Homological Algebra in Abelian Framed Bicategories: Exact Sequences and Embedding Theorems

TL;DR

This paper develops homology and cohomology theories within abelian framed bicategories, establishing exact sequences, a K"unneth theorem, and embedding theorems, thereby generalizing directed homology in algebraic structures.

Contribution

It introduces abelian framed bicategories and proves they support homology theories, exact sequences, K"unneth theorem, and embedding theorems, extending classical algebraic results to bicategorical contexts.

Findings

Exact sequences analogous to classical homology are established.

K"unneth theorem holds for closed monoidal abelian framed bicategories.

Embedding theorems relate these bicategories to bimodules over algebras.

Abstract

We introduce abelian framed bicategories, which are particular framed bicategories that are locally abelian, and show that they are suitable for developing homology and cohomology theories for directed structures. This means in particular that similar exact sequences as the relative homology and Mayer-Vietoris long exact sequences can be shown to hold. Also, for closed monoidal abelian framed bicategories, K\"unneth theorem holds as well. Finally, we prove embedding theorems similar to the Gabriel and Freyd-Mitchell theorems, for particular abelian framed bicategories, allowing to see those as bicategories of bimodules over algebras. This naturally links to the original motivation of this work, which was to generalize directed homology developed in the abelian framed bicategory of bimodules over (path) algebras.