PreprintCC BY-NC-SA 4.0arXiv
Groupoidal polygraphic homology
Abstract
We show that for a 1-category C, the (ω, k)-polygraphic homology of C for any k {\geq} 1, that is taken with cofibrant resolutions in strict (ω, k)- categories, does not depend on k and is canonically isomorphic to the homology of the classifying space of C. When C is a groupoid, we also show this for k = 0. In particular, this means that the classical homology of groups can be obtained by taking cofibrant resolutions in strict ω-groupoids. In order to show these results, we develop the theory of discrete Conduché fibrations in the category of strict (ω, k)-categories, building on previous work by the first-named author.
Related
BibTeX
@misc{guetta2026groupoidal,
title = {{Groupoidal polygraphic homology}},
author = {Léonard Guetta and François Métayer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29967},
eprint = {2609.29967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29967},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Groupoidal polygraphic homology
AU - Léonard Guetta
AU - François Métayer
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29967
UR - https://arxiv.org/abs/2609.29967
AB - We show that for a 1-category C, the (ω, k)-polygraphic homology of C for any k {\geq} 1, that is taken with cofibrant resolutions in strict (ω, k)- categories, does not depend on k and is canonically isomorphic to the homology of the classifying space of C. When C is a groupoid, we also show this for k = 0. In particular, this means that the classical homology of groups can be obtained by taking cofibrant resolutions in strict ω-groupoids. In order to show these results, we develop the theory of discrete Conduché fibrations in the category of strict (ω, k)-categories, building on previous work by the first-named author.
ER -