Characterisations of \(\Sigma\)-pure-injectivity in triangulated categories and applications to endoperfect objects

• Published in: arXiv.org 2021-03
• Main Author: Bennett-Tennenhaus, Raphael
• Format: Article
• Language: English
• Subjects: Categories; Embedding; Modules; Subgroups
• ISSN: 2331-8422

We provide various ways to characterise \(\Sigma\)-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 \(\Sigma\)-pure-injectives in triangulated categories are deduced from results about \(\Sigma\)-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.