Internal enriched categories

TL;DR

This paper introduces a unified theory of enrichment over internal monoidal categories, connecting and generalizing existing concepts of enriched, internal, indexed, and multicategory theories.

Contribution

It develops a new framework for internal enrichment that encompasses and relates various known generalizations of category enrichment.

Findings

Internal enrichment generalizes standard and internal categories.

Connections established between internal enrichment, indexed categories, and multicategories.

Provides a unified perspective on different enrichment theories.

Abstract

We introduce the theory of enrichment over an internal monoidal category as a common generalization of both the standard theories of enriched and internal categories. The aim of the paper is to justify and contextualize the new notion by comparing it to other known generalizations of enrichment: namely, those for indexed categories and for generalized multicategories. It turns out that both of these notions are closely related to internal enrichment and, as a corollary, to each other.