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.