TL;DR
This paper constructs categories of precosheaves and cosheaves valued in pro-k-modules on a Grothendieck site, proving their categorical properties and developing homology theories based on resolutions.
Contribution
It introduces and analyzes the categories of precosheaves and cosheaves with values in Pro(k), establishing their axiomatic properties and developing associated homology theories.
Findings
pCS(X,Pro(k)) satisfies AB4 and AB5* axioms
CS(X,Pro(k)) satisfies AB3 and AB5* axioms
Homology theories for cosheaves are constructed and studied
Abstract
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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Algebraic Geometry and Number Theory