论文标题
Pretopos I：一般理论中的稳定类别
The stable category of preorders in a pretopos I: general theory
论文作者
论文摘要
在最近的一篇文章中，Facchini和Finocchiaro认为是诱导相应稳定类别的预订集类别中的自然预修复理论。在目前的工作中，我们提出了一个类别$ \ mathsf {preord}（\ mathbb c）的稳定类别的替代构造（\ Mathbb c）$的内部预订$ \ Mathbb c $，启发了该概念的分类性质。当$ \ mathbb c $是一个Pretopos时，我们证明了从内部预订类别到相关稳定类别的商类函子保留有限的共同体。 Furthermore, we identify a wide class of pretoposes, including all $σ$-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.这些属性将在证明稳定类别的普遍属性方面发挥基本作用，这将成为有关该主题的第二篇文章的主题。
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 $σ$-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.
