Deconstructibility and the Hill lemma in Grothendieck categories

TL;DR

This paper explores the concept of deconstructibility in Grothendieck categories, establishing fundamental results and demonstrating its applications in structure theory, approximations, and model and t-structures in complex categories.

Contribution

It provides foundational results on deconstructible classes and illustrates their applications in constructing model and t-structures in Grothendieck categories.

Findings

Fundamental results on deconstructible classes established.

Framework for constructing model and t-structures outlined.

Connections to recent work by Gillespie, Enochs, Estrada, and others.

Abstract

A full subcategory of a Grothendieck category is called deconstructible if it consists of all transfinite extensions of some set of objects. This concept provides a handy framework for structure theory and construction of approximations for subcategories of Grothendieck categories. It also allows to construct model structures and t-structures on categories of complexes over a Grothendieck category. In this paper we aim to establish fundamental results on deconstructible classes and outline how to apply these in the areas mentioned above. This is related to recent work of Gillespie, Enochs, Estrada, Guil Asensio, Murfet, Neeman, Prest, Trlifaj and others.