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\section*{test category}

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

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

[[!include homotopy - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{homotopy_category}{Homotopy category}\dotfill \pageref*{homotopy_category} \linebreak
\noindent\hyperlink{model_category_structure}{Model category structure}\dotfill \pageref*{model_category_structure} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

In the 1980-s, [[Grothendieck]] in \emph{Pursuing Stacks} introduced test categories to make the variants of the [[homotopy theory]] based on the usage of combinatorial models with some kind of cell structure (e.g., [[simplicial set]]s, [[cubical set]]s and cellular sets) independent of a particular combinatorial model.

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

Given any small category $\mathcal{C}$, one considers $\mathcal{C}$-sets, hence [[presheaves]] on $\mathcal{C}$, hence [[contravariant functors]] from $\mathcal{C}$ to $Set$.

Given an [[object]] $c\in C$, one considers the [[representable functor]] $Hom_{\mathcal{C}}(-,c)=:\Delta^c$. If $X:\mathcal{C}^{op} \to Set$ is a $\mathcal{C}$-set, the elements of $X(c)$ are called the $c$-cells. By the [[Yoneda lemma]], they correspond to the [[natural transformations]] $\Delta^c\to X$.

Let the \textbf{cell category} of $X$, denoted $i_{\mathcal{C}} X$, be the full subcategory of the [[overcategory]] $\mathcal{C}Set/X$ whose objects are the transformations of the form $\Delta^c\to X$. The correspondence $X\mapsto i_{\mathcal{C}}X$ extends to a functor $i_{\mathcal{C}}:\mathcal{C}Set\to Cat$ which has a right adjoint $i_{\mathcal{C}}^*:Cat\to\mathcal{C}Set$ whose object part is given by the formula

\begin{displaymath}
i_{\mathcal{C}}^*(D)(c):= Hom_{Cat}(\mathcal{C}/c,D).
\end{displaymath}
Denote the counit of the adjunction $\epsilon : i_{\mathcal{C}}i_{\mathcal{C}}^*\to Id_{Cat}$.

Two $\mathcal{C}$-sets $X$ and $Y$ are \textbf{weakly equivalent} if there is a map $f:X\to Y$ inducing an equivalence $f_* : i_{\mathcal{C}} X\to i_{\mathcal{C}} Y$ of their cell categories, i.e., the induced map of [[nerve]]s (``[[classifying spaces]]'') $B(i_{\mathcal{C}} X)\to B(i_{\mathcal{C}} Y)$ is a weak equivalence of simplicial sets. The functor $i_{\mathcal{C}}:\mathcal{C}Set\to Cat$ induces a functor $i_{\mathcal{C}*}:Ho(\mathcal{C}Set)\to Ho(Cat)$ of the homotopy categories.

A \textbf{weak test category} is a small category $\mathcal{C}$ such that, for any category $D$ in $Cat$, the component of the counit $\epsilon_D : i_{\mathcal{C}}i_{\mathcal{C}}^* D \to D$ is an equivalence of categories.

A \textbf{test category} is any small category $\mathcal{A}$ such that

\begin{itemize}%
\item ($\mathcal{A}$ is aspherical) its ([[geometric realization]] of the) [[nerve]] (``[[classifying space]]'') $\vert \mathcal{A}\vert$ is [[contractible space|contractible]]


\item ($\mathcal{A}$ is a ``local test category'') for every [[object]] $a$ in $\mathcal{A}$ require the [[overcategory]] $\mathcal{A}/a$ to be a weak test category. Thus for any category $D$ in $Cat$, $\epsilon_D : i_{\mathcal{A}/a}i_{\mathcal{A}/a}^* D \to D$ is an [[equivalence of categories]].



\end{itemize}
A \textbf{strict test category} is a test category $\mathcal{A}$ such that

\begin{itemize}%
\item $i_{\mathcal{C}} : \mathcal{C}Set \to Cat$ preserves [[finite products]] up to weak equivalence,

\end{itemize}
or equivalently, such that

\begin{itemize}%
\item the induced functor $i_{\mathcal{C}*}:Ho(\mathcal{C}Set)\to Ho(Cat)$ preserves finite products.

\end{itemize}
Then one proceeds with $\mathcal{A}$-sets.

If $\mathcal{A}$ is a test category and $\mathcal{C}$ any small category whose classifying space is contractible (which may or may not be a test category itself), then their cartesian [[product]] $\mathcal{A}\times\mathcal{C}$ is a test category.

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

\hypertarget{homotopy_category}{}\subsubsection*{{Homotopy category}}\label{homotopy_category}

The [[homotopy category]] of a [[category of presheaves]] over a test category, as a [[category with weak equivalences]] is equivalent to the standard homotopy category of [[homotopy theory]]: that of the category of [[simplicial sets]]/[[topological spaces]] with weak equivalences being [[weak homotopy equivalences]].

In other words, presheaves over a test category are models for [[homotopy types]] of [[∞-groupoids]].

\hypertarget{model_category_structure}{}\subsubsection*{{Model category structure}}\label{model_category_structure}

The [[presheaf category]] over a test category with the above weak equivalences admits a [[model category]] structure: the [[model structure on presheaves over a test category]]. This is due to (\hyperlink{Cisinski}{Cisinski}) with further developments due to (\hyperlink{Jardine}{Jardine}).

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

Apart from the archeytpical example of the [[simplex category]] we have the following

\begin{itemize}%
\item The [[cube category]] is a test category (\hyperlink{Grothendieck}{Grothendieck}, \hyperlink{Cisinski}{Cisinski}), however not a strict one (\hyperlink{Kan}{Kan}). (The corresponding [[model category]] is discussed at [[model structure on cubical sets]].=)

The category of [[cubes]] equipped with [[connection on a cubical set]] is even a strict test category (\hyperlink{Maltsiniotis}{Maltsiniotis, 2008}).


\item The [[groupoid]]-analog $\tilde \Theta$ of the [[Theta category]] is a test category (\hyperlink{Ara}{Ara}).


\item The [[tree category]] $\Omega$ is a test category. This was proven in an unpublished note of Cisinski.



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

\begin{itemize}%
\item [[direct category]]


\item [[basic localizer]]



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

The notion of test category was introduced in

\begin{itemize}%
\item [[Alexandre Grothendieck]], \emph{[[Pursuing Stacks]]}, \href{http://people.math.jussieu.fr/~maltsin/groth/ps/Pursuing_Stacks.djvu}{djvu}

\end{itemize}
Various conjectures made there are proven in

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{Les pr\'e{}faisceaux comme mod\`e{}les des types d'homotopie}, Asterisque \textbf{308}, \href{http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf}{pdf}.

\end{itemize}
which moreover develops the main toolset and establishes the [[model structure on presheaves over a test category]].

General surveys include

\begin{itemize}%
\item [[Georges Maltsiniotis]], \emph{La th\'e{}orie de l'homotopie de Grothendieck}, Ast\'e{}risque, 301, pp. 1-140, (2005) (see \href{http://people.math.jussieu.fr/~maltsin/textes.html}{html})


\item [[Rick Jardine]], \emph{Categorical homotopy theory}, Homot. Homol. Appl. \textbf{8} (1), 2006, pp.71--144, (\href{http://www.intlpress.com/HHA/v8/n1/a3/}{HHA}, \href{http://www.intlpress.com/HHA/v8/n1/a3/v8n1a3.pdf}{pdf})



\end{itemize}
That the [[cube category]] is a test category is asserted without proof in (\hyperlink{Grothendieck}{Grothendieck}). A proof is spelled out in (\hyperlink{Cisinski}{Cisinski})

That it is not a strict test category is implicitly already in

\begin{itemize}%
\item [[Dan Kan]], \emph{Abstract homotopy. I} , Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 1092--1096. (\href{http://www.ncbi.nlm.nih.gov/pmc/articles/PMC528202/pdf/pnas00727-0082.pdf}{pdf})

\end{itemize}
and led to the preference for [[simplicial sets]] over [[cubical sets]].

That the category of [[cubes]] \emph{equipped with [[connection on a cubical set]]} forms a \emph{strict} test category is shown in

\begin{itemize}%
\item [[Georges Maltsiniotis]], \emph{La cat\'e{}gorie cubique avec connexions est une cat\'e{}gorie test stricte} . (French. English summary) Homology, Homotopy Appl. 11 (2009), no. 2, 309--326. (\href{http://www.intlpress.com/HHA/v11/n2/a15/}{web})

\end{itemize}
The test category nature of the groupoidal [[Theta category]] is discussed in

\begin{itemize}%
\item [[Dimitri Ara]], \emph{The groupoidal analogue $\tilde \Theta$ to Joyal's category $\Theta$ is a test category} (\href{http://arxiv.org/abs/1012.4319}{arXiv:1012.4319})

\end{itemize}
A short introduction can be found in

\begin{itemize}%
\item Chris Kapulkin, \emph{Introduction to Test Categories} \href{http://www.math.uwo.ca/faculty/kapulkin/notes/test_categories.pdf}{PDF}.

\end{itemize}
[[!redirects test categories]] [[!redirects strict test category]] [[!redirects strict test categories]] [[!redirects weak test category]] [[!redirects weak test categories]] [[!redirects Grothendieck homotopy theory]]



\end{document}