Homotopic morphisms and diagram theorems in extriangulated 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.