A locally internal category is an analogue of a large but locally small category relative to an elementary topos, when that topos is thought of as generalizing the category of sets.
More generally, locally internal categories can be defined over any category with finite limits. The notion is best-behaved when $E$ is locally cartesian closed (for instance, in that case the codomain fibration of $E$ is an example).
Given a category $E$ with finite limits and an object $X$ in $E$, one notices that the slice category $E/X$ is a symmetric monoidal category under fiber product in $E$. Hence we can consider categories enriched over $E/X$, i.e. $E/X$-categories.
A locally internal category $C$ over $E$ is given by
An $(E/X)$-enriched category $C_X$ for each object $X$ in $E$. This is thought of as the category of $X$-indexed families of objects of $E$.
For each morphism $f \colon X\to Y$ in $E$, an $(E/X)$-full embedding $\theta_f \colon f^* C_Y\to C_X$. Here $f^* C_Y$ means the $(E/X)$-enriched category obtained by applying the symmetric monoidal functor $f^* \colon E/Y \to E/X$ to the hom-objects of the $(E/Y)$-enriched category $C_Y$, and an enriched functor is a “full embedding” if it induces isomorphisms on hom-objects.
$f \mapsto \theta_f$ is functorial up to coherent isomorphism. This means certain diagrams commute analogous to those of a pseudofunctor, but with the functors $f^*$ applied at appropriate places to make them typecheck.
Alternatively, a category locally internal to $E$ is a category enriched in the bicategory $Span(E)$ admitting certain absolute colimits (see Betti–Walters).
In the stack semantics of $E$, a locally internal category “looks like” an ordinary locally small category.
Locally internal categories can also be identified with Grothendieck fibrations or indexed categories over $E$ which satisfy a certain “representability” or “comprehensibility” condition:
A Grothendieck fibration $p: C \to E$ is called locally small if, for every pair $A,B \in C$, there exists an object of $E_{pA \times pB}$, $(x,y) : I \to pA \times pB$, and a morphism $f: x^*A \to y^*B \in C_I$, which is terminal, in the sense that given another such datum $(J,z,w,g)$, there is a unique map $u: J \to I$ so that $xu = z, yu = w$, and the coherence isomorphisms identify $u^*f$ with $g$. (This is Elephant B.1.3.12).
An indexed category $E \to \operatorname{CAT}$ is called locally small if the associated fibration is locally small.
If we also take care of the appropriate morphisms have the following:
(1) The obvious forgetful functor from locally internal categories to $E$-indexed categories (equivalently, Grothendieck fibrations over $E$) is a fully faithful 2-functor. In particular, every indexed functor between locally internal categories is an enriched functor. Elephant, Proposition B2.2.2.
(2a) Let $S$ be a locally cartesian closed category, let $F:S\to S$ be an $S$-enriched functor whose underlying (ordinary) functor preserves pullbacks. Then $F$ extends to an $S$-indexed functor.
(2b) (Robert Pare) If this indexed functor preserves pullbacks (as an indexed functor) and if it induces the given enrichment, this extension is unique (up to a canonical isomorphism). Elephant B2.2.8.
J. Penon, Categories localement internes, C. R. Acad. Sci. Paris 278 (1974) A1577-1580
Locally internal categories, Appendix in: P. Johnstone, Topos theory, 1977
Chapter B2.2 of Sketches of an Elephant
Renato Betti, Robert F. C. Walters, Closed bicategories and variable category theory, Universita degli Studi di Milano (1985), reprinted in: Reprints in Theory and Applications of Categories, 26 (2020) 1-27 $[$tac:tr26$]$
Last revised on May 26, 2022 at 10:20:17. See the history of this page for a list of all contributions to it.