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Cosheafification

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Andrei V. Prasolov

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.

http://www.tac.mta.ca/tac/volumes/31/38/31-38.pdf

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