2020/04/15 by Raphael Bennett‐Tennenhaus, Bennett-Tennenhaus, Raphael
Mathematics · #03C60 (secondary) #18E45 #18G80 (primary) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras
paper · doi:10.48550/arxiv.2004.06854
openalex publication_date 2020/04/15 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
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.