Locally bounded enriched categories

Abstract

We define and study the notion of a locally bounded enriched category over a (locally bounded) symmetric monoidal closed category, generalizing the locally bounded ordinary categories of Freyd and Kelly. In addition to proving several general results for constructing examples of locally bounded enriched categories and locally bounded closed categories, we demonstrate that locally bounded enriched categories admit fully enriched analogues of many of the convenient results enjoyed by locally bounded ordinary categories. In particular, we prove full enrichments of Freyd and Kelly's reflectivity and local boundedness results for orthogonal subcategories and categories of models for sketches and theories. We also provide characterization results for locally bounded enriched categories in terms of enriched presheaf categories, and we show that locally bounded enriched categories admit useful adjoint functor theorems and a representability theorem. We also define and study the notion of $\alpha$-bounded-small weighted limit enriched in a locally $\alpha$-bounded closed category, which parallels Kelly's notion of $\alpha$-small weighted limit enriched in a locally $\alpha$-presentable closed category, and we show that enriched categories of models of $\alpha$-bounded-small weighted limit theories are locally $\alpha$-bounded.