nLab type-theoretic model category

Type-theoretic model categories


Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Type-theoretic model categories


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 MM that are often useful to include when constructing semantics of type theory are:

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.



Modeling type theory

(cf. categorical semantics of dependent types)

  • Since fibrations are closed under composition, MM 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 MM is right proper. And by adjointness, if pullback f *f^* along any fibration ff has a right adjoint f *f_*, then acyclic cofibrations are preserved by pullback along fibrations if and only if the functors f *f_* (for ff a fibration) preserve fibrations. This implies that MM models Π-types.

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

  • If acyclic cofibrations are preserved by pullback along fibrations, then for any map f:xyf:x\to y between fibrant objects, the functor f *:M/yM/xf^*: M/y \to M/x preserves acyclic cofibrations between fibrant objects of M/yM/y (i.e. fibrations over yy); see e.g. Shulman 17, lemma 7.2 (originally due to Joyal). This “Frobenius condition” implies that the path objects of MM model identity types.

  • Lumsdaine-Shulman shows that an excellent model category (in their sense, see above) models a wide class of higher inductive types.

Model-categorical constructions

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 (,1)(\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 /2\mathbb{Z}/2-action is a tribe with Π\Pi-types and a universe, but that the universe fails to be univalent and indeed that function extensionality fails to hold, even though the (,1)(\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.


Combination with the terminology of model topos to type-theoretic model topos:

reviewed in

Last revised on November 1, 2023 at 07:41:42. See the history of this page for a list of all contributions to it.