Enriched Duality in Double Categories: V-categories and V-cocategories

TL;DR

This paper develops a double categorical framework for V-categories and V-cocategories, generalizing the Sweedler theory, and explores their enrichment, duality, and fibration structures using V-matrices and proarrow equipments.

Contribution

It introduces a novel double categorical approach to V-categories and V-cocategories, extending the Sweedler theory and analyzing their enriched fibration structures.

Findings

Established an enrichment of V-categories in V-cocategories.

Demonstrated duality between categories and cocategories via double categories.

Applied the framework to categories of monads and comonads in fibrant double categories.

Abstract

In this work, we explore a double categorical framework for categories of enriched graphs, categories and the newly introduced notion of cocategories. A fundamental goal is to establish an enrichment of V-categories in V-cocategories, which generalizes the so-called Sweedler theory relatively to an enrichment of algebras in coalgebras. The language employed is that of V-matrices, and an interplay between the double categorical and bicategorical perspective provides a high-level flexibility for demonstrating essential features of these dual structures. Furthermore, we investigate an enriched fibration structure involving categories of monads and comonads in fibrant double categories a.k.a. proarrow equipments, which leads to natural applications for their materialization as categories and cocategories in the enriched matrices setting.