Homotopic morphisms and diagram theorems in extriangulated categories

TL;DR

This paper introduces homotopic morphisms in extriangulated categories, analyzes their properties, and explores diagram theorems like the 4x4 Lemma, connecting these concepts to triangulated categories.

Contribution

It develops the theory of homotopic morphisms in extriangulated categories and extends classical diagram theorems to this setting.

Findings

Any morphism of E-triangles can be decomposed into homotopic morphisms.

Analyzed 15 cases where morphisms are E-inflations or E-deflations.

Established relations between homotopic morphisms and good morphisms in triangulated categories.

Abstract

Homotopic morphisms of $\mathbb E$-triangles in extriangulated categories are introduced. Any morphism of $\mathbb E$-triangles is a composition of homotopic morphisms. Any morphism $(\alpha_1, \alpha_2, \alpha_3)$ of $\mathbb E$-triangles can be modified to be homotopic, by changing one of $\alpha_i$; moreover, all the 15 cases where $\alpha_i$ is an $\mathbb E$-inflation ($\mathbb E$-deflation) are analyzed. Some diagram theorems, especially $4\times 4$ Lemma and its $14$ variants, including $3\times 3$ diagram and Horseshoe Lemma, are investigated. A relation between homotopic morphisms and (middling) good morphisms in triangulated categories are given. Weakly idempotent complete extriangulated categories are characterized.