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
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive 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, (idemponent) 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

under construction


Homotopy type theory has categorical semantics in suitable homotopical categories which in turn present certain (infinity,1)-categories.

The concept of type-theoretic model category (Shulman 12a) or type-theoretic fibration category (Shulman 12b, def. 2.1) is one particular way of making this precise. Specifically every locally presentable locally cartesian closed (∞,1)-category has a presentation by a type-theoretic model category, hence provides higher categorical semantics for homotopy type theory (without possibly univalence). For more on this see also the respective sections at relation between type theory and category theory.


under construction

By decomposing the structure in homotopy type theory in layers as

  1. dependent type theory

  2. with identity types

  3. and univalent universe types.

A 1-category whose internal logic can interpret this needs to

  1. be a locally cartesian closed category

  2. equipped with a weak factorization system with stable path objects, such that acyclic cofibrations are preserved by pullback along fibrations between fibrant objects.

  3. (needs to be finished)


Function extensionality

(Shulman 12b, lemma 5.9)


The condition in prop. 1 holds in particular in right proper Cisinski model structures, since in these right base change along fibrations is a right Quillen functor (see e.g. the proof here).

Notice that every presentable locally Cartesian closed (∞,1)-category (by the discussion there) has a presentation by a right proper Cisinski model category. Accordingly we may say that every presentable locally Cartesian closed (∞,1)-category interprets HoTT+FunExt.


The definition originates in and a discussion of categorical semantics of homotopy type theory in a type-theoretic model category appears in

An exposition is in

Similar conisderations (using the term “typos” for something similar to a type-theoretic model category) are presented in

  • André Joyal, What is categorical type theory, various talks in 2013, (pdf)

Revised on January 13, 2017 16:20:51 by Urs Schreiber (