Recollements, coproducts and products in extriangulated categories

TL;DR

This paper introduces AET4 and AET4* conditions in extriangulated categories, generalizing homological properties from abelian categories, and explores their implications in recollements and t-structures.

Contribution

It defines AET4 and AET4* conditions in extriangulated categories and establishes their equivalences, extending homological tools beyond abelian categories.

Findings

AET4 and AET4* generalize AB4 conditions in extriangulated categories.

Recollement conditions imply AET4 properties transfer between categories.

Relations between t-structure smashing conditions and AET4 in extended hearts.

Abstract

We introduce a notion similar to the AB4 (resp. AB4{*}) condition for abelian categories but in the context of extriangulated categories. We will refer to this notion as AET4 (resp. AET4{*}). One of our main results shows equivalent statements for AET4 (resp. AET4{*}), which generalize statements commonly used in homological constructions in abelian categories. As an application, we will give conditions for a recollement $(\mathcal{A},\mathcal{B},\mathcal{C})$ of extriangulated categories with $\mathcal{B}$ AET4 (resp. AET4{*}) to imply that the categories $\mathcal{A}$ and $\mathcal{C}$ are AET4 (resp. AET4{*}); and we will show a relation between the $n$-smashing (resp. $n$-co-smashing) condition for a $t$-structure and the AET4 (resp. AET4{*}) condition of the extended hearts of the $t$-structure. It is also included an appendix where we study in detail the properties of adjoint pairs between extriangulated categories which are necessary for the development of the paper, including some special properties for higher extension groups.