Cosheaves, an introduction

Wednesday 16 March 2016 h. 14:30, Room 2BC30
Pietro Polesello (Dip. Mat.)
"Cosheaves, an introduction"

Abstract
It is well known that locally defined distributions glue together, that is, they define a sheaf. In fact, this follows immediately from the fact that test functions (i.e. smooth functions with compact support) form a cosheaf, which is the dual notion of a sheaf. By definition, cosheaves on a space X and with values in category C are dual to sheaves on X with values in the opposite category C'. For this reason, cosheaves did not attract much attention, being considered as part of sheaf theory. However, passing from C to C', may cause difficulties, as in general C and C' do not share the same good properties needed for sheaf theory. Moreover, dealing with cosheaves may be more convenient, as they appear naturally in analysis (as the compactly supported sections of c-soft sheaves, such as smooth functions or distributions), in algebraic analysis (e.g. as the subanalytic cosheaf of Schwartz functions), in topology (in relation with Fox's theory of topological branched coverings), and in tops theory. Moreover, as sheaves are the natural coefficient spaces for cohomology theories, cosheaves play the same role for homology theories, such as Cech homology, and they are (hidden) ingredients of Poincare' duality (recently, cosheaves infiltrated Poincare'-Verdier duality in the context of Lurie's "higher topos theory"). In this seminar, I will give a brief introduction to cosheaves, giving examples and explaining the relation with sheaves and with Fox's theory.