On the obscure axiom for one-sided exact categories

TL;DR

This paper investigates the obscure axiom in one-sided exact categories, exploring its failure, homological implications, and conditions under which it holds, thereby deepening understanding of their structural properties and derived categories.

Contribution

It introduces three versions of the obscure axiom for one-sided exact categories, analyzes their homological consequences, and relates their validity to the embedding into the exact hull.

Findings

Failure of the obscure axiom is controlled by the embedding into the exact hull.

Equivalent homological properties are established for the three versions of the obscure axiom.

Each version's closure preserves the bounded derived category up to triangle equivalence.

Abstract

One-sided exact categories are obtained via a weakening of a Quillen exact category. Such one-sided exact categories are homologically similar to Quillen exact categories: a one-sided exact category $\mathcal{E}$ can be (essentially uniquely) embedded into its exact hull ${\mathcal{E}}^{\textrm{ex}}$; this embedding induces a derived equivalence $\textbf{D}^b(\mathcal{E}) \to \textbf{D}^b({\mathcal{E}}^{\textrm{ex}})$. Whereas it is well known that Quillen's obscure axioms are redundant for exact categories, some one-sided exact categories are known to not satisfy the corresponding obscure axiom. In fact, we show that the failure of the obscure axiom is controlled by the embedding of $\mathcal{E}$ into its exact hull ${\mathcal{E}}^{\textrm{ex}}.$ In this paper, we introduce three versions of the obscure axiom (these versions coincide when the category is weakly idempotent complete) and establish equivalent homological properties, such as the snake lemma and the nine lemma. We show that a one-sided exact category admits a closure under each of these obscure axioms, each of which preserves the bounded derived category up to triangle equivalence.