Dualities between Cellular Sheaves and Cosheaves

TL;DR

This paper proves a conjecture that the derived categories of cellular sheaves and cosheaves are equivalent, providing new insights into their dualities and applications to sheaf cohomology.

Contribution

It establishes a derived equivalence between cellular sheaves and cosheaves, confirming MacPherson's conjecture and offering a new perspective on sheaf cohomology.

Findings

Derived category of cellular sheaves is equivalent to that of cellular cosheaves.

Classical dualities induce an exchange between sheaves and cosheaves.

New description of compactly supported sheaf cohomology using the equivalence.

Abstract

This paper affirms a conjecture of MacPherson: that the derived category of cellular sheaves is equivalent to the derived category of cellular cosheaves. We give a self-contained treatment of cellular sheaves and cosheaves and note that certain classical dualities give rise to an exchange of sheaves with cosheaves. Following a result of Pitts that states that cosheaves are cocontinuous functors on the category of sheaves, we use the derived equivalence provided here to gain a novel description of compactly supported sheaf cohomology.