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

Abstract

We define an intrinsic symmetric bi-right-exact (and for varieties, bi-cocontinuous) bilinear product on objects of a semi-abelian category, constructed as the cosmash product in the two-nilpotent reflection. When applied to abelian objects, this recovers classical tensor products in many cases. A recognition theorem states that any symmetric bi-cocontinuous bifunctor on an abelian variety of algebras is realised as the bilinear product in the variety of algebras over a suitable 2-nilpotent symmetric operad in the monoidal category of abelian groups. For abelian groups replaced with any commutative ring, the bilinear product of algebras over such an operad is associative as long as the only unary operations are given by multiplication with scalars, but not in general. This relies on a right-exactness theorem for cross-effects of bifunctors, and consequently for cosmash products. We develop basic properties, compare the bilinear product to the Brown-Loday non-abelian tensor product, and prove a categorical version of Ganea's six-term exact homology sequence. We further characterise abelian extensions via internal action cores, obtaining explicit descriptions of bilinear products in categories of representations; in particular, the bilinear product of the associated Beck modules generalises the classical tensor product of representations for groups and Lie algebras.