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

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


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

(∞,1)-topos theory





Extra stuff, structure and property



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 extensionaltype 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 statement would be that the internal logic of (∞,1)-toposes is homotopy type theory, though there is fine print involved.

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. Fore 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

  • Steve Awodey, Homotopy and Type Theory, grant proposal project description (pdf)

  • Steve Awodey, Type theory and homotopy, Epistemology versus Ontology. Springer Netherlands, 2012. 183-201. (2010) arXiv:1010.1810

A pronounced statement of the weaker version was highlighted in

and stated more precisely in


The proof of the weaker version (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 Example 2.16 of


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

Last revised on March 17, 2019 at 23:06:50. See the history of this page for a list of all contributions to it.