Homological algebra over non-unital rings and algebras, with applications to $(\infty, 1)$-categories

TL;DR

This paper develops homological algebra for modules over non-unital rings and algebras, applying it to define and analyze homology theories of $( abla,1)$-categories and directed spaces, including relative homology.

Contribution

It introduces homological algebra frameworks over non-unital rings and applies them to $( abla,1)$-categories, expanding tools for higher category theory and directed topology.

Findings

Defined homology for $( abla,1)$-categories

Established exact sequences for relative homology

Extended homological methods to non-unital algebraic structures

Abstract

The article is developing homological algebra in modules over non-unital rings and algebras. The main application is the definition and study of (directed) homology of $(\infty,1)$-categories and of directed spaces, including relative homology and its exact sequence.