
TL;DR
This paper proves the existence of a coreflection called cosheafification for precosheaves on Grothendieck sites, providing explicit constructions and connections to shape theory, especially when values are in pro-objects.
Contribution
It introduces a cosheafification process for precosheaves valued in locally presentable categories, with explicit constructions for pro-objects and applications to topological spaces.
Findings
Existence of cosheafification for precosheaves in certain categories.
Any precosheaf with values in pro-objects is smooth and locally isomorphic to a cosheaf.
Connections established between cosheaves and shape theory.
Abstract
It is proved that for any Grothendieck site , there exists a coreflection (called ) from the category of precosheaves on with values in a category , to the full subcategory of cosheaves, provided either or is locally presentable. If is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category of pro-objects in . In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in is , i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Algebraic structures and combinatorial models
