Cosheaf

In topology, a branch of mathematics, a cosheaf with values in an ∞-category C that admits colimit is a functor F from the category of open subsets of a topological space X (more precisely its nerve) to C such that

(1) The F of the empty set is the initial object.
(2) For any increasing sequence $U_{i}$ of open subsets with union U, the canonical map $\varinjlim F(U_{i})\to F(U)$ is an equivalence.
(3) $F(U\cup V)$ is the pushout of $F(U\cap V)\to F(U)$ and $F(U\cap V)\to F(V)$ .

The basic example is $U\mapsto C_{*}(U;A)$ where on the right is the singular chain complex of U with coefficients in an abelian group A.

Example:^[1] If f is a continuous map, then $U\mapsto f^{{-1}}(U)$ is a cosheaf.

Notes

↑ http://www.math.harvard.edu/~lurie/282ynotes/LectureIX-NPD.pdf

References

http://www.math.harvard.edu/~lurie/282ynotes/LectureVIII-Poincare.pdf
http://arxiv.org/pdf/1303.3255v1.pdf , section 3, in particular Thm 3.10 p.34

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Cosheaf

See also

Notes

References