Characterizing (Co)Free Dagger Categories
Abstract
For any category, there exists both a free dagger category and a cofree dagger category over it. A natural question to ask is: given a dagger category, how can we tell if it is free or cofree without specifying an external base category? In this paper, we provide characterizations of both free dagger categories and cofree dagger categories via internal dagger category structure. To characterize cofree dagger categories, we use rectangular bands and show that a dagger category is cofree if and only if it is enriched over rectangular bands. For free dagger categories, we define the notion of a zigzag dagger category, and then show that a dagger category is free if and only if it is a zigzag dagger category. We also show that free dagger categories can be characterized as the coalgebras of the induced comonad from the free dagger category adjunction, and similarly that cofree free dagger categories can be characterized as the algebras of the induced monad from the cofree dagger category adjunction.
Related
- Models for dagger (∞,1)-categories III: Univalent flagged dagger (∞,1)-categories and unitarily complete dagger Segal anima
- Models for dagger (∞,1)-categories I: Dagger simplicial sets and unitary cores
- Models for dagger (∞,1)-categories II: Dagger complete Segal spaces and the Joyal--Tierney equivalence
- Dagger Categories in Riemannian Geometry
- The coanalyticity of the reduced K-theory on modules over free tensor algebras of dualizable t-categories
BibTeX
@misc{lemay2026characterizing,
title = {{Characterizing (Co)Free Dagger Categories}},
author = {Jean-Simon Pacaud Lemay},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14285},
eprint = {2609.14285},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14285},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Characterizing (Co)Free Dagger Categories
AU - Jean-Simon Pacaud Lemay
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14285
UR - https://arxiv.org/abs/2609.14285
AB - For any category, there exists both a free dagger category and a cofree dagger category over it. A natural question to ask is: given a dagger category, how can we tell if it is free or cofree without specifying an external base category? In this paper, we provide characterizations of both free dagger categories and cofree dagger categories via internal dagger category structure. To characterize cofree dagger categories, we use rectangular bands and show that a dagger category is cofree if and only if it is enriched over rectangular bands. For free dagger categories, we define the notion of a zigzag dagger category, and then show that a dagger category is free if and only if it is a zigzag dagger category. We also show that free dagger categories can be characterized as the coalgebras of the induced comonad from the free dagger category adjunction, and similarly that cofree free dagger categories can be characterized as the algebras of the induced monad from the cofree dagger category adjunction.
ER -