Australian Category Seminar

(I) The right-hand associativity for $\square$ , r : A -> A $\square$ A is natural in A and satisfies the triangle law. There is a strict symmetric monoidal functor r' : a$\square$ -> A but it is not natural. (II) On generalization of the notion "monoid” is "category". Another is "monoidal category". In the same way "abelian group" becomes "additive category" or "symmetric monoidal closed category". There is a full embedding Ab -> SMon mapping A tensor B to A $\square$ B. (III) Definition. Let A be a SMC and V be SMon. Then a V-norm for A is a symmetric monoidal functor | | : A -> V satisfying the following condition, |A(A, B)| $\cong$ |A(B, A)|. |A| is the V-cat with the same objects as A but with |A|(A, B) = |A(A, B)| etc. Say | | is cauchy complete iff |A| is.