## Definable Enriched Categories

### Giacomo Tendas – 1 May 2019

Picking up where we left off at my previous talk, we introduce an enriched notion of definable category. Then, if our base for enrichment is a symmetric monoidal finitary variety, we can prove a duality between the 2-category V-Ex of small exact V-categories, and V-Def of definable V-categories, generalizing the ordinary result of Kuber-Rosicky, and the additive version of Prest-Rajani. We then use this to give an explicit description of the free exact completions over finitely complete V-categories and over regular V-categories.

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