A Morse Mayer-Vietoris Sequence for Cosheaf Homology on Simplicial Complexes

TL;DR

This paper develops a Morse Mayer-Vietoris sequence for cosheaf homology on simplicial complexes, linking discrete Morse theory with classical homological tools to enhance understanding of cosheaf structures.

Contribution

It introduces a Morse Mayer-Vietoris sequence for cosheaf homology, establishing a quasi-isomorphism with the standard sequence and connecting categorical perspectives with discrete Morse theory.

Findings

Constructed a Morse Mayer-Vietoris sequence for cosheaf homology.

Proved a quasi-isomorphism between Morse and standard sequences.

Connected categorical and topological perspectives on cosheaves.

Abstract

We investigate the homology of cosheaves over finite simplicial complexes. After constructing the Mayer-Vietoris short exact sequence for this homology theory, we apply discrete Morse theory to this setting, defining the associated Morse chain complex. The main result is the construction of a Morse Mayer-Vietoris short exact sequence of Morse chain complexes, for which we establish a quasi-isomorphism to the standard Mayer-Vietoris sequence. Discussions relating poset-based (co)sheaf definitions to topological ones via Kan extensions and exploring a modern categorical perspective on discrete Morse theory are also included.