nLab externalization

Idea

Many notions internal to a category $C$ (e.g.. internal groups) may be described alternatively 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, an internal groupoid (or even an internal category) $G$ in a finitely complete category $C$ gives rise to a Grothendieck fibration. A Grothendieck fibration equivalent to the externalization of an internal category is called small fibration.

References

• John 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