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\begin{document}

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\section*{basic localizer}

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

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

[[!include category theory - contents]]

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

[[!include homotopy - contents]]

\hypertarget{basic_localizers}{}\section*{{Basic localizers}}\label{basic_localizers}

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{asphericity_and_local_equivalences}{Asphericity and local equivalences}\dotfill \pageref*{asphericity_and_local_equivalences} \linebreak
\noindent\hyperlink{selfduality}{Self-duality}\dotfill \pageref*{selfduality} \linebreak
\noindent\hyperlink{cisinskis_theorem}{Cisinski's theorem}\dotfill \pageref*{cisinskis_theorem} \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}

By a \emph{basic localizer} one means a [[localizer]] on the category [[Cat]] of [[categories]], hence a choice of a [[class]] of [[functors]] to be called the \emph{[[weak equivalences]]}, subject to some conditions.

These conditions ensure in particular that a basic localizer always contains the weak equivalences of the [[Thomason model structure]] on [[Cat]] (see \hyperlink{Maltsiniotis11}{Maltsiniotis 11, 1.2.1}), the [[localization]] at which is equivalent to the standard [[homotopy category]]. On the other hand, the standard [[equivalences of categories]], which are the weak equivalences in the [[canonical model structure]] on [[Cat]], do not form a basic localizer.

Hence basic localizers are a tool for [[homotopy theory]] modeled on [[category theory]]. In fact, their introduction by [[Grothendieck]] was motivated from the study of [[test categories]] (see remark \ref{CitationOnTestCategories} below).

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

The definition is due to [[Grothendieck]]:

\begin{defn}
\label{BasicLocalizer}\hypertarget{BasicLocalizer}{}
A \textbf{basic localizer} is a [[class]] $W$ of [[morphisms]] in [[Cat]] such that

\begin{enumerate}%
\item $W$ contains all [[identities]], satisfies the [[2-out-of-3 property]] and is closed under [[retracts]] (in the literature this is sometimes called being \emph{weakly saturated}),


\item If $A$ has a [[terminal object]], then the functor $A\to 1$ is in $W$, and


\item Given a commutative triangle in $Cat$:

\begin{displaymath}
\itexarray{ A & & \overset{u}{\to} & & B\\ & _v \searrow & & \swarrow_w \\ & & C }
\end{displaymath}
if each induced functor $v/c \to w/c$ between [[comma categories]] is in $W$, then $u$ is also in $W$.



\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The term in French is \textbf{localisateur fondamental}, which is sometimes translated as \textbf{fundamental localizer}.

\end{remark}
\begin{remark}
\label{CitationOnTestCategories}\hypertarget{CitationOnTestCategories}{}
In [[Pursuing Stacks]] [[Grothendieck]] wrote about def. \ref{BasicLocalizer} the following:

\begin{quote}%
These conditions are enough, I quickly checked this night, in order to validify all results developed so far on [[test categories]], [[weak test categories]], [[strict test categories]], weak test functors and test functors (with values in $(Cat)$) (of notably the review in par. 44, page 79--88), provided in the case of test functors we restrict to the case of loc. cit. when each of the categories $i(a)$ has a final object. All this I believe is justification enough for the definition above.


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

\begin{itemize}%
\item The class of \emph{all} functors between small categories is, of course, the maximal basic localizer.


\item The class of functors inducing an [[isomorphism]] on [[connected components]] is a basic localizer.


\item The class of functors whose [[nerve]] is a [[weak homotopy equivalence]] is a basic localizer. (These are the [[weak equivalences]] in the [[Thomason model structure]].)


\item For any [[derivator]] $D$, the class of $D$-equivalences is a basic localizer. This includes all the previous examples.


\item The class of [[equivalences of categories]] is \emph{not} a basic localizer (it fails the second condition). (These are the weak equivalences of the [[canonical model structure]].)



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

\hypertarget{asphericity_and_local_equivalences}{}\subsubsection*{{Asphericity and local equivalences}}\label{asphericity_and_local_equivalences}

If $W$ is a basic localizer, we define the following related classes. We sometimes refer to functors in $W$ as \emph{weak equivalences}.

\begin{itemize}%
\item A category $A$ is \textbf{($W$-)aspherical} if $A\to 1$ is in $W$. Thus the second axiom says exactly that any category with a terminal object is aspherical.
\item A functor $u\colon A\to B$ is \textbf{($W$-)aspherical} if for all $b\in B$, the comma category $u/b$ is aspherical.
\item When the hypotheses of the third axiom are satisfied, we say that $u$ is a \textbf{local weak equivalence over $C$}. Thus the third axiom says exactly that every local weak equivalence is a weak equivalence.

\end{itemize}
\begin{example}
\label{}\hypertarget{}{}
If $W = \pi_0$-equivalences, then a category is aspherical iff it is connected, and a functor is aspherical iff it is [[initial functor|initial]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
If $W =$ nerve equivalences, then a category is aspherical iff its nerve is contractible, and a functor is aspherical iff it is [[homotopy initial functor|homotopy initial]].

\end{example}
We observe the following.

\begin{itemize}%
\item A category $A$ is aspherical iff the functor $A\to 1$ is aspherical, since the only comma category involved in the latter assertion is $A$ itself.


\item An aspherical functor is a weak equivalence. For if $u\colon A\to B$ is aspherical, then consider the triangle

\begin{displaymath}
\itexarray{ A & & \overset{u}{\to} & & B\\ & _u \searrow & & \swarrow \\ & & B }
\end{displaymath}
The third axiom tells us to consider, for a given $b\in B$, the functor $u/b \to B/b$. But $u/b$ is aspherical by assumption, while $B/b$ is aspherical by the second axiom since it has a terminal object. Thus, by 2-out-of-3, the functor $u/b \to B/b$ is in $W$, and thus by the third axiom $u$ is in $W$.


\item If $u$ has a right adjoint, then it is aspherical. For in this case, each category $u/b$ has a terminal object, and thus is aspherical.


\item If $I$ denotes the [[interval category]], then for any category $A$ the projection $A\times I\to A$ has a right adjoint, hence is aspherical and thus a weak equivalence. By 2-out-of-3, the two injections $A \rightrightarrows A\times I$ are also weak equivalences, so $A\times I$ is a [[cylinder object]] for $W$. It follows that if we have a natural transformation $f\to g$, then $f$ is in $W$ if and only if $g$ is. Moreover, if $f$ is a ``homotopy equivalence'' in the sense that it has an ``inverse'' $g$ such that $f g$ and $g f$ are connected to identities by arbitrary natural zigzags, then $f$ is a weak equivalence.


\item In particular, any left or right adjoint is a weak equivalence.



\end{itemize}
\hypertarget{selfduality}{}\subsubsection*{{Self-duality}}\label{selfduality}

It is a non-obvious fact that the notion of basic localizer is self-dual.

\begin{theorem}
\label{}\hypertarget{}{}
A functor $u : A \to B$ is in a basic localizer $W$ if and only if $u^{op} : A^{op} \to B^{op}$ is in $W$.

\end{theorem}
\begin{proof}
See Proposition 1.2.6 in (\hyperlink{Cisinski04}{Cisinski 04}).

\end{proof}
\hypertarget{cisinskis_theorem}{}\subsubsection*{{Cisinski's theorem}}\label{cisinskis_theorem}

Since the definition consists merely of closure conditions, the intersection of any family of basic localizers is again a basic localizer. It follows that there is a unique \emph{smallest} basic localizer. The following was conjectured by [[Grothendieck]] and proven by [[Denis-Charles Cisinski]].

\begin{theorem}
\label{}\hypertarget{}{}
The class of functors whose [[nerve]] is a [[weak homotopy equivalence]] is the smallest basic localizer.

\end{theorem}
\begin{proof}
See Th\'e{}or\`e{}me 2.2.11 in (\hyperlink{Cisinski04}{Cisinski 04}).

\end{proof}
Note that this is a larger class than the class of ``[[homotopy equivalences]]'' considered above. For instance, the category generated by the graph

\begin{displaymath}
\bullet \leftarrow \bullet \rightarrow \bullet \leftarrow \bullet \rightarrow \bullet \leftarrow \bullet \rightarrow \cdots
\end{displaymath}
has a contractible nerve, but its identity functor is not connected to a constant one by any natural zigzag.

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

\begin{itemize}%
\item [[canonical model structure]], [[Thomason model structure]] on [[Cat]]


\item [[Cisinski model structure]]



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

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{[[joyalscatlab:Les préfaisceaux comme type d'homotopie]]}, Ast\'e{}risque, Volume 308, Soc. Math. France (2006), 392 pages (\href{http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{Le localisateur fondamental minimal}

\end{itemize}
See also at \emph{[[Cisinski model structure]]}.

\begin{itemize}%
\item [[Georges Maltsiniotis]], \emph{Homotopical exact squares and derivators} (\href{http://arxiv.org/abs/1101.4144}{arXiv:1101.4144})

\end{itemize}
[[!redirects basic localizor]] [[!redirects basic localizors]] [[!redirects basic localizers]] [[!redirects basic localiser]] [[!redirects basic localisers]] [[!redirects fundamental localizer]] [[!redirects fundamental localizers]] [[!redirects fundamental localiser]] [[!redirects fundamental localisers]] [[!redirects localisateur fondamental]]



\end{document}