locally internal category


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.

Given a topos EE and an object XX in EE, one notices that the slice category E/XE/X is a symmetric monoidal category; hence we can consider categories enriched over E/XE/X, i.e. E/XE/X-categories.

A locally internal category CC over EE is given by

  • an E/XE/X-category C XC_X for each object XX in EE (which is thought of as being XX-indexed families of objects of EE)
  • for each morphism f:XYf: X\to Y in EE an E/XE/X-full embedding θ f:f *C YC X\theta_f: f^* C_Y\to C_X such that fθ ff\mapsto \theta_f is functorial up to coherent isomorphisms

In the stack semantics of EE, 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 EE which satisfy a certain “representability” or “comprehensibility” condition:

A Grothendieck fibration p:CEp: C \to E is called locally small if, for every pair A,BCA,B \in C, there exists an object of E pA×pBE_{pA \times pB}, (x,y):IpA×pB(x,y) : I \to pA \times pB, and a morphism f:x *Ay *BC If: x^*A \to y^*B \in C_I, which is terminal, in the sense that given another such datum (J,z,w,g)(J,z,w,g), there is a unique map u:JIu: J \to I so that xu=z,yu=wxu = z, yu = w, and the coherence isomorphisms identify u *fu^*f with gg. (This is Elephant B.1.3.12).

An indexed category ECATE \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 EE-indexed categories (equivalently, Grothendieck fibrations over EE) 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 SS be a locally cartesian closed category, let F:SSF:S\to S be an SS-enriched functor whose underlying (ordinary) functor preserves pullbacks. Then FF extends to an SS-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.

Of course, EE does not have to be a topos. For the definition, it suffices for EE to have finite limits, although the notion is best-behaved when EE is locally cartesian closed (for instance, in that case the codomain fibration of EE is an example).


  • 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

Last revised on May 28, 2021 at 05:14:06. See the history of this page for a list of all contributions to it.