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The present exploratory paper deals with tensor products in the localityframework {developed in previous work}, a natural setting for an algebraicformulation of the locality principle in quantum field theory. Locality tensorproducts of locality vector spaces raise challenging questions, such as whetherthe locality tensor product of two locality vector spaces is a locality vectorspace. A related question is whether the quotient of locality vector spaces isa locality vector space, which we first reinterpret in a group theoreticlanguage and then in terms of short exact sequences. We prove a universalproperty for the locality tensor algebra and for the locality envelopingalgebra, the analogs in the locality framework of the tensor algebra and of theenveloping algebra. These universal properties hold under compatibilityassumptions between the locality and the multilinearity underlying theconstruction of tensor products which we formulate in the form of conjecturalstatements. Assuming they hold true, we generalise the Milnor-Moore theorem tothe locality setup and discuss some of its consequences.

Tensor products and the Milnor-Moore theorem in the locality setup