Transfer of homological objects in exact categories via adjoint triples. Applications to functor categories
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911945413623808 |
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| author | Estrada, Sergio Cortés-Izurdiaga, Manuel Odabasi, Sinem |
| author_facet | Estrada, Sergio Cortés-Izurdiaga, Manuel Odabasi, Sinem |
| contents | 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 $(\mbox{Add}(\mathcal{A}, R\mbox{-Mod}); \mathcal{E})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04012 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transfer of homological objects in exact categories via adjoint triples. Applications to functor categories Estrada, Sergio Cortés-Izurdiaga, Manuel Odabasi, Sinem Category Theory Representation Theory 18G35, 18G25, 18E99 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 $(\mbox{Add}(\mathcal{A}, R\mbox{-Mod}); \mathcal{E})$. |
| title | Transfer of homological objects in exact categories via adjoint triples. Applications to functor categories |
| topic | Category Theory Representation Theory 18G35, 18G25, 18E99 |
| url | https://arxiv.org/abs/2407.04012 |