
  const(P X)(\mathbb{A}^0)
\end{displaymath}
where $X \overset{j_X}{\longrightarrow} P X$ is a [[fibrant replacement]] (\href{geometry+of+physics+--+categories+and+toposes#FibrantCofibrantReplacementFunctorToHomotopyCategory}{this Def.}). But $const(-)(\mathbb{A}^0)$ is clearly the [[identity functor]] and the plain adjunction unit is the [[identity morphism]], so that this composite is just $j_X$ itself, which is indeed a weak equivalence.

For the other case, let $\mathbf{X} \in [\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}^1}$ be fibrant. This means (by \href{geometry+of+physics+--+categories+and+toposes#ExistenceOfLeftBousfieldLocalization}{this Prop.}) that $\mathbf{X}$ is fibrant in the injective [[model structure on simplicial presheaves]] as well as in the local model structure, and is a derived-$\mathbb{A}^1$-[[local object]] (\href{geometry+of+physics+--+categories+and+toposes#DerivedLocalObjects}{this Def.}), in that the [[derived hom-functor]] out of any $\mathbb{A}^n \times \mathbb{A}^1 \overset{p_1}{\longrightarrow} \mathbb{A}^n$ into $\mathbf{X}$ is a [[weak homotopy equivalence]]:

\begin{displaymath}
\mathbb{R}Hom( p_1 )
  \;\colon\;
  \mathbb{R}Hom( \mathbb{A}^n , \mathbf{X})
  \overset{\in W}{\longrightarrow}
  \mathbb{R}Hom( \mathbb{A}^n \times \mathbb{A}^1 , \mathbf{X})
\end{displaymath}
But since $\mathbf{X}$ is fibrant, this derived hom is equivalent to the ordinary [[hom-functor]] (\href{geometry+of+physics+--+categories+and+toposes#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory}{this Lemma}), and hence with the [[Yoneda lemma]] (\href{geometry+of+physics+--+categories+and+toposes#YonedaLemma}{this Prop.}) we have that

\begin{displaymath}
\mathbf{X}(p_1)
  \;\colon\;
  \mathbf{X}(\mathbb{A}^n)
  \overset{\in W}{\longrightarrow}
  \mathbf{X}(\mathbb{A}^{n+1})
\end{displaymath}
is a weak equivalence, for all $n \in \mathbb{N}$. By [[induction]] on $n$ this means that in fact

\begin{displaymath}
\mathbf{X}(\mathbb{A}^0)
   \overset{\in W}{\longrightarrow} 
  \mathbf{X}(\mathbb{A}^n)
\end{displaymath}
is a weak equivalence for all $n \in \mathbb{N}$. But these are just the components of the [[adjunction counit]]

\begin{displaymath}
const (\mathbf{X}(\mathbb{A}^0))
   \underoverset{\in W}{\epsilon}{\longrightarrow}
  \mathbf{X}
\end{displaymath}
which is hence also a weak equivalence. Hence for the [[derived adjunction counit]]

\begin{displaymath}
const (Q \mathbf{X})(\mathbb{A}^0)
    \overset{const(p_{\mathbf{X}}(\mathbb{A}^0))}{\longrightarrow}
  const (\mathbf{X}(\mathbb{A}^0))
   \underoverset{\in W}{\epsilon}{\longrightarrow}
  \mathbf{X}
\end{displaymath}
to be a weak equivalence, it is now sufficient to see that the value of a [[cofibrant replacement]] $p_{\mathbf{X}}$ on $\mathbb{A}^0$ is a weak equivalence. But by definition of the weak equivalences of simplicial presheaves these are objectwise weak equivalences.

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

\begin{itemize}%
\item [[Fabien Morel]], [[Vladimir Voevodsky]], Def. 3.1 in \emph{$\mathbb{A}^1$-homotopy theory of schemes}, Publications Mathématiques de l'IHÉS, Volume 90 (1999), p. 45-143 (\href{http://www.numdam.org/item/?id=PMIHES_1999__90__45_0}{Numdam:PMIHES\_1999\_\_90\_\_45\_0} \href{http://www.math.uiuc.edu/K-theory/0305/}{K-Theory:0305} )

\end{itemize}
For more references see also at \emph{[[motivic homotopy theory]]}.

[[!redirects homotopy localization]] [[!redirects homotopy localizations]]

[[!redirects homotopy localization at an object]] [[!redirects homotopy localizations at an object]]

[[!redirects localization at an object]] [[!redirects localizations at an object]]

[[!redirects localization at objects]] [[!redirects localizations at objects]]



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