Many notions internal to a category CC (e.g. internal groups) may be described instead in terms of functors (presheaves, pseudofunctors etc.) with domain CC, fibrations over CC and so on. The process of replacing the internal structures on object or families of objects in CC by such “external” structures involving the whole category CC is called the externalization. Sometimes external definitions give large versions (in the sense of set-theoretic size) of some internal notions.

For example, a groupoid (or even a category) GG internal to a finitely complete category CC gives rise to a Grothendieck fibration. A Grothendieck fibration equivalent to an externalization of an internal category is called small fibration.

  • J. Duskin, An outline of non-abelian cohomology in a topos : (I) The theory of bouquets and gerbes, Cahiers de Topologie et Géométrie Différentielle Catégoriques 23 no. 2 (1982), 165–191 numdam

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