nLab
locally cartesian closed (infinity,1)-category

Contents

Definition

An (∞,1)-category CC has finite (∞,1)-limits if it has a terminal object and for every object xXx \in X, the over-(∞,1)-category C /xC_{/x} has finite products.

We say that such a CC is locally cartesian closed if moreover C /xC_{/x} is a cartesian closed (∞,1)-category for every object xx. This is equivalent to asking that the pullback functor f *:C yC xf^*\colon C_{\y} \to C_{\x}, for any f:xyf\colon x\to y in CC, has a right adjoint Π f\Pi_f.

Examples

Properties

Presentations

The following theorem should be compared with the fact that every locally presentable (∞,1)-category admits a presentation by a Cisinski model category, indeed by a left Bousfield localization of a global model structure on simplicial presheaves.

Theorem

For a locally presentable (,1)(\infty,1)-category CC, the following are equivalent.

  1. CC is locally cartesian closed.
  2. (∞,1)-Colimits in CC are stable under pullback.
  3. CC admits a presentation by a combinatorial locally cartesian closed model category.
  4. CC admits a presentation by a right proper Cisinski model category.
  5. CC admits a presentation by a right proper left Bousfield localization of an injective model category of simplicial presheaves.
Proof

Since left adjoints preserve colimits, the first condition implies the second. The converse holds by the adjoint functor theorem since each slice of CC is locally presentable.

Suppose MM is a right proper Cisinski model category. Then pullback along a fibration preserves cofibrations (since they are the monomorphisms) and weak equivalences (since MM is right proper). Since MM is a locally cartesian closed 1-category, pullback also has a right adjoint, so it is a left Quillen functor; thus the fourth condition implies the third. Since left Quillen functors preserve homotopy colimits, the third condition implies the second.

Clearly the fifth condition implies the fourth, so it suffices to show that the second condition implies the fifth. For that, see this blog comment by Denis-Charles Cisinski. An alternative proof can be found in (Gepner-Kock 12, Thm 7.10).

Remark

Further equivalent characterizations of locally cartesian closed (,1)(\infty,1)-categories are in (Lurie, prop. 6.1.1.4, lemma 6.1.3.3)

Remark

Comparing the third and the fifth item in theorem 1 notice that the projective and the injective model structure on simplicial presheaves are Quillen equivalent (as discussed at model structure on functors.)

Internal logic and homotopy type theory

It is expected that the internal logic of locally cartesian closed (,1)(\infty,1)-categories should be a sort of homotopy type theory (specifically, that with intensional identity types and dependent products), in higher analogy with the relation between type theory and category theory.

Ideally, this should involve two directions: interpretation of type theory in such (,1)(\infty,1)-categories, and construction of such (,1)(\infty,1)-categories from syntax.

For the first direction, Theorem 1 says in particular that every locally presentable and locally cartesian closed (,1)(\infinity,1)-category has a presentation by a type-theoretic model category. As discussed there, this provides the categorical semantics for homotopy type theory (without in general the univalence axiom). Together, this yields:

Theorem

(Cisinski, Shulman) Every locally presentable locally cartesian closed (,1)(\infinity,1)-category admits an interpretation of type theory with Σ\Sigma-types, Π\Pi-types, and Id\mathrm{Id}-types.

This includes in particular all (∞-stack-) (∞,1)-toposes, which should in addition possess univalent universes. See also internal logic of an (∞,1)-topos.

An explicit statement in the second direction was conjectured in (Joyal 2011), and established in Chris Kapulkin’s PhD thesis (see Kapulkin 14):

Theorem

(Kapulkin) If T is any dependent type theory with (at least) Σ\Sigma-types, Π\Pi-types, and Id\mathrm{Id}-types, then the simplicial localisation of its classifying category Cl(T)\mathrm{Cl}(\mathbf{T}) is a locally cartesian closed (,1)(\infty,1)-category.

However, this “syntactic (,1)(\infty,1)-category” will not in general be locally presentable, lacking appropriate colimits. There is thus a mismatch between the two best statements we have at present, for the two directions of the internal language correspondence.

References

Characterizations of locally presentable locally cartesian closed (,1)(\infty,1)-categories (as locally presentable (∞,1)-categories with universal colimits) are discussed in section 6.1 of

Early discussion in the context of homotopy type theory is in

proposing homotopy type theory as an internal language for locally cartesian closed \infty-categories.

Denis-Charles Cisinski’s argument in Theorem 1 above, that every locally presentable locally cartesian closed (,1)(\infinity,1)-category admits a presentation by a type-theoretic model category, originally appeared on the Cafélog comment](http://golem.ph.utexas.edu/category/2012/05/the_mysterious_nature_of_right.html#c041306)) and is mentioned in print in Examples 2.16 of

which gives a detailed account of the categorical semantics of homotopy type theory in type-theoretic model categories such as those presenting locally cartesian closed (,1)(\infty,1)-categories.

For more on this see also the relevant sections at relation between type theory and category theory.

Discussion of the converse direction, obtaining locally cartesian closed (,1)(\infty,1)-categories as syntactic categories of homotopy type theories is in

  • Chris Kapulkin, Type theory and locally cartesian closed quasicategories, Oxford 2014 (video)

A discussion of object classifiers, univalent families, and model category presentations is the context of (,1)(\infty,1)-categories (and hence in categorical semantics for what should be homotopy type theory with univalent universes “weakly a la Tarski”) appeared also in

Revised on May 27, 2015 08:37:04 by Peter Le Fanu Lumsdaine (130.237.198.45)