Cosheaf

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In topology, a branch of mathematics, a cosheaf is a dual notion to that of a sheaf that is useful in studying Borel-Moore homology.

Contents

Definition

We associate to a topological space its category of open sets , whose objects are the open sets of , with a (unique) morphism from to whenever . Fix a category . Then a precosheaf (with values in ) is a covariant functor , i.e., consists of

Suppose now that is an abelian category that admits small colimits. Then a cosheaf is a precosheaf for which the sequence

is exact for every collection of open sets, where and . (Notice that this is dual to the sheaf condition.) Approximately, exactness at means that every element over can be represented as a finite sum of elements that live over the smaller opens , while exactness at means that, when we compare two such representations of the same element, their difference must be captured by a finite collection of elements living over the intersections .

Equivalently, is a cosheaf if

Examples

A motivating example of a precosheaf of abelian groups is the singular precosheaf, sending an open set to , the free abelian group of singular -chains on . In particular, there is a natural inclusion whenever . However, this fails to be a cosheaf because a singular simplex cannot be broken up into smaller pieces. To fix this, we let be the barycentric subdivision homomorphism and define to be the colimit of the diagram

In the colimit, a simplex is identified with all of its barycentric subdivisions. One can show using the Lebesgue number lemma that the precosheaf sending to is in fact a cosheaf.

Fix a continuous map of topological spaces. Then the precosheaf (on ) of topological spaces sending to is a cosheaf. [2]

Notes

  1. Bredon, Glen E. (24 January 1997). Sheaf Theory. Springer. ISBN   9780387949055.
  2. Lurie, Jacob. "Tamagawa Numbers via Nonabelian Poincare Duality, Lecture 9: Nonabelian Poincare Duality in Algebraic Geometry" (PDF). School of Mathematics, Institute for Advanced Study.

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