三角范畴中$Σ$ -纯内射性的刻画及其在内完全对象中的应用
Characterisations of $Σ$-pure-injectivity in triangulated categories and applications to endoperfect objects
摘要：
我们提供了各种方法来构造$Σ$ -纯内射对象在一个复杂生成的三角范畴。这些特征模拟了模块的模型论的类似的众所周知的结果。证明涉及两种方法。在第一种方法中，我们从模块理论的设置调整参数。在这里，固定环上的模的单排序语言被替换为规范的多排序语言，其排序由紧对象给出。在整个过程中，我们使用Yoneda嵌入的一个变体，称为resticted Yoneda functor，它将多排序结构与每个对象相关联。第二种方法是使用此函子翻译语句。特别地，从Grothendieck范畴中关于$Σ$ -内射对象的结果出发，得到了三角范畴中关于$Σ$-纯内射对象的结果.结合这两种方法突出了函子范畴中生成元的有序pp-可定义子群和零化子子群之间的联系。我们的特征化激发了我们所谓的内完成对象的引入，它概括了内有限对象。
英文摘要：
We provide various ways to characterise $Σ$-pure-injective objects in a compactly generated triangulated category. These characterisations mimic analogous well-known results from the model theory of modules. The proof involves two approaches. In the first approach we adapt arguments from the module-theoretic setting. Here the one-sorted language of modules over a fixed ring is replaced with a canonical multi-sorted language, whose sorts are given by compact objects. Throughout we use a variation of the Yoneda embedding, called the resticted Yoneda functor, which associates a multi-sorted structure to each object. The second approach is to translate statements using this functor. In particular, results about $Σ$-pure-injectives in triangulated categories are deduced from results about $Σ$-injective objects in Grothendieck categories. Combining the two approaches highlights a connection between sorted pp-definable subgroups and annihilator subobjects of generators in the functor category. Our characterisation motivates the introduction of what we call endoperfect objects, which generalise endofinite objects.
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