From: "Ellis D. Cooper" <xtalv1@netropolis.net>
To: Steve Lack <steve.lack@mq.edu.au>
Cc: categories@mta.ca
Subject: Severe Strict Monoidal Category Naivete
Date: Fri, 03 Dec 2010 10:51:20 -0500 [thread overview]
Message-ID: <E1POhE7-0002RO-S1@mlist.mta.ca> (raw)
In-Reply-To: <8C65074A-894A-4F7A-B47D-9D8411A9CFC3@mq.edu.au>
Dear Steve,
At 09:54 PM 12/2/2010, you wrote:
If I understand correctly, you have arrows f:X->Y and g:Y->Z and you
are comparing the tensor products f@g:X@Y->Y@Z and g@f:Y@X->Z@Y. They
have different domain and codomain, so cannot be equal.
I was thinking more about f:X->Y and g:Y->Z and tensoring f with the
identity morphism of Y to get f@1_Y:X@Y->Y@Y, and also tensoring 1_Y
with g to get 1_Y@g:Y@Y->Y@Z. So I get the composition f@1_Y followed
by 1_Y@g is a morphism from
X@Y->Y@Z, and you made me realize that f followed by g as a morphism
X->Y cannot possibly equal f@1_Y followed by 1_Y@g.
Then again, if the ambient strict monoidal category is symmetric, so
that the latter composition is a morphism X@Y->Z@Y, then to my mind
somehow this is pretty much the same as the composition X->Z of f
followed by g, basically because only the identity morphism of Y is involved.
The context of my inquiry is chemical reaction, as suggested by John
Baez a while ago. That is, if f and g are chemical reactions that
transform X to Y and Y to Z, respectively, then the net effect is
just transformation of X to Y, since the Y produced by f is
completely consumed by g. Bottom line: I would like a correct way to
say that tensoring f with an identity morphism is somehow no different from f.
Ellis
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next prev parent reply other threads:[~2010-12-03 15:51 UTC|newest]
Thread overview: 5+ messages / expand[flat|nested] mbox.gz Atom feed top
2010-12-02 14:55 Ellis D. Cooper
2010-12-03 2:54 ` Steve Lack
[not found] ` <8C65074A-894A-4F7A-B47D-9D8411A9CFC3@mq.edu.au>
2010-12-03 15:51 ` Ellis D. Cooper [this message]
2010-12-04 14:00 ` Ellis D. Cooper
[not found] <E1POk5N-0004lN-Lv@mlist.mta.ca>
[not found] ` <alpine.LRH.2.00.1012041110000.9194@mlist.mta.ca>
2010-12-04 23:44 ` David Roberts
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