Factorizable enriched categories and applications

TL;DR

This paper introduces a generalized construction called the twisted tensor product for enriched categories, unifying various algebraic product concepts through simple twisting systems and matched pairs.

Contribution

It defines the twisted tensor product of enriched categories, generalizing known algebraic products, and introduces simple twisting systems and matched pairs for concrete examples.

Findings

Unified framework for algebraic products via twisted tensor products

Examples include categories, posets, and groupoids

Connections to bicrossed products and double cross products

Abstract

We define the twisted tensor product of two enriched categories, which generalizes various sorts of `products' of algebraic structures, including the bicrossed product of groups, the twisted tensor product of (co)algebras and the double cross product of bialgebras. The key ingredient in the definition is the notion of simple twisting systems between two enriched categories. To give examples of simple twisted tensor products we introduce matched pairs of enriched categories. Several other examples related to ordinary categories, posets and groupoids are also discussed.