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.

See also

• sheaf (mathematics)

Notes

1. 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