Applications of enriched categories

Vincent Schmitt – 16 August 2000

Lawvere introduced the notion Cauchy completion for enrichments over V where V is a moinoidal closed category. He showed how this completion process captures the common notion of metric space completion. I want to present the following results. Quasi uniform spaces may be represented as enrichments over monoidal categories. Moreover (quasi-)uniformly continuous may be also represented by enriched functors. The quasi-uniform space completion is also a Cauchy-completion. To obtain these results I needed to introduce ``super'' monoidal functors. A 2-category with objects enrichments over diffrent bases needed be defined. In this general context, the Cauchy-completion still corresponds to a reflection.