Cosheaf - Alchetron, The Free Social Encyclopedia

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 lim → ⁡ F ( U i ) → F ( U ) is an equivalence.

(3) F ( U ∪ V ) is the pushout of F ( U ∩ V ) → F ( U ) and F ( U ∩ V ) → F ( V ) .

The basic example is U ↦ C ∗ ( U ; A ) where on the right is the singular chain complex of U with coefficients in an abelian group A.

Example: If f is a continuous map, then U ↦ f − 1 ( U ) is a cosheaf.

References

Cosheaf Wikipedia

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