Enrichment as Categorical Delooping I: Enrichment Over Iterated Monoidal Categories

TL;DR

This paper explores how categories enriched over k-fold monoidal categories form higher monoidal structures, revealing a categorical analogue of topological delooping and extending enrichment concepts.

Contribution

It introduces a framework connecting enrichment over k-fold monoidal categories with higher monoidal 2-categories, generalizing the notion of delooping in category theory.

Findings

V-Cat is symmetric only when V is symmetric.

Enrichment over a k-fold monoidal category yields a (k-1)-fold monoidal 2-category.

Defines the 3-category of categories enriched over V-Cat.

Abstract

The 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception is the case in which V is symmetric, which leads to V-Cat being symmetric as well. This paper describes how these facts are related to a categorical analogue of topological delooping. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k-fold monoidal categories and their higher dimensional counterparts. The main result is that for V a k-fold monoidal category, V-Cat becomes a (k-1)-fold monoidal 2-category in a canonical way. I indicate how this process may be iterated by enriching over V-Cat, along the way defining the 3-category of categories enriched over V-Cat.