Cellular cosheaf homology are cosheaf homology

arXiv2202.03659MaRDI QIDQ6390458

Daisuke Kishimoto, Yasutomo Yushima

Publication date: 8 February 2022

Abstract: A cosheaf is the dual notion of a sheaf, but we cannot define its homology as the formal dual of sheaf cohomology, in general, because of the lack of the cosheafification. A cellular cosheaf is a contravariant functor from the face poset of a CW complex to the category of abelian groups. We show that given a cellular cosheaf $F$, there is a natural way to associate a cosheaf $widehatF$, for which we can define homology as the formal dual of sheaf cohomology, such that the Borel-Moore homology of $F$ is isomorphic to the homology of $widehatF$ whenever the underlying CW complex of $F$ is a simplicial complex.

Mathematics Subject Classification ID

Sheaf cohomology in algebraic topology (55N30) Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) (18F20)

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