## Locally bounded enriched categories

### Jason Parker – 10 November 2021

We define and study the notion of a locally bounded category enriched over a (locally bounded) symmetric monoidal closed category, generalizing the locally bounded ordinary categories of Freyd and Kelly. Along with providing many new examples of 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 show that locally bounded enriched categories satisfy useful representability and adjoint functor theorems. We also define and study the notion of an alpha-bounded-small limit with respect to a locally alpha-bounded closed category, which parallels Kelly’s notion of alpha-small limit with respect to a locally alpha-presentable closed category. This is joint work with Rory Lucyshyn-Wright at Brandon University, contained in the following preprint: https://arxiv.org/abs/2110.07072.

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