# Semantics

A strict symmetric premonoidal category (proposed by Power and Robinson) is a category P together with:

- A strict symmetric monoidal category C with the same objects as P (called the centre of P).

- An identity on objects inclusion C P.

- Two functors:

  : C P P
  : P C P

  such that:

  - the three functors , and coincide on objects,
  - the three `obvious' functors from C C to P coincide, and
  - the symmetry in C is a natural isomorphism X Y Y X in P.