natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
Homotopy type theory has categorical semantics in suitable homotopical categories which in turn present certain (∞,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 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 locally cartesian closed (∞,1)-category has a presentation by a type-theoretic model category. For more on this see also the respective sections at relation between type theory and category theory.
Some of the additional assumptions on a model category $M$ that are often useful to include when constructing semantics of type theory are:
$M$ is right proper.
$M$ is a locally cartesian closed category, or at least that pullback along fibrations has a right adjoint.
$M$ is combinatorial, or at least accessible and/or cofibrantly generated.
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.
Acyclic cofibrations are preserved by pullback along fibrations.
The underlying category of $M$ is a Grothendieck topos, or even a presheaf topos.
$M$ is a simplicial model category.
Definitions in the literature include:
In 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.
In 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.
In 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.
In 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.
The classical model structure on simplicial sets satisfies all the above properties, as does the injective model structure on simplicial presheaves.
Every locally presentable locally Cartesian closed (∞,1)-category (by the discussion there) has a presentation by a 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.
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.
The type-theoretic model structure on the presheaves on a small category with an atomic interval object. This gives examples on various type of cartesian cubical sets.
Since fibrations are closed under composition, $M$ always models Σ-types.
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.
These $\Pi$-types satisfy function extensionality if and only if $f_*$ also preserves acyclic fibrations, or equivalently if $f^*$ preserves cofibrations (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.
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. Shulman 17, lemma 7.2 (originally due to Joyal). This “Frobenius condition” implies that the path objects of $M$ model identity types.
Lumsdaine-Shulman shows that an excellent model category (in their sense, see above) models a wide class of higher inductive types.
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 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 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.
Peter Arndt and Krzysztof Kapulkin, 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, doi
Michael Shulman, Univalence for inverse diagrams and homotopy canonicity, Mathematical Structures in Computer Science, Volume 25, Issue 5 ( From type theory and homotopy theory to Univalent Foundations of Mathematics ) June 2015 (arXiv:1203.3253, doi:/10.1017/S0960129514000565)
Mike Shulman, Minicourse on Homotopy Type Theory part 3, Categorical models of homotopy type theory, April 2012 (pdf)
Denis-Charles Cisinski, Univalent universes for elegant models of homotopy types (arXiv:1406.0058)
Anthony Bordg, On the inadequacy of the projective structure with respect to the Univalence Axiom, arxiv:1712.02652
Mike Shulman, Univalence for inverse EI diagrams. Homology, Homotopy and Applications, 19:2 (2017), p219–249, DOI, arXiv:1508.02410.
Peter LeFanu Lumsdaine and Mike Shulman, Semantics of higher inductive types, arxiv:1705.07088.
David Gepner and Joachim Kock, Univalence in locally cartesian closed categories, arxiv:1208.1749