Cellular sheaf cohomology in Polymake

TL;DR

This paper introduces cellular sheaves and cosheaves on polyhedral complexes, details their cohomology, and presents an extension for polymake with applications in toric and tropical geometry.

Contribution

It provides the first implementation of cellular sheaves in polymake, linking geometric and algebraic cohomologies with practical computational tools.

Findings

Extension enables computation of cellular sheaf cohomologies in polymake

Links between cellular sheaf cohomologies and algebraic variety cohomologies

Illustrative examples from toric and tropical geometry

Abstract

This chapter provides a guide to our polymake extension cellularSheaves. We first define cellular sheaves on polyhedral complexes in Euclidean space, as well as cosheaves, and their (co)homologies. As motivation, we summarise some results from toric and tropical geometry linking cellular sheaf cohomologies to cohomologies of algebraic varieties. We then give an overview of the structure of the extension cellularSheaves for polymake. Finally, we illustrate the usage of the extension with examples from toric and tropical geometry.