locally cartesian closed (infinity,1)-category



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.




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.


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 structure on sSet-enriched presheaves over some small sSet-site.

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.

The fifth condition implies the fourth, since the model structure therein is Cisinski. The only nonobvious part of this is that its underlying category is a topos, which follows from the fact that sSetsSet-enriched presheaves on a sSetsSet-enriched category CC can be identified with internal diagrams on CC regarded as an internal category in sSetsSet, and the category of internal diagrams on any internal category in a topos (such as sSetsSet) is again a topos.

It remains 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).


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


Comparing the third and the fifth item in theorem \ref{Presentations} 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. More specifically, the following are hoped to be true:

  • From any type theory, its syntactic category is (or gives rise to) an (,1)(\infty,1)-category, which is the initial object in an appropriate (,1)(\infty,1)-category (or perhaps (,2)(\infty,2)-category) of locally cartesian closed (,1)(\infty,1)-categories (perhaps with additional structure corresponding to additional axioms or rules in the type theory). Thus, type theory can be “interpreted” in any lcc (,1)(\infty,1)-category by way of the universal morphism from this initial object.

  • From any locally cartesian closed (,1)(\infty,1)-category CC, one can construct a type theory with rules or axioms corresponding to its objects and morphisms, which has a canonical interpretation in CC itself. This would be the “internal language” of CC.

  • These construction should set up some sort of adjunction, or perhaps equivalence, between some (,1)(\infty,1)- or (,2)(\infty,2)-category of type theories and that of lcc (,1)(\infty,1)-categories.

Some partial results in these directions are known. For instance, Theorem \ref{Presentations} says in particular that every locally presentable and locally cartesian closed (,1)(\infty,1)-category has a presentation by a type-theoretic model category. As discussed there, this provides categorical semantics for homotopy type theory (without in general the univalence axiom), at least if we assume the initiality of the syntactic 1-category (which is known in a special case and expected to be true in all cases). Together, this yields:


(Cisinski, Shulman) Assuming the initiality of the syntactic 1-category, every locally presentable locally cartesian closed (,1)(\infty,1)-category admits an interpretation of Martin-Löf type theory with dependent sum types, dependent product types, and identity types (“homotopy type theory”).

Moreover it also interprets function extensionality (Shulman 12, lemma 5.9, see this this discussion.

Hence every presentable locally cartesian closed \infty-category interprets HoTT+FunExt.

This includes in particular all (∞-stack-) (∞,1)-toposes. The latter should in addition possess univalent universes, but that is at present known only weakly except for a restricted class. See also internal logic of an (∞,1)-topos.

This statement is not fully satisfactory for several reasons. Firstly, it assumes local presentability. Secondly, rather than staying in the world of (,1)(\infty,1)-categories, it goes by way of a strict presentation. Thus, the existence and behavior of a “universal model” is unclear.

A step in the latter direction was conjectured in (Joyal 2011), and established in Chris Kapulkin’s PhD thesis (see Kapulkin 14, Kapulkin 15):


(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.

Furthermore, nothing is yet known about the initiality of this syntactic (,1)(\infty,1)-category, which is a central aspect of what it means to have an internal language. Finally, no one has yet explicitly constructed an “internal language” type theory from an (,1)(\infty,1)-category, and so of course the desired adjunction/equivalence cannot yet even be stated as a precise conjecture.


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 \ref{Presentations} 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 Café (this blog comment) 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

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 January 13, 2017 16:13:50 by Urs Schreiber (