Enriched accessible categories

Carmen Quinteiro – 4 June 1997

We consider category theory enriched in a locally finitely presentable symmetric monoidal closed category V. We define V-filtered colimits to be those colimits weighted by a V-flat presheaf and consider the corresponding notion of V-accessible category. We prove that the V-accessible categories coincide with the categories of V-flat presheaves, and also with the categories of V-points of categories of V-presheaves. The V-locally presentable categories are exactly the V-cocomplete accessible ones. We define V-sketches and we prove that the V-accessible categories are, exactly, the categories of models of a V-sketch.