type-theoretic model category


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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
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


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:



Modeling type theory

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.