model of type theory in an (infinity,1)-topos (Rev #4, changes)

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It is expected that every elementary (infinity,1)-topos admits 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 $(\infty,1)$-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*.

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 $f^* \dashv f_*$ for a fibration $f$ are Quillen adjunctions, so that the $(\infty,1)$-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 sum type, ~~dependent product type?~~dependent product type, and identity type (the latter by using path objects). By applying the local universe 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 $(\infty,1)$-category can be presented by a right proper Cisinski model category. Therefore, any such $(\infty,1)$-category admits a model of type theory.

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

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 $U$ together with a fibration $\tilde{U}\to U$, which is “closed under the type-forming operations” up to isomorphism. For instance, if $B\to A$ is a pullback of $\tilde{U}$ along some map $A\to U$, and likewise $C\to B$ is a pullback of $\tilde{U}$ along some map $B\to U$, then the composite $C\to A$ is a pullback of $\tilde{U}$ along some map $A\to U$. In particular, we can apply this to the “generic” composable pair of fibrations over $U^{(1)}$, the “type of composable fibrations”, to get a classifying map $U^{(1)}\to U$ of the composite that interprets the $\Sigma$-type former on the universe. The requirements for $\Pi$-types and identity types are similar.

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

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

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

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

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

Thus, if we use the local universe coherence theorem, $U$ 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 $El(\Sigma A B)$ is *equivalent* to $\Sigma (El A) (El B)$, and so on. However, we can still define a map $Id_U(A,B) \to Equiv(El(A),El(B))$, 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”.