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\begin{document}

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\section*{model structure on cubical sets}

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

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - contents]]

\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{HomotopyTheory}{Homotopy theory}\dotfill \pageref*{HomotopyTheory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

There is a [[model category]] structure on the [[category]] $[\Box^{op},Set]$ of [[cubical set]]s whose [[homotopy theory]] is that of the [[classical model structure on simplicial sets]].

Using this version of the [[homotopy hypothesis]]-theorem, cubical sets are a way to describe the [[homotopy type]] of [[∞-groupoid]]s using of all the [[geometric shapes for higher structures]] the [[cube]].

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

There is an evident [[simplicial set]]-valued [[functor]]

\begin{displaymath}
\Box \to sSet
\end{displaymath}
from the [[cube category]] to [[sSet]], which sends the cubical $n$-cube to the simplicial $n$-cube

\begin{displaymath}
\mathbf{1}^n \mapsto (\Delta[1])^{\times n}
  \,.
\end{displaymath}
Similarly there is a canonical [[Top]]-valued functor

\begin{displaymath}
\Box \to Top
\end{displaymath}
\begin{displaymath}
\mathbf{1}^n \mapsto (\Delta^1_{Top})^n
  \,.
\end{displaymath}
The corresponding [[nerve and realization]] [[adjunction]]

\begin{displaymath}
(|-| \dashv Sing_\Box) : Top \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing_\Box}{\to}} Set^{\Box^{op}}
\end{displaymath}
is the cubical analogue of the simplicial nerve and realization discussed \hyperlink{ForKanComplexes}{above}.

\begin{theorem}
\label{}\hypertarget{}{}
There is a [[model structure on cubical sets]] $Set^{\Box^{op}}$ whose

\begin{itemize}%
\item weak equivalences are the morphisms that become weak equivalences under geometric realization $|-|$;


\item cofibrations are the [[monomorphism]]s.



\end{itemize}
\end{theorem}
This is (\hyperlink{Jardine}{Jardine, section 3}).

Explicitly, a set of generating cofibrations is given by the boundary inclusions $\partial \Box^n \to \Box^n$, and a set of generating acyclic cofibrations is given by the horn inclusions $\sqcap_{k,\epsilon}^n \to \Box^n$. This is (\hyperlink{Cisinski}{Cisinski, Thm 8.4.38}). Thus, as a consequence of Cisinski's work, the fibrations are exactly cubical Kan fibrations.

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

\hypertarget{HomotopyTheory}{}\subsubsection*{{Homotopy theory}}\label{HomotopyTheory}

The following theorem establishes a form of the [[homotopy hypothesis]] for cubical sets.

\begin{theorem}
\label{}\hypertarget{}{}
The [[unit of an adjunction|unit of the adjunction]]

\begin{displaymath}
A \to Sing_\Box(|A|)
\end{displaymath}
is a weak equivalence in $Set^{{\Box}^{op}}$ for every cubical set $A$.

The counit of the adjunction

\begin{displaymath}
|Sing_\Box X| \to X
\end{displaymath}
is a weak equivalence in $Top$ for every topological space $X$.

It follows that we have an [[equivalence of categories]] induced on the [[homotopy categories]]

\begin{displaymath}
Ho(Top) \simeq Ho(Set^{\Box^{op}})
  \,.
\end{displaymath}
\end{theorem}
This is (\hyperlink{Jardine}{Jardine, theorem 29, corollary 30}).

In fact, by the discussion at \emph{[[adjoint (∞,1)-functor]]} it follow that the [[derived functors]] of the adjunction exhibit the [[simplicial localizations]] of cubical sets equivalent to that of simplicial sets, hence makes their [[(∞,1)-categories]] [[equivalence of (∞,1)-categories|equivalent]] (hence equivalent to [[∞Grpd]]).

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

Using that the [[cube category]] is a [[test category]] a model structure on cubical sets follows as a special case of the [[model structure on presheaves over a test category]], due to

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{[[joyalscatlab:Les préfaisceaux comme type d'homotopie]]}, Asterisque \textbf{308}.

\end{itemize}
Cisinski also derives explicit generating cofibrations and generating acyclic cofibrations using his theory of [[generalized Reedy category]], or \emph{categories skelettiques}. See Section 8.4.

The model structure on cubical sets as above is given in detail in

\begin{itemize}%
\item [[Rick Jardine]], \emph{Model structure on cubical sets} (2002) (\href{http://hopf.math.purdue.edu/Jardine/cubical2.pdf}{pdf})

\end{itemize}
There is also the old work

\begin{itemize}%
\item Victor Gugenheim, \emph{On supercomplexes} Trans. Amer. Math. Soc. 85 (1957), 35--51 \href{http://www.ams.org/journals/tran/1957-085-01/S0002-9947-1957-0086299-1/S0002-9947-1957-0086299-1.pdf}{PDF}

\end{itemize}
in which ``supercomplexes'' are discussed, that combine [[simplicial sets]] and cubical sets (def 5). There are functors from simplicial sets to supercomplexes (after Defn 5) and, implicitly, from supercomplexes to cubical sets (in Appendix II). This was written in 1956, long before people were thinking as formally as nowadays and long before Quillen model theory, but a comparison of the homotopy categories might be in there.

[[!redirects model structures on cubical sets]]



\end{document}