The stable category of preorders in a pretopos I: general theory
| Autor: | Francis Borceux, Federico Campanini, Marino Gran |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: | |
| Popis: | In a recent article Facchini and Finocchiaro considered a natural pretorsion theory in the category of preordered sets inducing a corresponding stable category. In the present work we propose an alternative construction of the stable category of the category $\mathsf{PreOrd} (\mathbb C)$ of internal preorders in any coherent category $\mathbb C$, that enlightens the categorical nature of this notion. When $\mathbb C$ is a pretopos we prove that the quotient functor from the category of internal preorders to the associated stable category preserves finite coproducts. Furthermore, we identify a wide class of pretoposes, including all $\sigma$-pretoposes and all elementary toposes, with the property that this functor sends any short $\mathcal Z$-exact sequences in $\mathsf{PreOrd} (\mathbb C)$ (where $\mathcal Z$ is a suitable ideal of trivial morphisms) to a short exact sequence in the stable category. These properties will play a fundamental role in proving the universal property of the stable category, that will be the subject of a second article on this topic. Comment: 36 pages |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|