# Directed Containers as Categories

arXiv: https://arxiv.org/abs/1604.01187 · HTML: https://mlsanity.com/paper/1604.01187/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2016.

## Built on by (most cited first, 2 of 2)

- [Learners' Languages](https://mlsanity.com/paper/2103.01189/) (2021)
- [Compositional Active Inference II: Polynomial Dynamics. Approximate Inference Doctrines](https://mlsanity.com/paper/2208.12173/) (2022)

## Abstract (arXiv)

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.