nLab model of type theory in an (infinity,1)-topos



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



It is expected that every elementary (infinity,1)-topos may serve as a model of homotopy type theory with the univalence axiom and higher inductive types. However, so far there is not a standard definition of elementary (,1)(\infty,1)-topos. We do almost know that every Grothendieck (infinity,1)-topos admits such a model; the caveat is that we only get univalent universes that are weakly a la Tarski (see universe).

Right proper Cisinski model categories

A Cisinski model category is a model category on a Grothendieck 1-topos whose cofibrations are the monomorphisms. It is right proper if weak equivalences are preserved by pullback along fibrations. This ensures that the adjunctions f *f *f^* \dashv f_* for a fibration ff are Quillen adjunctions, so that the (,1)(\infty,1)-category presented by the model category is locally cartesian closed.

The fibrant objects in a right proper Cisinski model category can be shown to form a comprehension category, one of the structures used for semantics of type theory, which admits all the structure necessary to model the type-forming operations of the dependent sum type, dependent product type, and identity type (the latter by using path objects). By applying the local universe model? or natural model coherence theorem, we see that any right proper Cisinski model category models type theory.

Cisinski and Gepner-Kock have also shown that any locally presentable, locally cartesian closed (,1)(\infty,1)-category can be presented by a right proper Cisinski model category. Therefore, any such (,1)(\infty,1)-category admits a model of type theory.

Higher inductive types

Models of higher inductive types can be constructed in any simplicial combinatorial model category; for now, see the note at higher inductive types.

Strong univalent universes

In some cases we can construct a univalent universe which satisfies the rules to be a “strong” universe a la Tarski. This means we have an object UU together with a fibration U˜U\tilde{U}\to U, which is “closed under the type-forming operations” up to isomorphism. For instance, if BAB\to A is a pullback of U˜\tilde{U} along some map AUA\to U, and likewise CBC\to B is a pullback of U˜\tilde{U} along some map BUB\to U, then the composite CAC\to A is a pullback of U˜\tilde{U} along some map AUA\to U. In particular, we can apply this to the “generic” composable pair of fibrations over U (1)U^{(1)}, the “type of composable fibrations”, to get a classifying map U (1)UU^{(1)}\to U of the composite that interprets the Σ\Sigma-type former on the universe. The requirements for Π\Pi-types and identity types are similar.

In practice, the way we ensure these requirements is to show that a fibration is a pullback of U˜U\tilde{U}\to U if and only if its “fibers” are bounded by some cardinal number κ\kappa (used in defining UU). Then as long as κ\kappa is inaccessible, such fibrations will be closed under the type-forming operations.

Strong univalent universes are known to exist in the Reedy model category sSet R opsSet^{R^op} of simplicial presheaves on an elegant Reedy category RR; see this paper and this paper.

Weak univalent universes

Suppose we have any right proper Cisinski model category that presents a Grothendieck (,1)(\infty,1)-topos; we will show that it models a univalent universe which is “weakly a la Tarski”. (Since any Grothendieck (,1)(\infty,1)-topos is locally cartesian closed, such a model category always exists as remarked above.)

It is known that for any regular κ\kappa, an (,1)(\infty,1)-topos has an object classifier for κ\kappa-small morphisms, i.e. a κ\kappa-small morphism V˜V\tilde{V}\to V such that for any object AA, the space of maps AVA\to V is naturally equivalent to the core of the category of κ\kappa-small morphisms into AA.

Let κ\kappa be inaccessible, and let U˜U\tilde{U} \to U be a fibration between fibrant objects of the model category that represents V˜V\tilde{V}\to V. Then since κ\kappa-small morphisms in the (,1)(\infty,1)-topos are closed under composition, diagonals, and dependent products, the fibrations in the model category that are homotopy pullbacks of U˜U\tilde{U}\to U are also so closed. In particular, we can again build the universal composable pair over U (1)U^{(1)} and obtain a map U (1)UU^{(1)}\to U which classifies its composite up to homotopy (i.e. so that the composite is a homotopy pullback of U˜U\tilde{U}\to U along it). Dependent products and identity types are similar.

Thus, if we use the local universe coherence theorem, UU represents a “universe” which comes with type-forming “operations”, but these operations do not literally respect the actual type-formers on actual types. Instead we can say only that El(ΣAB)El(\Sigma A B) is equivalent to Σ(ElA)(ElB)\Sigma (El A) (El B), and so on. However, we can still define a map Id U(A,B)Equiv(El(A),El(B))Id_U(A,B) \to Equiv(El(A),El(B)), and the full universal property of the object classifier implies that it is an equivalence (see for instance Gepner-Kock). Thus, we have a univalent universe which is “weakly a la Tarski”.


Full proof of Awodey's conjecture that \infty-stack \infty-toposes model univalent homotopy type theory:


Last revised on June 17, 2022 at 14:31:16. See the history of this page for a list of all contributions to it.