Australian Category Seminar

Following a previous talk, a new proof is given of the existence of $A \square B$, for symmetric monoidal categories A and B. Proposition 1: SMon(A,SMon(B,C)) $\cong$ SMon(B,SMon(A,C)) 2-naturally in all variables $\square$ Note that by replacing SMon by SMon_strong or SMon_strict ie replace "all monoidal functors B -> C" by "all strong (resp. strict)..." many parallel theorems result. Proposition 2: Let $\alpha$ be a 2-cell in SMon then SMon($\alpha$, -) is defined. Theorem 3: SMon(B,-) : SMon_strict -> SMon has a left 2-adjoint $-\square B$ which is 2-natural in B. Hence $\square$ : SMon^2 -> SMon is a 2-functor. Proposition 4: i: Smon_strict(A $\square$ B,C) -> SMon(A $\square$ B,C) has a right adjoint S. Proof. S is constructed on the gencrators of A $\square$ B given in the previous lecture. Corollary 5: There are 2-natural strict monoidal functors a and a' between (A $\square$ B) $\square$ C and A $\square$ (B $\square$ C) which both satisfy the pentagon law for monoidal categories. Note 1) a and a’ are not inverse. Note 2) Prettier calculations may be made with strong monoidal functors (eg. a becomes an equivalence) but a 'norm' is not strong im general. Note 3) There is a third version of the theorem for bicategorics and homomorphisms. For morphisms of bicategories and homomorphisms. For morphisms of bicategories the naturality conditions break down.