Cosheaf

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.

[further explanation needed] We associate to a topological space

its category of open sets

, whose objects are the open sets of

, with a (unique) morphism from

Fix a category

Then a precosheaf (with values in

) is a covariant functor

consists of Suppose now that

is an abelian category that admits small colimits.

Then a cosheaf is a precosheaf

of open sets, where

(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

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

is a cosheaf if A motivating example of a precosheaf of abelian groups is the singular precosheaf, sending an open set

, the free abelian group of singular

In particular, there is a natural inclusion

However, this fails to be a cosheaf because a singular simplex cannot be broken up into smaller pieces.

be the barycentric subdivision homomorphism and define

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

is in fact a cosheaf.

Fix a continuous map

of topological spaces.

) of topological spaces sending

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