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We provide various ways to characterise $\Sigma$-pure-injective objects in acompactly generated triangulated category. These characterisations mimicanalogous well-known results from the model theory of modules. The proofinvolves two approaches. In the first approach we adapt arguments from themodule-theoretic setting. Here the one-sorted language of modules over a fixedring is replaced with a canonical multi-sorted language, whose sorts are givenby compact objects. Throughout we use a variation of the Yoneda embedding,called the resticted Yoneda functor, which associates a multi-sorted structureto each object. The second approach is to translate statements using thisfunctor. In particular, results about $\Sigma$-pure-injectives in triangulatedcategories are deduced from results about $\Sigma$-injective objects inGrothendieck categories. Combining the two approaches highlights a connectionbetween sorted pp-definable subgroups and annihilator subobjects of generatorsin the functor category. Our characterisation motivates the introduction ofwhat we call endoperfect objects, which generalise endofinite objects.

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