coherence for symmetric monoidal and (co)affine categories

[Paul Blain Levy <P.B.Levy@cs.bham.ac.uk>]

Dear all,

Given a category C, define symm(C) to be the following category:

- an object is a finite family [or finite sequence, if preferred] of
C-objects

- a morphism from (C_i | i in I) to (D_j | j in J) consists of an
bijection f : I --> J and, for each i in I, a C-morphism C_i --> D_fi.

Define coaff(C) likewise but with "injection" instead of "bijection".

It seems to be folklore that

(1) symm(C) is the free symmetric monoidal category on C

(2) coaff(C) is the free coaffine category (symmetric monoidal category
with initial unit) on C.

In the special case where C is discrete, these statements follow from
the coherence arguments in Mac Lane's "Natural associativity and
commutativity" and Petric's "Coherence in substructural categories".

But for general C, where are these statements proved?

Paul



-- 
Paul Blain Levy
School of Computer Science, University of Birmingham
http://www.cs.bham.ac.uk/~pbl



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