The functoriality of moves on graphs and extended path homomorphisms
Abstract
Combinatorics of graphs is a very powerful tool to unravel various properties of graph algebras. In particular, isomorphisms between graph algebras are often implemented by moves between their graphs. In this paper, we make these combinatorial methods functorial. Our categorical approach not only reveals the relationship between moves, but also leads to unexpected extensions of known moves (shifts and reductions) and the characterization of out-splits as admissible epimorphisms admitting a lifting of the source map. The former allowed us to go beyond isomorphisms and Morita equivalences, and the latter yielded a factorization lemma for out-splits. Better still, the functorial understanding of maximal out-splits permitted combining them with the derived-graph construction, which resulted in computing new proper endomorphisms of Cuntz algebras. Finally, the functorial implementation of balanced in-splits required the construction of a new category of graphs, where morphisms are given as homomorphisms of graph inverse semigroups. Thus, the combinatorics of moves brought a new tool to the realm of noncommutative topology where directed graphs are viewed as quantum spaces.
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BibTeX
@misc{castro2026functoriality,
title = {{The functoriality of moves on graphs and extended path homomorphisms}},
author = {Gilles G. de Castro and Piotr M. Hajac and Mateusz Lowiel and Elizabeth A. Pacheco},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.15439},
eprint = {2606.15439},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.15439},
}
RIS
TY - UNPB
TI - The functoriality of moves on graphs and extended path homomorphisms
AU - Gilles G. de Castro
AU - Piotr M. Hajac
AU - Mateusz Lowiel
AU - Elizabeth A. Pacheco
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2606.15439
UR - https://arxiv.org/abs/2606.15439
AB - Combinatorics of graphs is a very powerful tool to unravel various properties of graph algebras. In particular, isomorphisms between graph algebras are often implemented by moves between their graphs. In this paper, we make these combinatorial methods functorial. Our categorical approach not only reveals the relationship between moves, but also leads to unexpected extensions of known moves (shifts and reductions) and the characterization of out-splits as admissible epimorphisms admitting a lifting of the source map. The former allowed us to go beyond isomorphisms and Morita equivalences, and the latter yielded a factorization lemma for out-splits. Better still, the functorial understanding of maximal out-splits permitted combining them with the derived-graph construction, which resulted in computing new proper endomorphisms of Cuntz algebras. Finally, the functorial implementation of balanced in-splits required the construction of a new category of graphs, where morphisms are given as homomorphisms of graph inverse semigroups. Thus, the combinatorics of moves brought a new tool to the realm of noncommutative topology where directed graphs are viewed as quantum spaces.
ER -