nLab base of enrichment

Idea

In enriched category theory, a base of enrichment is some kind of category theoretic structure $V$ – most often a monoidal category, but potentially something more general like a bicategory, virtual double category, or skew-monoidal category – over which one intends to consider $V$-enriched categories.

Frequently, but not always, $V$ will be a Bénabou cosmos, which provides sufficient infrastructure to carry out enriched versions of most of the standard category theoretic constructions, for example of enriched functor categories, tensor products of enriched categories, enriched presheaf categories, Eilenberg-Moore categories, specific weighted limits and weighted colimits, and so on.

In this context, “change of base” or “change of base of enrichment”, refers to a 2-functor $V\text{-}Cat \longrightarrow W\text{-}Cat$ between (very large) categories of enriched categories that is induced by a lax monoidal functor $V \to W$ between bases of enrichment. See also the section Base change at enriched category.

References

• Max Kelly, p. 9 of: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, pdf]

• Francis Borceux, §6.4 of: Handbook of Categorical Algebra Vol 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]