nLab Awodey's proposal



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


(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos




In Awodey 09, Awodey 10 was first expressed the idea that dependent type theory with intensional identity types (Martin-Löf dependent type theory), viewed as homotopy type theory, is in similar relation to the concept of (∞,1)-toposes as extensional type theory is to the ordinary concept of toposes (as discussed at relation between type theory and category theory).

From Awodey 09, p. 13, Awodey 10, p. 15:

The homotopy interpretation of Martin-Löf type theory into Quillen model categories, and the related results on type-theoretic constructions of higher groupoids, are analogous to basic results interpreting extensional type theory and higher-order logic in (1-)toposes, and clearly indicate that the logic of higher toposes, and therewith of higher homotopy theory, is a form of intensional type theory.

A concise re-statement would be that:

  1. the internal logic of (∞,1)-toposes is univalent homotopy type theory

    (though there is fine print involved, e.g. the initiality conjecture);

  2. there is a model of (univalent) homotopy type theory in any ( , 1 ) (\infty,1) -topos

    (this version has a proof, see below);

  3. homotopy type theory is synthetic homotopy theory

    (this may be read as a suggestive colloquial version of the previous statement, remaining vague on whether univalence is considered or not).

Following this suggestion, the weaker form of this idea, ignoring the univalent type universe and relating to the broader class of locally Cartesian closed (∞,1)-categories, was stated more concretely as a conjecture in Joyal 11. For more precision see Kapulkin-Lumsdaine 16, p. 9.

Roughly, this is about the following table of correspondences (for more see at relation between type theory and category theory):

internal logic/type theoryhigher category theory
type theorylocally Cartesian closed categories
homotopy type theorylocally Cartesian closed (∞,1)-categories
homotopy type theory with univalent type universeselementary (∞,1)-toposes

Fore more precision see Kapulkin-Lumsdaine 16, p. 9.


A proof of the weaker version of the conjecture, in form of the statement that every locally presentable locally Cartesian closed (∞,1)-category is presented by a suitable type theoretic model category which provides categorical semantics for homotopy type theory, was proven in Shulman 12, Example 2.16, following Cisinski 12.

Generalizing this to a proof of the full conjecture required finding “strict” models for the object classifier by strict type universes. A series of article (Shulman 12, Shulman 13) showed that this is possible in an increasing class of special cases.

A proof of the general case was finally announced in Shulman 19.

For more see at model of type theory in an (infinity,1)-topos.



The idea is due to

A pronounced statement of the weaker version was highlighted in

and stated more precisely in


The proof of the weaker version of Awodey’s conjecture (that every locally Cartesian closed (∞,1)-category has a presentation by a suitable type-theoretic model category which provides categorical semantics for homotopy type theory) is due, independently, to


The proof of the stronger version (including univalent type universes modelling object classifier of (∞,1)-toposes) was found for the special case of (∞,1)-presheaf (∞,1)-toposes over elegant Reedy categories in

A general proof was announced in

and appeared in

It is reviewed in:

Last revised on July 26, 2022 at 14:21:22. See the history of this page for a list of all contributions to it.