Intrinsic tensor products and a Ganea-type extension of the five-term exact sequence

Authors

Bo Shan Deval, Manfred Hartl, Tim Van der Linden

Categories

Abstract

We establish a right-exactness theorem for the cross-effects of bifunctors, and consequently for cosmash products, in Janelidze-Márki-Tholen semi-abelian categories. This result motivates an intrinsic definition of a bilinear product, a tensor-like operation on objects of a category, constructed in terms of limits and colimits. Given two objects in the category, their bilinear product is the abelian object obtained as the cosmash product in the category of two-nilpotent objects of the reflections of these objects. In many concrete cases, this operation, applied to a pair of abelian objects, captures a classical tensor product. We explain this by proving a recognition theorem, which states that any symmetric, bi-cocontinuous bifunctor on an abelian variety of algebras can be recovered as the bilinear product within a suitable semi-abelian variety, namely of algebras over a certain two-nilpotent operad. In other words, the extra structure carried by such a bifunctor on the abelian variety (for instance, a tensor product, known in the literature) is encoded by means of a surrounding semi-abelian variety whose abelian core is the original variety. We illustrate the construction with several examples, develop its basic properties, and compare it to the semi-abelian analogue of the Brown-Loday non-abelian tensor product. As an application, we present a categorical version of Ganea's six-term exact homology sequence. Finally, we characterise abelian extensions via internal action cores, yielding explicit descriptions of cosmash products and bilinear products in certain categories of representations.