Classifying covering types in homotopy type theory
This work provides a formalization of a fundamental result in algebraic topology within homotopy type theory, which is significant for the foundations of mathematics and computer science.
The authors formalize the Galois correspondence between covering spaces and subgroups of the fundamental group in homotopy type theory, and develop an n-dimensional generalization of covering spaces. They demonstrate applicability by classifying coverings of lens spaces and constructing the Poincaré homology sphere.
Covering spaces are a fundamental tool in algebraic topology because of the close relationship they bear with the fundamental groups of spaces. Indeed, they are in correspondence with the subgroups of the fundamental group: this is known as the Galois correspondence. In particular, the covering space corresponding to the trivial group is the universal covering, which is a "1-connected" variant of the original space, in the sense that it has the same homotopy groups, except for the first one which is trivial. In this article, we formalize this correspondence in homotopy type theory, a variant of Martin-Löf type theory in which types can be interpreted as spaces (up to homotopy). Along the way, we develop an n-dimensional generalization of covering spaces. Moreover, in order to demonstrate the applicability of our approach, we formally classify the covering of lens spaces and explain how to construct the Poincaré homology sphere.