Some aspects of cosheaves on diffeological spaces

Alireza Alireza Ahmadi,Akbar Dehghan Nezhad

doi:10.29252/cgasa.12.1.123

Abstract

We define a notion of cosheaves on diffeological spaces by cosheaves on the site of plots. This provides a framework to describe diffeological objects such as internal tangent bundles, the Poincar'{e} groupoids, and furthermore, homology theories such as cubic homology in diffeology by the language of cosheaves. We show that every cosheaf on a diffeological space induces a cosheaf in terms of the D-topological structure. We also study quasi-cosheaves, defined by pre-cosheaves which respect the colimit over covering generating families, and prove that cosheaves are quasi-cosheaves. Finally, a so-called quasi-v{C}ech homology with values in pre-cosheaves is established for diffeological spaces.

Highlights

We investigate cosheaves on diffeological spaces, defined as cosheaves on the site of plots (Definition 3.2)
One can suggest a version of other homology theories defined on manifolds for diffeological spaces
While a cosheaf on a diffeological space, is in essence, nothing more than an assignment of ordinary cosheaves to plots, we prove that every cosheaf on a diffeological space gives rise to a cosheaf with respect to the D-topology of the space (Theorem 6.2)

Summary

Introduction

This result may be considered as the counterpart of the van Kampen’s Theorem in diffeology In this manner, one can suggest a version of other homology theories defined on manifolds for diffeological spaces. Quasi-Čech homology can be regarded as the diffeological counterpart of Čech homology This provides a combinatorial approach to determine further data on a diffeological space form given data over covering generating families

Preliminaries

Cosheaves on diffeological spaces

The Poincaré groupoids as cosheaves

Cosheaves and D-topology

Quasi-cosheaves and quasi-Čech homology