coherence for symmetric monoidal and (co)affine categories

coherence for symmetric monoidal and (co)affine categories

[Paul Blain Levy]

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



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: coherence for symmetric monoidal and (co)affine categories

[Claudio Hermida]

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/ ]

Re: coherence for symmetric monoidal and (co)affine categories

[John Baez]

Hi -

Paul Blain Levy writes:

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.
>

>  [....]

But for general C, where are these statements proved?


I'll admit to having used the first statement without proof on page 14 here:

http://arxiv.org/abs/q-alg/9702014

As I get older I feel worse about doing such things.  Perhaps an efficient
proof of both statements could be developed from here:

Symmetric monoidal sketches
Martin Hyland and John Power
https://www.dpmms.cam.ac.uk/~martin/Research/Oldpapers/hp00.pdf

Theorem 2.5 gives the free symmetric monoidal category on a "symmetric
monoidal sketch".

Best,
jb


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]