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\begin{document}

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\section*{cubical-type model category}

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

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{cubicaltype_model_categories}{}\section*{{Cubical-type model categories}}\label{cubicaltype_model_categories}

\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{assumptions}{Assumptions}\dotfill \pageref*{assumptions} \linebreak
\noindent\hyperlink{instances_of_the_assumptions}{Instances of the assumptions}\dotfill \pageref*{instances_of_the_assumptions} \linebreak
\noindent\hyperlink{equivalence_with_simplicial_sets}{Equivalence with simplicial sets}\dotfill \pageref*{equivalence_with_simplicial_sets} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{general}{}\subsection*{{General}}\label{general}

These are a class of model structures on certain categories of [[cubical sets]], due to [[Christian Sattler]], motivated by [[cubical type theory]].

\hypertarget{assumptions}{}\subsubsection*{{Assumptions}}\label{assumptions}

Let $\mathbb{C}$ be a [[small category]] with [[finite products]]. Assume there is a (cartesian) [[interval object]] $\mathbb{I}$, i.e., an object $\mathbb{I}$ with two morphisms $0,1 : e \to \mathbb{I}$, where $e$ is the terminal object in $\mathbb{C}$.

Assume further that we have [[connection on a cubical set|connection maps]] $\vee,\wedge : \mathbb{I}\times\mathbb{I} \to \mathbb{I}$ such that $x \vee 0 = 0 \vee x = x$ and $x \wedge 1 = 1 \wedge x = x$.

We also need a face lattice $\mathbb{F}$, that is, a sublattice of the [[subobject classifier]] $\Omega$ in the [[presheaf category]] $cSet =
\mathbb{C}^{op} \to Set$ containing the endpoints of each cylinder $J^+ = J \times \mathbb{I}$ for each object $J$ of $\mathbb{C}$, and an operation $\forall : \mathbb{F}^\mathbb{I} \to \mathbb{F}$ right adjoint to the projection in the sense that $\psi \le \forall \delta$ iff $\psi p \le \delta$, where $p : \mathbb{F} \times \mathbb{I} \to
\mathbb{F}$.

If we want the model to be effective, we also require that each proposition $\psi = 1$ with $\psi \in \mathbb{F}(I)$ is decidable. (This rules out taking $\mathbb{F} = \Omega$.)

\hypertarget{instances_of_the_assumptions}{}\subsubsection*{{Instances of the assumptions}}\label{instances_of_the_assumptions}

A simple and central example is the full subcategory of the category of [[posets]] $Pos$ on powers of $\mathbb{I} = (0 \lt 1)$. This is also the [[Lawvere theory]] of [[distributive lattices]]. The [[idempotent completion]] is the full subcategory of $Pos$ on finite lattices.

Other prominent examples are the Lawvere theories of [[de Morgan algebra|de Morgan]], [[Kleene algebra|Kleene]], and [[boolean algebra|boolean]] algebras.

For computational purposes we can take $\mathbb{F}$ to be the smallest lattice containing the cylinder endpoints (the faces of cubes). To get [[Cisinski model structures]] we can take $\mathbb{F} = \Omega$.

\hypertarget{equivalence_with_simplicial_sets}{}\subsection*{{Equivalence with simplicial sets}}\label{equivalence_with_simplicial_sets}

One may wonder whether these models structures are equivalent to the model in [[simplicial sets]]. This is not the case for Cartisian cubes; see \href{https://groups.google.com/d/msg/homotopytypetheory/RQkLWZ_83kQ/tAyb3zYTBQAJ}{mailing-list}. However, this model does form a Grothendieck [[(infinity,1)-topos]], because it satisfies the Giraud axioms. For deMorgan cubical sets the geometric realization is not a Quillen equivalence (because of the reversal map); the counterexample is the unit interval quotient by the symmetry); see \hyperlink{sattlerHIM}{Sattler}. Whether they are equivalent by another map is not yet excluded. The question whether the geometric realization for the cubical sets based on distribute lattices is an equivalence is still open.

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

\begin{itemize}%
\item [[Thierry Coquand]], \emph{A model structure on some presheaf categories}, \href{http://www.cse.chalmers.se/~coquand/mod2.pdf}{pdf}


\item [[Thierry Coquand]], \emph{Some examples of complete Cisinski model structures}, \href{http://www.cse.chalmers.se/~coquand/cis3.pdf}{pdf}


\item [[Christian Sattler]], \emph{Do cubical models of type theory also model homotopy types}, lecture at Hausdorff Trimester Program: Types, Sets and Constructions, \href{https://www.youtube.com/watch?v=wkPDyIGmEoA}{youtube}



\end{itemize}
[[!redirects type-theoretic model structures]] [[!redirects type theoretic model structures]] [[!redirects type theoretic model structure]] [[!redirects type-theoretic model structure]]



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