Locally Cartesian Closed Quasicategories

• Locally cartesian closed quasicategories are (∞,1)-categories with finite limits where every pullback functor has a right adjoint, enabling dependent product formation.
• They bridge categorical semantics and type theory by establishing a DK-equivalence with π-tribes and supporting models of Martin-Löf type theory with functional extensionality.
• These structures underpin key advances in univalent foundations and higher topos theory, offering a robust framework for synthetic homotopy constructions.

A locally cartesian closed quasicategory is a quasicategory $C$ that admits finite limits and such that, for every morphism $f:Y\to X$, the pullback functor $f^*:C_{/X}\to C_{/Y}$ has a right adjoint $\Pi_f:C_{/Y}\to C_{/X}$. In this sense, local cartesian closure is a slice-wise exponentiability condition internal to the $(\infty,1)$-categorical setting, and it is the quasicategorical form of the structure needed to interpret dependent products. Recent work places these objects at the center of an internal-language correspondence: the paper "Internal languages of locally cartesian closed $(\infty,1)$-categories" establishes a DK-equivalence between $\pi$-tribes and locally cartesian closed quasicategories, and from this derives that Martin-Löf type theory with dependent sums, intensional identity types, and dependent products satisfying functional extensionality is the internal language of locally cartesian closed $(\infty,1)$-categories (Cherradi, 3 Sep 2025).

1. Definition and basic categorical structure

A quasicategory, or $(\infty,1)$-category, is a simplicial set $C\in sSet$ such that for every $f:Y\to X$0 the inner-horn inclusion

$f:Y\to X$1

admits fillers. Equivalently, every diagram of simplices shaped like an inner horn

$f:Y\to X$2

extends to a simplex $f:Y\to X$3. Concretely, if $f:Y\to X$4, an inner horn is a collection of $f:Y\to X$5 faces of an $f:Y\to X$6-simplex missing the $f:Y\to X$7-th face, and the horn-filling condition requires a coherent composition of morphisms up to higher homotopy (Cherradi, 3 Sep 2025).

Given an object $f:Y\to X$8, the slice quasicategory $f:Y\to X$9 is defined by the pullback

$f^*:C_{/X}\to C_{/Y}$0

where $f^*:C_{/X}\to C_{/Y}$1 evaluates at the target of the $f^*:C_{/X}\to C_{/Y}$2-simplex and $f^*:C_{/X}\to C_{/Y}$3 picks out $f^*:C_{/X}\to C_{/Y}$4. It is again a quasicategory. For any morphism $f^*:C_{/X}\to C_{/Y}$5, pullback along $f^*:C_{/X}\to C_{/Y}$6 defines a functor

$f^*:C_{/X}\to C_{/Y}$7

A quasicategory $f^*:C_{/X}\to C_{/Y}$8 is locally cartesian closed if it satisfies two conditions: it admits finite limits, and for every map $f^*:C_{/X}\to C_{/Y}$9, the pullback functor $\Pi_f:C_{/Y}\to C_{/X}$0 has a right adjoint $\Pi_f:C_{/Y}\to C_{/X}$1 (Cherradi, 3 Sep 2025).

The adjunction is expressed by natural equivalences of mapping spaces: $\Pi_f:C_{/Y}\to C_{/X}$2 This formula is the higher-categorical form of dependent product formation. It is also the point at which local cartesian closure diverges from ordinary cartesian closure: the relevant exponentials live in slices rather than only in the ambient category.

A standard equivalent characterization is that each slice $\Pi_f:C_{/Y}\to C_{/X}$3 is itself a cartesian closed quasicategory. When $\Pi_f:C_{/Y}\to C_{/X}$4 is locally cartesian closed, every slice $\Pi_f:C_{/Y}\to C_{/X}$5 admits finite limits and internal homs; limits in $\Pi_f:C_{/Y}\to C_{/X}$6 are computed by first forgetting down to $\Pi_f:C_{/Y}\to C_{/X}$7 and then forming homotopy limits in $\Pi_f:C_{/Y}\to C_{/X}$8 (Cherradi, 2022). This equivalence is often the most practical way to recognize the condition in concrete constructions.

A common misconception is to identify local cartesian closure with ordinary cartesian closure plus pullbacks. The supplied results do not support that simplification. What is required is the existence of right adjoints to all base-change functors between slices, or equivalently cartesian closure in every slice, not merely in the ambient quasicategory (Cherradi, 2022).

2. Slices, internal homs, and exponentials in the higher setting

The slice formulation is central because dependent type formers are inherently relative to a base object. If $\Pi_f:C_{/Y}\to C_{/X}$9 is locally cartesian closed, then each slice $(\infty,1)$0 is again locally cartesian closed, and in particular cartesian closed. For $(\infty,1)$1, there is an internal hom object

$(\infty,1)$2

representing the right fibration

$(\infty,1)$3

The usual adjunction $(\infty,1)$4 restricts to the corresponding slice-internal-hom adjunction

$(\infty,1)$5

(Gepner et al., 2012).

In the presentable setting, local cartesian closure interacts tightly with universality of colimits. For a presentable $(\infty,1)$6-category $(\infty,1)$7, the existence of all right adjoints $(\infty,1)$8 is equivalent to the single condition that colimits in $(\infty,1)$9 are universal (Gepner et al., 2012). This is a structural criterion rather than a quasicategory-specific construction, but it explains why locally cartesian closed examples are abundant among higher-categorical localizations and topoi.

The representability viewpoint provides another equivalent formulation. An $(\infty,1)$0-category with pullbacks is cartesian closed if and only if, for every $(\infty,1)$1, the right fibration

$(\infty,1)$2

sending $(\infty,1)$3 to $(\infty,1)$4 is representable; the representing object is the usual internal hom $(\infty,1)$5. By the same token, local cartesian closure is equivalent to representability of the corresponding slice-wise mapping fibrations (Gepner et al., 2012).

This suggests a useful conceptual distinction. Ordinary exponentials $(\infty,1)$6 control maps out of products in a fixed ambient category, whereas the objects $(\infty,1)$7 control reindexing along arbitrary morphisms and therefore encode genuinely dependent behavior. In the literature summarized here, that distinction is what allows the passage from categorical semantics to intensional type theory (Cherradi, 3 Sep 2025).

3. Constructions, localizations, and canonical examples

Several major classes of locally cartesian closed quasicategories are identified in the cited work. Every presheaf $(\infty,1)$8-category $(\infty,1)$9 is locally cartesian closed, with pointwise limits and $\pi$0 given by the right Kan-extension formula. Presentable locally cartesian closed $\pi$1-categories are exactly accessible locally cartesian localizations of presheaf $\pi$2-categories (Gepner et al., 2012). More precisely, an $\pi$3-category $\pi$4 is presentable and locally cartesian closed if and only if it is an accessible locally cartesian localization of $\pi$5 for some small $\pi$6 (Gepner et al., 2012).

If

$\pi$7

is an accessible localization of a presentable locally cartesian closed $\pi$8-category $\pi$9, then $(\infty,1)$0 is called locally cartesian when it commutes with pullback along maps between local objects. The conditions that characterize this are equivalent: pullback functors preserve local equivalences, or base-change sends local objects to local objects. When these conditions hold, $(\infty,1)$1 is again a presentable locally cartesian closed $(\infty,1)$2-category (Gepner et al., 2012).

This framework includes $(\infty,1)$3-topoi and $(\infty,1)$4-topoi. Every $(\infty,1)$5-topos is presentable and locally cartesian closed, and truncation localizations are locally cartesian, so $(\infty,1)$6-topoi are also presentable locally cartesian closed $(\infty,1)$7-categories (Gepner et al., 2012). The same paper also introduces certain $(\infty,1)$8-quasitopoi, described as full subcategories of $(\infty,1)$9-separated objects associated to a stable factorization system and a left-exact localization, and shows that these are again presentable locally cartesian closed $(\infty,1)$0-categories even though they need not be $(\infty,1)$1-topoi (Gepner et al., 2012).

A distinct construction starts not from a presheaf or localization presentation but from a given quasicategory $(\infty,1)$2. Using a higher version of the Yoneda embedding

$(\infty,1)$3

one constructs a simplicially enriched category $(\infty,1)$4 as a full simplicial subcategory of a simplicial model category $(\infty,1)$5, spanned by injective fibrant replacements of representables. The simplicial nerve $(\infty,1)$6 is equivalent to the original quasicategory $(\infty,1)$7, and when $(\infty,1)$8 is locally cartesian closed, $(\infty,1)$9 can be endowed with enough structure to provide a strict simplicial model of dependent type theory (Cherradi, 2022).

These two modes of construction serve different purposes. The localization picture is especially suited to structural and presentability arguments; the Yoneda-based rigidification provides a strict enriched presentation amenable to type-theoretic semantics (Gepner et al., 2012, Cherradi, 2022).

4. Type theory, contextual categories, and the semantic direction

A major source of locally cartesian closed quasicategories is Martin-Löf type theory. The paper "Locally Cartesian Closed Quasicategories from Type Theory" proves that the quasicategories arising from models of Martin-Löf type theory via simplicial localization are locally cartesian closed (Kapulkin, 2015). The setting is a contextual category equipped with $C\in sSet$0-, $C\in sSet$1-, and $C\in sSet$2-structure, together with the $C\in sSet$3-$C\in sSet$4 rule and functional extensionality. Writing

$C\in sSet$5

for a simplicial localization landing in quasicategories, the main theorem states that for any such categorical model $C\in sSet$6,

$C\in sSet$7

is a locally cartesian closed quasicategory (Kapulkin, 2015).

The proof proceeds through the quasicategory of frames $C\in sSet$8 of a fibration category $C\in sSet$9. Every categorical model $f:Y\to X$00 carries a fibration category structure in which fibrations are iterated dependent projections and weak equivalences are the bi-invertible maps. Szumiło’s construction yields a finitely complete quasicategory $f:Y\to X$01, and for any fibration category there is a canonical categorical equivalence

$f:Y\to X$02

The crucial further steps are that slicing is preserved, adjoints are preserved, and local cartesian closure can first be proved at the level of fibration categories and then transferred to quasicategories (Kapulkin, 2015).

At the fibration-category level, local cartesian closure means that for every fibration $f:Y\to X$03, the pullback functor

$f:Y\to X$04

admits a right adjoint $f:Y\to X$05 which is exact. For a dependent projection $f:Y\to X$06, the right adjoint is defined by

$f:Y\to X$07

with unit and counit supplied by the usual $f:Y\to X$08-$f:Y\to X$09 laws. In this way, dependent products in type theory become right adjoints to pullback in the associated quasicategory (Kapulkin, 2015).

The same source emphasizes the role of identity types and functional extensionality in matching weak equivalences and homotopies in the underlying fibration category with quasicategorical homotopies. This is why the semantic output is not merely finitely complete but locally cartesian closed in the full quasicategorical sense (Kapulkin, 2015).

One misconception addressed by this body of work is that type theory furnishes only syntactic examples of higher categories. The result of (Kapulkin, 2015) is more specific: the simplicial localizations of categorical models of intensional type theory carry the exact slice-adjoint structure required for local cartesian closure.

5. Internal languages and the DK-equivalence with $f:Y\to X$10-tribes

The converse direction, formulated as an internal-language theorem, is developed in "Internal languages of locally cartesian closed $f:Y\to X$11-categories" (Cherradi, 3 Sep 2025). The paper introduces $f:Y\to X$12-tribes as tribes $f:Y\to X$13 in which, for every fibration $f:Y\to X$14, the pullback functor

$f:Y\to X$15

admits a right adjoint $f:Y\to X$16, and $f:Y\to X$17 preserves anodyne maps. A tribe is a category $f:Y\to X$18 together with classes of maps, fibrations and anodyne maps, such that all isomorphisms and pullbacks of fibrations are fibrations, every map factors functorially as anodyne followed by fibration, and anodyne maps have the left lifting property with respect to all fibrations (Cherradi, 3 Sep 2025).

The main theorem is a Dwyer–Kan equivalence: $f:Y\to X$19 where $f:Y\to X$20 sends a $f:Y\to X$21-tribe $f:Y\to X$22 to the quasicategory obtained by simplicial localization of $f:Y\to X$23 inverting the trivial fibrations. A functor of relative categories is a DK-equivalence if it induces an equivalence on homotopy categories and weak equivalences on simplicial mapping spaces (Cherradi, 3 Sep 2025). Thus the theorem identifies $f:Y\to X$24-tribes and locally cartesian closed quasicategories as two presentations of the same homotopy theory.

The construction uses a frame functor

$f:Y\to X$25

to semi-cubical tribes, together with a canonical path object functor $f:Y\to X$26. It is shown that $f:Y\to X$27 and $f:Y\to X$28 admit fibration-category structures in which weak equivalences are the DK-equivalences, so that Cisinski’s approximation theorem applies. A rigidification lemma then states that if $f:Y\to X$29 is a $f:Y\to X$30-tribe functor preserving internal products up to weak equivalence, there is a span $f:Y\to X$31 in which both legs strictly preserve $f:Y\to X$32 (Cherradi, 3 Sep 2025). Essential surjectivity is obtained by constructing, for every locally cartesian closed quasicategory $f:Y\to X$33, a $f:Y\to X$34-tribe $f:Y\to X$35 whose localization is equivalent to $f:Y\to X$36, via “free fibration” and descent in the Rezk model (Cherradi, 3 Sep 2025).

From this follows one of the internal languages conjecture: Martin-Löf type theory with dependent sums, intensional identity types, and dependent products satisfying functional extensionality is the internal language of locally cartesian closed $f:Y\to X$37-categories (Cherradi, 3 Sep 2025). Equivalently, the composite

$f:Y\to X$38

is a DK-equivalence (Cherradi, 3 Sep 2025).

The correspondence of judgments is explicit. Contexts $f:Y\to X$39 correspond to objects of the tribe, types over $f:Y\to X$40 correspond to fibrations $f:Y\to X$41, terms correspond to sections, substitution is pullback, $f:Y\to X$42-, $f:Y\to X$43-, and $f:Y\to X$44-formation correspond respectively to dependent sums, path objects, and dependent products, and judgmental equality corresponds to homotopies in slices (Cherradi, 3 Sep 2025).

6. Univalence, higher topoi, and further consequences

Locally cartesian closed quasicategories are also the ambient setting for higher-categorical formulations of univalence. The paper "Univalence in locally cartesian closed infinity-categories" develops the basic theory of locally cartesian localizations of presentable locally cartesian closed $f:Y\to X$45-categories, establishes representability of equivalences, and shows that univalent families form a poset isomorphic to the poset of bounded local classes (Gepner et al., 2012). It follows that every $f:Y\to X$46-topos has a hierarchy of universal univalent families, indexed by regular cardinals, and that $f:Y\to X$47-topoi have univalent families classifying $f:Y\to X$48-truncated maps (Gepner et al., 2012).

Univalent families are preserved and detected by right adjoints to locally cartesian localizations, and this yields canonical univalent families in certain $f:Y\to X$49-quasitopoi (Gepner et al., 2012). The same work also shows that any presentable locally cartesian closed $f:Y\to X$50-category is modeled by a combinatorial type-theoretic model category, and conversely that the $f:Y\to X$51-category underlying a combinatorial type-theoretic model category is presentable and locally cartesian closed (Gepner et al., 2012). Under this correspondence, univalent families in presentable locally cartesian closed $f:Y\to X$52-categories correspond to univalent fibrations in combinatorial type-theoretic model categories (Gepner et al., 2012).

A related semantic strengthening appears in the Yoneda approach to interpreting type theory in a quasicategory. If $f:Y\to X$53 is an elementary higher topos, defined by finite limits and colimits, local cartesian closure, a subobject classifier $f:Y\to X$54, and object classifiers closed under $f:Y\to X$55, $f:Y\to X$56, and fibered limits, then the resulting simplicial category $f:Y\to X$57 admits, in addition to $f:Y\to X$58, $f:Y\to X$59, and $f:Y\to X$60, a strict subobject classifier and a univalent universe $f:Y\to X$61 classifying all small fibrations. Therefore $f:Y\to X$62 is a model of homotopy type theory with $f:Y\to X$63-types, $f:Y\to X$64-types, $f:Y\to X$65-types, finite coproducts, pushouts, $f:Y\to X$66-types whenever they exist in $f:Y\to X$67, one or a tower of univalent universes, and propositional resizing (Cherradi, 2022).

The internal-language paper states further corollaries in this direction: any locally cartesian closed $f:Y\to X$68-category admits an interpretation of HoTT with univalent $f:Y\to X$69-types and $f:Y\to X$70; transport along paths in the quasicategory corresponds to identity-type induction in the internal language; and synthetic homotopy constructions in a quasicategory $f:Y\to X$71 can be carried out in the corresponding type theory $f:Y\to X$72 (Cherradi, 3 Sep 2025). The last point indicates a methodological consequence rather than a change in foundational content: type-theoretic reasoning becomes available as a native calculus for general locally cartesian closed $f:Y\to X$73-categories.

These developments also delimit the scope of the theory. Some results require only the bare quasicategorical definition of local cartesian closure, while others assume presentability, elementary higher topos structure, or additional classifier hypotheses. A plausible implication is that “locally cartesian closed quasicategory” now functions less as a single isolated definition than as the common structural denominator linking slice exponentials, dependent type theory, and univalent semantics across several models of higher category theory.