Transfer of homological objects in exact categories via adjoint triples. Applications to functor categories

TL;DR

This paper explores how homological objects transfer across exact categories via adjoint triples, providing new descriptions of cotorsion pairs and characterizations of projective/injective objects in functor categories.

Contribution

It introduces a method to transfer cotorsion pairs in exact categories using adjoint triples, with applications to functor categories and characterizations of projective/injective objects.

Findings

Cotorsion pairs can be transferred without injective/projective hypotheses.

Evaluation and stalk functors induce cotorsion pairs in functor categories.

Intrinsic characterization of projective/injective objects in additive functor categories.

Abstract

For a given family $\{(\mathrm{q}_i, \mathrm{t}_i, \mathrm{p_i} )\}_{i \in I}$ of adjoint triples between exact categories $\mathcal{C}$ or $\mathcal{D}$, we show that any cotorsion pair in $\mathcal{C}$ and $\mathcal{D}$ yield two canonical cotorsion pairs providing a concrete description of objects without using any injectives/projectives object hypothesis. We firstly apply this result for the evaluation functor on the functor category $\operatorname{Add}(\mathcal{A}, R \mbox{-Mod})$ equipped with an exact structure $\mathcal{E}$. Under mild conditions on $\mathcal{A}$, we introduce the stalk functor at any object of $\mathcal{A}$, and subsequently, we investigate cotorsion pairs induced by stalk functors. Finally, we use them to present an intrinsic characterization of projective/injective objects in $(\mbox{Add}(\mathcal{A}, R\mbox{-Mod}); \mathcal{E})$.