Flow equivalence of diagram categories and Leavitt path algebras

TL;DR

This paper generalizes Morita equivalence results for Leavitt path algebras, showing they depend only on categorical direct sums, not on the field or linear algebra specifics, thus broadening their applicability.

Contribution

It extends Morita equivalence theorems for Leavitt path algebras to arbitrary categories with binary coproducts, beyond vector spaces over a field.

Findings

Morita equivalence depends only on categorical direct sums.

Results are independent of the coefficient field F.

Framework may apply to other Morita equivalence problems.

Abstract

Several constructions on directed graphs originating in the study of flow equivalence in symbolic dynamics (e.g., splittings and delays) are known to preserve the Morita equivalence class of Leavitt path algebras over any coefficient field F. We prove that many of these equivalence results are not only independent of F, but are largely independent of linear algebra altogether. We do this by formulating and proving generalisations of these equivalence theorems in which the category of F-vector spaces is replaced by an arbitrary category with binary coproducts, showing that the Morita equivalence results for Leavitt path algebras depend only on the ability to form direct sums of vector spaces. We suggest that the framework developed in this paper may be useful in studying other problems related to Morita equivalence of Leavitt path algebras.