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Recollements, coproducts and products in extriangulated categories

Alejandro Argudín-Monroy, Octavio Mendoza, Carlos E. Parra

TL;DR

This paper defines and analyzes AET4 and AET4$^*$, natural analogues of AB4/AB4$^*$ in the framework of extriangulated categories. It provides multiple equivalent characterizations via natural maps on higher extension groups, universal $ ext{E}$-extensions, and coproduct-compatibility, and develops a robust adjoint-pair/coproduct machinery to study product- and coproduct-compatibility in extriangulated settings. The authors apply these ideas to hearts of intervals, $t$-structures, and smashing/co-smashing phenomena, establishing conditions under which extended hearts inherit AET4(AET4$^*$) and relating these properties to recollements of extriangulated categories. An appendix develops higher extension theory and adjoint-pair interactions without assuming projective/injective objects, facilitating dualizable results and broader applicability in exact, triangulated, and tilted/heart-structured contexts.

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.

Recollements, coproducts and products in extriangulated categories

TL;DR

This paper defines and analyzes AET4 and AET4, natural analogues of AB4/AB4 in the framework of extriangulated categories. It provides multiple equivalent characterizations via natural maps on higher extension groups, universal -extensions, and coproduct-compatibility, and develops a robust adjoint-pair/coproduct machinery to study product- and coproduct-compatibility in extriangulated settings. The authors apply these ideas to hearts of intervals, -structures, and smashing/co-smashing phenomena, establishing conditions under which extended hearts inherit AET4(AET4) and relating these properties to recollements of extriangulated categories. An appendix develops higher extension theory and adjoint-pair interactions without assuming projective/injective objects, facilitating dualizable results and broader applicability in exact, triangulated, and tilted/heart-structured contexts.

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 of extriangulated categories with AET4 (resp. AET4{*}) to imply that the categories and are AET4 (resp. AET4{*}); and we will show a relation between the -smashing (resp. -co-smashing) condition for a -structure and the AET4 (resp. AET4{*}) condition of the extended hearts of the -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.

Paper Structure

This paper contains 18 sections, 53 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functors, coproducts and matrices
  4. Recollements
  5. Extriangulated categories and functors
  6. Extriangulated categories with negative first extension
  7. The AET4 condition for extriangulated categories
  8. The product category of an extriangulated category
  9. The adjoint pair associated to the coproduct
  10. The AET4 condition for extriangulated categories
  11. Universal $\mathbb{E}$-extensions
  12. AET4 for coproduct-compatible extriangulated categories with negative first extension
  13. AET4 for hearts of intervals of $s$-torsion pairs
  14. AET4 for hearts and extended hearts of $t$-structures
  15. An example of a non-AET4* extended heart
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $A=\coprod_{i=1}^{n}A_{i}$ and $B=\coprod_{i=1}^{m}B_{i}$ in $\mathcal{C}.$ Then, the map $\Phi_{B,A}^{F}:F(A,B)\to\mathrm{Mat}_{m\times n}^{F}(A,B),$ defined by $[\Phi_{B,A}^{F}(x)]_{i,j}:=\pi_{i}^{B}\cdot x\cdot\mu_{j}^{A},$ is an isomorphism of abelian groups whose inverse is $(\Phi_{B,A}^{F}

Theorems & Definitions (124)

  • Definition 1.1
  • Lemma 2.1
  • Proposition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 114 more