Categories enriched over symmetric closed multicategories

Abstract

We construct a machine which takes as input a locally small symmetric closed complete multicategory $\mathsf{V}$. And its output is again a locally small symmetric closed complete multicategory $\mathsf{V}\text{-}\mathcal{C}at$, the multicategory of small $\mathsf{V}$-categories and multi-entry $\mathsf{V}$-functors. An example of such $\mathsf{V}$ is provided by short spaces (vector spaces with a system of seminorms) and short maps. When the ground multicategory $\mathsf{V}$ is $\mathsf{Set}$ we obtain strict 2-categories and their surroundings by iterating twice the construction of categories.

76 pp, diagrams.sty (version 3.96). Missing information for proofs can be recovered with some work or with the help of version 4. Corrected typos and typographically corrected