cosheaf

In topology, a branch of mathematics, a cosheaf with values in an ∞-category ''C'' that admits colimits 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:http://www.math.harvard.edu/~lurie/282ynotes/LectureIX-NPD.pdf If ''f'' is a continuous map, then $U \mapsto f^(U)$ is a cosheaf.

See also

*sheaf (mathematics)

Notes

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

Algebraic topology
Category theory
Sheaf theory

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