externalization

Many notions internal to a category $C$ (e.g. internal groups) may be described instead in terms of functors (presheaves, pseudofunctors etc.) with domain $C$, fibrations over $C$ and so on. The process of replacing the internal structures on object or families of objects in $C$ by such “external” structures involving the whole category $C$ 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) $G$ internal to a finitely complete category $C$ 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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