Cellular Cohomology in Homotopy Type Theory

TL;DR

This paper develops cellular cohomology within homotopy type theory, enabling easier computation of cohomology groups for finite cell complexes, formalized in Agda.

Contribution

It formalizes cellular cohomology in homotopy type theory and proves its equivalence to other cohomology theories for finite cell complexes.

Findings

Cellular cohomology can be computed from combinatorial data.

The formalization was completed in the Agda proof assistant.

Cohomology theories like Eilenberg-MacLane are included.

Abstract

We present a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage over homotopy groups in that these abelian groups of many common spaces are easier to compute. Cellular cohomology is a special kind of cohomology designed for cell complexes: these are built in stages by attaching spheres of progressively higher dimension, and cellular cohomology defines the groups out of the combinatorial description of how spheres are attached. Our main result is that for finite cell complexes, a wide class of cohomology theories (including the ones defined through Eilenberg-MacLane spaces) can be calculated via cellular cohomology. This result was formalized in the Agda proof assistant.