Homotopy Path Algebras

TL;DR

This paper introduces homotopy path algebras, explores their properties, and connects them to topological stratifications, mirror symmetry, and algebraic resolutions, providing new insights into their structure and applications.

Contribution

It defines homotopy path algebras, proves they always admit cellular resolutions, and links them to mirror symmetry and topological stratifications, revealing new structural properties.

Findings

Homotopy path algebras always admit cellular resolutions.

Shellability implies Koszulity and minimal cellular resolutions.

Certain algebras are Koszul and have minimal resolutions when directable.

Abstract

We define a basic class of algebras which we call homotopy path algebras. We find that such algebras always admit a cellular resolution and detail the intimate relationship between these algebras, stratifications of topological spaces, and entrance/exit paths. As examples, we prove versions of homological mirror symmetry due to Bondal-Ruan for toric varieties and due to Berglund-H\"ubsch-Krawitz for hypersurfaces with maximal symmetry. We also demonstrate that a form of shellability implies Koszulity and the existence of a minimal cellular resolution. In particular, when the algebra determined by the image of the toric Frobenius morphism is directable, then it is Koszul and admits a minimal cellular resolution.