From: Ronnie Brown <ronnie.profbrown@btinternet.com>
To: Tom Leinster <tl@maths.gla.ac.uk>
Cc: categories@mta.ca
Subject: Re: Tensor of monads
Date: Sun, 01 Aug 2010 20:49:25 +0100 [thread overview]
Message-ID: <E1OfuBA-0006nE-Rg@mlist.mta.ca> (raw)
In-Reply-To: <E1OfK0j-0001ua-W0@mlist.mta.ca>
Looking at the two ways of evaluating (say with . vertically and *
horizontally)
1 e
e 1
is what gives 1=e but it seems to need two sided units.
Ronnie
Tom Leinster wrote:
> On Fri, 30 Jul 2010, Prof. Peter Johnstone wrote:
>
>> On Fri, 30 Jul 2010, Sergey Goncharov wrote:
>>
>>> This looks fine! But I guess, Eckmann-Hilton argument does not apply
>>> to your previous example because it presupposes that the monoidal
>>> structures share the unit, which was not the case there, was it?
>>>
>> Yes, it was: the fact that the unit for each semilattice structure is
>> a homomorphism for the other forces them to be the same.
>
> In any case, it doesn't matter: the Eckmann-Hilton argument *doesn't*
> presuppose that the monoid structures share the unit.
>
> Here are the weakest hypotheses I know for the elementary Eckmann-Hilton
> argument:
>
> Let A be a set.
> Let . be a binary operation on A with two-sided unit 1.
> Let * be a binary operation on A with two-sided unit e.
> Suppose that
>
> (a * b) . (a' * b') = (a . a') * (b . b')
>
> for all a, b, a', b' in A.
> Then . = *, 1 = e, and (A, ., 1) is a commutative monoid.
>
> So equality of the units and associativity, as well as the more familiar
> stuff, come for free. (I bet you can weaken "two-sided", too.)
>
> Tom
>
>
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next prev parent reply other threads:[~2010-08-01 19:49 UTC|newest]
Thread overview: 24+ messages / expand[flat|nested] mbox.gz Atom feed top
2010-07-28 14:02 Sergey Goncharov
2010-07-29 8:21 ` N.Bowler
2010-07-29 9:18 ` Prof. Peter Johnstone
2010-07-29 10:29 ` Michael Barr
2010-07-31 8:45 ` Richard Garner
[not found] ` <AANLkTinxyVQ1fXu7DLWu4CUF3AP2KPX6PLQFDB+zG4Ef@mail.gmail.com>
2010-07-31 12:48 ` Michael Barr
[not found] ` <alpine.LRH.2.00.1007291006210.5174@siskin.dpmms.cam.ac.uk>
2010-07-29 13:24 ` Prof. Peter Johnstone
[not found] ` <alpine.LRH.2.00.1007291422370.5174@siskin.dpmms.cam.ac.uk>
2010-07-30 1:02 ` Sergey Goncharov
2010-07-31 20:34 ` Eckmann-Hilton (Was: Tensor of monads) Toby Bartels
[not found] ` <4C5224A4.4000105@informatik.uni-bremen.de>
2010-07-30 10:37 ` Tensor of monads Prof. Peter Johnstone
2010-07-30 22:41 ` Tom Leinster
2010-08-01 19:49 ` Ronnie Brown [this message]
2010-08-02 9:47 ` Ronnie Brown
2010-08-01 0:31 ` Richard Garner
2010-08-02 19:55 ` Paul Levy
2010-08-03 6:39 ` Richard Garner
[not found] ` <AANLkTimd202AX=3hUqU9ABkKUy9Z4Loh1RXTiDgVZ3Ku@mail.gmail.com>
2010-08-03 11:03 ` Paul Levy
2010-08-09 20:26 ` Paul Levy
2010-08-05 20:06 ` Sergey Goncharov
2010-08-08 19:24 ` Gordon Plotkin
2010-07-30 3:44 ` Joyal, André
[not found] ` <AANLkTin5+paq8sP-eVjdf8rZOyA-z=t6QzAhCqVUsyQi@mail.gmail.com>
2010-08-01 9:13 ` Richard Garner
2010-08-02 14:17 ` Prof. Peter Johnstone
[not found] ` <alpine.LRH.2.00.1008021514290.18118@siskin.dpmms.cam.ac.uk>
2010-08-02 21:12 ` Richard Garner
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