  The present exploratory paper deals with tensor products in the locality
framework {developed in previous work}, a natural setting for an algebraic
formulation of the locality principle in quantum field theory. Locality tensor
products of locality vector spaces raise challenging questions, such as whether
the locality tensor product of two locality vector spaces is a locality vector
space. A related question is whether the quotient of locality vector spaces is
a locality vector space, which we first reinterpret in a group theoretic
language and then in terms of short exact sequences. We prove a universal
property for the locality tensor algebra and for the locality enveloping
algebra, the analogs in the locality framework of the tensor algebra and of the
enveloping algebra. These universal properties hold under compatibility
assumptions between the locality and the multilinearity underlying the
construction of tensor products which we formulate in the form of conjectural
statements. Assuming they hold true, we generalise the Milnor-Moore theorem to
the locality setup and discuss some of its consequences.
