Lefschetz fixed-object and fixed-morphism theorems for acyclic categories
Algebraic Topology
2024-04-11 v1 Category Theory
Abstract
We introduce two novel complementary notions of the Lefschetz number for a functor from a finite acyclic category to itself and we prove a Lefschetz fixed-object theorem and a Lefschetz fixed-morphism theorem. In order to do so, we use the connection between these type of categories and simplicial structures, such as trisps or delta complexes. Through the use of a pair of functors that, when composed, form the barycentric subdivision, we are not only able to identify fixed objects but also fixed chains of morphisms.
Cite
@article{arxiv.2404.06573,
title = {Lefschetz fixed-object and fixed-morphism theorems for acyclic categories},
author = {Samuel Castelo-Mourelle and Enrique Macías-Virgós and David Mosquera-Lois},
journal= {arXiv preprint arXiv:2404.06573},
year = {2024}
}
Comments
8 pages with 3 figures