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\section*{type-theoretic model category}

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

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

[[!include type theory - contents]]

\hypertarget{typetheoretic_model_categories}{}\section*{{Type-theoretic model categories}}\label{typetheoretic_model_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{modeling_type_theory}{Modeling type theory}\dotfill \pageref*{modeling_type_theory} \linebreak
\noindent\hyperlink{modelcategorical_constructions}{Model-categorical constructions}\dotfill \pageref*{modelcategorical_constructions} \linebreak
\noindent\hyperlink{a_warning}{A warning}\dotfill \pageref*{a_warning} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

[[homotopy type theory|Homotopy type theory]] has [[categorical semantics]] in suitable [[homotopical categories]] which in turn present certain [[(infinity,1)-categories]]. The additional structure of type theory corresponds to structure on these homotopical categories that makes them into a certain kind of [[fibration category]], known as a [[type-theoretic fibration category]] or a [[tribe]].

In practice, however, semantic examples of tribes naturally sit inside certain [[Quillen model categories]]. The concept of \emph{type-theoretic model category} refers to a model category with additional structure that in particular ensures that its subcategory of fibrant objects is a tribe, but also includes additional conditions that make it easier to use model-categorical tools to prove things about the type-theoretic behavior of that tribe. At present it is not clear whether there is a unique ``correct'' notion of ``type-theoretic model category''; instead there is a range of stronger or weaker hypotheses that are often useful in proofs of this sort.

Regardless, one purpose of the notion(s) is to ensure that all [[(∞,1)-categories]] with sufficient structure can be presented by a type-theoretic model category, and hence provide higher [[categorical semantics]] for [[homotopy type theory]] (without possibly [[univalence]]). Specifically, every [[locally presentable (∞,1)-category|locally presentable]] [[locally cartesian closed (∞,1)-category]] has a presentation by a type-theoretic model category. For more on this see also the respective sections at \emph{[[relation between type theory and category theory]]}.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Some of the additional assumptions on a model category $M$ that are often useful to include when constructing semantics of type theory are:

\begin{itemize}%
\item $M$ is [[right proper]].


\item $M$ is a [[locally cartesian closed category]], or at least that pullback along [[fibrations]] has a right adjoint.


\item $M$ is [[combinatorial model category|combinatorial]], or at least [[accessible model category|accessible]] and/or [[cofibrantly generated model category|cofibrantly generated]].


\item The cofibrations in $M$ are the monomorphisms; or at least that they are closed under limits, preserved by pullbacks, all monomorphisms are cofibrations, and/or all fibrant objects are cofibrant.


\item Acyclic cofibrations are preserved by pullback along fibrations.


\item The underlying category of $M$ is a [[Grothendieck topos]], or even a [[presheaf topos]].


\item $M$ is a [[simplicial model category]].


\item $M$ is a [[simplicially locally cartesian closed category]].



\end{itemize}
Definitions in the literature include:

\begin{itemize}%
\item In \hyperlink{AK}{Arndt-Kapulkin} a ``logical model category'' was defined to be a model category in which pullback along any fibration has a right adjoint and acyclic cofibrations are preserved by pullback along fibrations.


\item In \hyperlink{Shulman15}{Shulman 15} a ``type-theoretic model category'' was defined to be a right proper model category in which pullback along any fibration has a right adjoint and cofibrations are closed under limits.


\item In \hyperlink{GK}{Gepner-Kock} a ``combinatorial type-theoretic model category'' was defined to be a right proper combinatorial locally cartesian closed model category whose cofibrations are the monomorphisms.


\item In \hyperlink{LS}{Lumsdaine-Shulman} a ``good model category'' was defined to be a simplicial, right proper, simplicially locally cartesian closed model category in which every monomorphism is a cofibration and cofibrations are closed under limits, while an ``excellent model category'' (no relation to the similarly-named [[excellent model category]]) was defined to be a good model category that is additionally combinatorial.



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

\begin{itemize}%
\item The [[classical model structure on simplicial sets]] satisfies \emph{all} the above properties, as does the [[injective model structure on simplicial presheaves]].


\item Every [[presentable (∞,1)-category|locally presentable]] [[locally Cartesian closed (∞,1)-category]] (by the discussion ) has a presentation by a [[right proper model category|right proper]] [[Cisinski model category]] (a combinatorial model structure on a Grothendieck topos whose cofibrations are the monomorphisms), indeed one whose underlying category is a presheaf topos. Thus very strong conditions may be assumed without significantly restricting the class of (∞,1)-categories that can be presented.


\item Another (counter-)example to keep in mind, however, is the [[canonical model structure on groupoids]], which is combinatorial and simplicial, all objects are fibrant and cofibrant, pullback along fibrations has a right adjoint, cofcibrations are closed under limits, and all monomorphisms are cofibrations; but the category $Gpd$ is not locally cartesian closed (hence not a Grothendieck topos), and not every cofibration is a monomorphism.


\item The [[type-theoretic model structure]] on the presheaves on a small category with an [[atomic object|atomic]] [[interval object]]. This gives examples on various type of cartesian [[cubical sets]].



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

\hypertarget{modeling_type_theory}{}\subsubsection*{{Modeling type theory}}\label{modeling_type_theory}

\begin{itemize}%
\item Since fibrations are closed under composition, $M$ always models [[Σ-types]].


\item If cofibrations are preserved by pullback (such as if they are the monomorphisms), then acyclic cofibrations are preserved by pullback along fibrations if and only if $M$ is right proper. And by adjointness, if pullback $f^*$ along any fibration $f$ has a right adjoint $f_*$, then acyclic cofibrations are preserved by pullback along fibrations if and only if the functors $f_*$ (for $f$ a fibration) preserve fibrations. This implies that $M$ models [[Π-types]].


\item These $\Pi$-types satisfy [[function extensionality]] if and only if $f_*$ also preserves acyclic fibrations, or equivalently if $f^*$ preserves cofibrations (\hyperlink{Shulman15}{Shulman 15, lemma 5.9}). In particular, if $M$ is right proper, cofibrations are preserved by pullback, and pullback along any fibration has a right adjoint, then $M$ models $\Pi$-types with function exensionality.


\item If acyclic cofibrations are preserved by pullback along fibrations, then for any map $f:x\to y$ between fibrant objects, the functor $f^*: M/y \to M/x$ preserves acyclic cofibrations between fibrant objects of $M/y$ (i.e. fibrations over $y$); see e.g. \hyperlink{Shulman17}{Shulman 17, lemma 7.2} (originally due to Joyal). This ``Frobenius condition'' implies that the [[path objects]] of $M$ model [[identity types]].


\item \hyperlink{LS}{Lumsdaine-Shulman} shows that an excellent model category (in their sense, see above) models a wide class of [[higher inductive types]].



\end{itemize}
\hypertarget{modelcategorical_constructions}{}\subsubsection*{{Model-categorical constructions}}\label{modelcategorical_constructions}

\begin{itemize}%
\item By \hyperlink{GK}{Gepner-Kock, Proposition 7.8}, a [[semi-left-exact reflection|semi-left-exact]] [[left Bousfield localization]] of a combinatorial type-theoretic model category (in their sense, see above) is again a combinatorial type-theoretic model category.

\end{itemize}
\hypertarget{a_warning}{}\subsubsection*{{A warning}}\label{a_warning}

In general one wants to think of the interpretation of type theory in the underlying [[tribe]] of a type-theoretic model category as ``living in'' the $(\infty,1)$-category presented by the model category. However, this is not automatic merely from the fact that the subcategory of fibrant objects in a model category is a tribe; one needs some stronger conditions such as those above to ensure that the 1-categorical constructions present the relevant $\infty$-categorical ones.

For instance, in \hyperlink{Bordg17}{Bordg 17} it is shown that the category of fibrant objects in the [[projective model structure]] on the category of [[groupoids]] with $\mathbb{Z}/2$-action is a tribe with $\Pi$-types and a universe, but that the universe fails to be [[univalence axiom|univalent]] and indeed that [[function extensionality]] fails to hold, even though the $(\infty,1)$-category presented by this model structure is locally cartesian closed and has an [[object classifier]] for discrete objects.

Generally one wants at least to require that all fibrant objects are cofibrant, in order that the underlying tribe of fibrant objects has the same [[simplicial localization]] as the model category itself.

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

\begin{itemize}%
\item [[Peter Arndt]] and [[Krzysztof Kapulkin]], \emph{Homotopy-Theoretic Models of Type Theory}, In: Ong L. (eds) Typed Lambda Calculi and Applications. TLCA 2011. Lecture Notes in Computer Science, vol 6690. Springer, Berlin, Heidelberg, \href{https://doi.org/10.1007/978-3-642-21691-6_7}{doi}


\item [[Michael Shulman]], \emph{Univalence for inverse diagrams and homotopy canonicity}, Mathematical Structures in Computer Science, Volume 25, Issue 5 ( \emph{From type theory and homotopy theory to Univalent Foundations of Mathematics} ) June 2015 (\href{https://arxiv.org/abs/1203.3253}{arXiv:1203.3253}, \href{https://doi.org/10.1017/S0960129514000565}{doi:/10.1017/S0960129514000565})


\item [[Mike Shulman]], \emph{\href{https://home.sandiego.edu/~shulman/hottminicourse2012/}{Minicourse on Homotopy Type Theory}} part 3, \emph{Categorical models of homotopy type theory}, April 2012 (\href{https://home.sandiego.edu/~shulman/hottminicourse2012/03models.pdf}{pdf})


\item [[Denis-Charles Cisinski]], \emph{Univalent universes for elegant models of homotopy types} (\href{http://arxiv.org/abs/1406.0058}{arXiv:1406.0058})


\item [[Anthony Bordg]], \emph{On the inadequacy of the projective structure with respect to the Univalence Axiom}, \href{https://arxiv.org/abs/1712.02652}{arxiv:1712.02652}


\item [[Mike Shulman]], \emph{Univalence for inverse EI diagrams}. Homology, Homotopy and Applications, 19:2 (2017), p219–249, \href{http://dx.doi.org/10.4310/HHA.2017.v19.n2.a12}{DOI}, \href{https://arxiv.org/abs/1508.02410}{arXiv:1508.02410}.


\item [[Peter LeFanu Lumsdaine]] and [[Mike Shulman]], \emph{Semantics of higher inductive types}, \href{https://arxiv.org/abs/1705.07088}{arxiv:1705.07088}.


\item [[David Gepner]] and [[Joachim Kock]], \emph{Univalence in locally cartesian closed categories}, \href{https://arxiv.org/abs/1208.1749}{arxiv:1208.1749}



\end{itemize}
[[!redirects presentation of homotopy type theory]] [[!redirects categorical semantics of homotopy type theory]]

[[!redirects type-theoretic model category]] [[!redirects type-theoretic model categories]] [[!redirects type-theoretic model cateory]]

[[!redirects type theoretic model category]] [[!redirects type theoretic model categories]] [[!redirects type theoretic model cateory]]



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