The Nonunital Homology

Standard homological algebra assumes rings have a unit element — an identity for multiplication. This is convenient (projective modules, free resolutions, derived functors all work cleanly) but restrictive. Many natural algebraic structures — ideals, null algebras, algebras of compact operators — are non-unital. And (∞,1)-categories, when viewed algebraically, naturally produce non-unital structures.

Homological algebra extends to the non-unital setting, including directed homology of (∞,1)-categories and directed spaces (arXiv:2603.11937). Relative homology and exact sequences survive the loss of the unit. The theory applies to directed spaces — topological spaces with a notion of “allowed direction” — where the homology should respect the direction.

The structural observation: the unit in a ring is primarily a technical convenience for homological algebra, not a structural necessity. The long exact sequence of a pair, the connecting homomorphism, the five-lemma — these work because of the abelian group structure of the modules, not because of the ring’s unit. Removing the unit removes some constructions (free modules become harder, for instance) but the core homological machinery survives. For (∞,1)-categories, where unitality is often only homotopical, the non-unital theory is the natural one, and the unital version is a special case.