Higher Dimensional Enrichment

TL;DR

This paper generalizes the concept of enriched 2-categories to higher dimensions using k-fold monoidal categories, providing a recursive framework and establishing a 3-category structure for V-2-categories.

Contribution

It introduces a recursive definition of V-n-categories for higher dimensions and proves that V-2-categories form a 3-category structure.

Findings

Generalization of enriched 2-categories to V-n-categories

Recursive definition framework for V-n-categories

Proof that V-2-categories form a 3-category

Abstract

Lyubashenko has described enriched 2-categories as categories enriched over V-Cat, the 2-category of categories enriched over a symmetric monoidal V. I have generalized this to the k-fold monoidal V. The symmetric case can easily be recovered. The introduction of this paper proposes a recursive definition of V-n-categories and their morphisms. Then I consider the special case of V-2-categories and give the details of the proof that with their morphisms these form the structure of a 3-category.