It is proved that for any small Grothendieck site X, there exists a coreflection (called \emph{cosheafification}) from the category of precosheaves on X with values in a category $K$, to the full subcategory of cosheaves, provided either $K$ or $K^{op}$ is locally presentable. If $K$ is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category $Pro(K)$ of pro-objects in $K$. In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in $Pro(K)$ is smooth, i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.
Keywords: Cosheaves, smooth precosheaves, cosheafification, pro-category, cosheaf homology, locally presentable categories, accessible categories
2010 MSC: Primary 18F10, 18F20; Secondary 55P55, 55Q07, 14F20
Theory and Applications of Categories, Vol. 31, 2016, No. 38, pp 1134-1175.
Published 2016-12-31.
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