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arXiv · 2604.22186

Homotopic morphisms and diagram theorems in extriangulated categories

Chencheng Zhang·Xue-Song Lu·Pu Zhang
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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.

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Chencheng Zhang, Xue-Song Lu, Pu Zhang. 2026-04-24. Homotopic morphisms and diagram theorems in extriangulated categories. https://arxiv.org/abs/2604.22186

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