The categories pCS(X,Pro(k)) of precosheaves and CS(X,Pro(k)) of cosheaves on a small Grothendieck site X, with values in the category Pro(k) of pro-k-modules, are constructed. It is proved that pCS(X,Pro(k)) satisfies the AB4 and AB5* axioms, while CS(X,Pro(k)) satisfies AB3 and AB5*. Homology theories for cosheaves and precosheaves, based on quasi-projective resolutions, are constructed and investigated.
Keywords: Cosheaves, precosheves, cosheafification, pro-category, cosheaf homology, precosheaf homology, Čech homology, shape theory, pro-homology, pro-homotopy, locally presentable categories.
2020 MSC: Primary 18F10, 18F20, 18G05, 18G10; Secondary 55P55, 55Q07, 14F20
Theory and Applications of Categories, Vol. 37, 2021, No. 33, pp 1080-1148.