On Structures in Arrow Categories

TL;DR

This paper explores how various categorical structures such as monoidal, rigid, pivotal, and algebraic objects are inherited or characterized within arrow categories, providing new insights into their structural properties.

Contribution

It demonstrates that monoidal equivalences extend to arrow categories and characterizes algebraic objects like (co)algebras and Hopf algebras within arrow categories.

Findings

Monoidal equivalences induce monoidal equivalences in arrow categories

Conditions under which arrow categories are rigid and pivotal are identified

Descriptions of (co)algebra, bialgebra, and Hopf algebra objects in arrow categories

Abstract

In this article we investigate which categorical structures of a category C are inherited by its arrow category. In particular, we show that a monoidal equivalence between two categories gives rise to a monoidal equivalence between their arrow categories. Moreover, we examine under which circumstances an arrow category is rigid and pivotal. Finally, we derive what the (co)algebra, bialgebra and Hopf algebra objects are in an arrow category.