Directed Containers as Categories

TL;DR

This paper explores the structure of directed containers, revealing their equivalence to small categories and deriving implications for comonads, including constructions like opposite categories and groupoids.

Contribution

It establishes an equivalence between directed containers and small categories, providing new insights into comonad structures and their categorical properties.

Findings

Directed containers are equivalent to small categories.

Directed container morphisms resemble monoid morphisms in reverse.

Implications for comonads include constructions like opposite categories and groupoids.

Abstract

Directed containers make explicit the additional structure of those containers whose set functor interpretation carries a comonad structure. The data and laws of a directed container resemble those of a monoid, while the data and laws of a directed container morphism those of a monoid morphism in the reverse direction. With some reorganization, a directed container is the same as a small category, but a directed container morphism is opcleavage-like. We draw some conclusions for comonads from this observation, considering in particular basic constructions and concepts like the opposite category and a groupoid.