2022/02/08 by Daisuke Kishimoto, Kishimoto, Daisuke, Yasutomo Yushima +1
Mathematics · Medicine · #18F20 #55N30 #Algebraic Topology (math.AT) #Botulinum Toxin and Related Neurological Disorders #Category Theory (math.CT) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2202.03659
openalex publication_date 2022/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.