Characterisations of ${\varSigma }$-pure-injectivity in triangulated categories and applications to endocoperfect objects
Volume 261 / 2023
Abstract
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 restricted 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 pp-definable subgroups of a particular sort and annihilator subobjects of the image of this sort under the restricted Yoneda functor. Our characterisation motivates the introduction of what we call endocoperfect objects, which generalise endofinite objects.
