Dear Paul, On 2015-08-22 11:06 AM, Paul Blain Levy wrote:
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
The 'coaffine envelope' construction of [HT12, Cor 2.12] provides the free coaffine category 'D' over a symmetric one D (wrt strong symmetric functors). So combining this construction with Symm, one gets Coaff(C) = 'Symm(C)' The explicit construction in ibid does not spell out this special case, except for C =1, where one gets the familiar category of finite sets and injections (with sum as tensor). The general description yields Coaff(C) ( X, Y) = [ \Coprod_{W\in Symm(C)} Symm(C)(X \tensor W, Y)]_~ that is, morphisms f:X \tensor W -> Y, indexed by W, subject to an equivalence relation via directed paths. It reduces to the explicit description of Coaff(C) above by noting that every equivalence class has a unique underlying injection i:|X| -> |Y| between the underlying index sets, which gives a canonical representative of ~-classes with W = (|W|, {Y_w}_{w\in |W|}) where |W| is the complement of the image of i in |Y|. As for Symm(C), I am not positive about the original reference, but surely it is worthwhile referring to the classical [JS93] which gives an explicit description of the free braided monoidal category on a given category (Prop. 2.2(b) and Cor. 2.4). One easily gets Symm(C) by restricting to braidings which are symmetries (so the construction involves permutation groups rather braid groups). The general construction (a Grothendieck construction) is introduced in Kelly's theory of clubs [K74]. The general theory that encompasses these monadic constructions is that of *polynomial monads*, as in the recent [W15] (and references therein for this rich theory), which includes the construction of Symm among the many examples. Claudio REFERENCES [HT12] - Hermida, Claudio, and Robert D. Tennent. "Monoidal indeterminates and categories of possible worlds."/Theoretical Computer Science/430 (2012): 3-22. [JS93] - Joyal, Andr??, and Ross Street. "Braided tensor categories."/Advances in Mathematics/102.1 (1993): 20-78. [K74] - Kelly, Gregory Max. "On clubs and doctrines."/Category seminar/. Springer Berlin Heidelberg, 1974. [W15] - Weber, Mark. "Polynomials in categories with pullbacks."/Theory and Applications of Categories/30.16 (2015): 533-598. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]