* Re: Severe Strict Monoidal Category Naivete [not found] ` <alpine.LRH.2.00.1012041110000.9194@mlist.mta.ca> @ 2010-12-04 23:44 ` David Roberts 0 siblings, 0 replies; 5+ messages in thread From: David Roberts @ 2010-12-04 23:44 UTC (permalink / raw) To: categories Hi Ellis, to push the chemical reaction analogy further, tensoring with an identity morphism 1_Y is like having a chemical present that doesn't take part in the reaction: it's there are the beginning and end, but doesn't change. But you can't take it away (and no, it's not like a catalyst, in that your original arrow f was there to begin with), unless perhaps Y has some sort of dual or (weak) tensor inverse, and by now the analogy is stretched beyond breaking point. David [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Severe Strict Monoidal Category Naivete @ 2010-12-02 14:55 Ellis D. Cooper 2010-12-03 2:54 ` Steve Lack ` (2 more replies) 0 siblings, 3 replies; 5+ messages in thread From: Ellis D. Cooper @ 2010-12-02 14:55 UTC (permalink / raw) To: categories (1) Is strict monoidal category the same as monoid in category of categories? (2) Is it not true that in a strict monoidal category if $X\xrightarrow{f}Y\xrightarrow{g}Z$ then $f\square g= g\circ f$? (3) Is the pentagon axiom automatically satisfied in a strict monoidal category? Many thanks for your patience and pointers. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: Severe Strict Monoidal Category Naivete 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> 2010-12-04 14:00 ` Ellis D. Cooper 2 siblings, 0 replies; 5+ messages in thread From: Steve Lack @ 2010-12-03 2:54 UTC (permalink / raw) To: Ellis D. Cooper; +Cc: categories Dear Ellis, On 03/12/2010, at 1:55 AM, Ellis D. Cooper wrote: > (1) Is strict monoidal category the same as monoid in category of categories? Yes. > (2) Is it not true that in a strict monoidal category if > $X\xrightarrow{f}Y\xrightarrow{g}Z$ then $f\square g= g\circ f$? 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. If you considered a commutative monoid in the category of categories, then these arrows would be equal. But such commutative monoids are very rare. > (3) Is the pentagon axiom automatically satisfied in a strict > monoidal category? > Yes. In that case it asserts that two identity arrows with the same domain and codomain are equal. Steve Lack. > Many thanks for your patience and pointers. > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
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* Severe Strict Monoidal Category Naivete [not found] ` <8C65074A-894A-4F7A-B47D-9D8411A9CFC3@mq.edu.au> @ 2010-12-03 15:51 ` Ellis D. Cooper 0 siblings, 0 replies; 5+ messages in thread From: Ellis D. Cooper @ 2010-12-03 15:51 UTC (permalink / raw) To: Steve Lack; +Cc: categories 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/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Severe Strict Monoidal Category Naivete 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> @ 2010-12-04 14:00 ` Ellis D. Cooper 2 siblings, 0 replies; 5+ messages in thread From: Ellis D. Cooper @ 2010-12-04 14:00 UTC (permalink / raw) To: droberts; +Cc: categories Hi David, At 11:50 PM 12/3/2010, you wrote: >to push the chemical reaction analogy further, tensoring with an >identity morphism 1_Y is like having a chemical present that doesn't >take part in the reaction: it's there are the beginning and end, but >doesn't change. But you can't take it away (and no, it's not like a >catalyst, in that your original arrow f was there to begin with). I agree completely. All I am saying is that since there is no effect on the reaction by the presence of a neutral chemical, it might just as well not be mentioned. Perhaps what I am getting at is a quotient category in which X@Y->Z@Y is identified with X->Z in this specific situation. I do believe this is how chemists think of their applications of Hess' Law, which is what my inquiry is really all about. Ellis [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
end of thread, other threads:[~2010-12-04 23:44 UTC | newest] Thread overview: 5+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- [not found] <E1POk5N-0004lN-Lv@mlist.mta.ca> [not found] ` <alpine.LRH.2.00.1012041110000.9194@mlist.mta.ca> 2010-12-04 23:44 ` Severe Strict Monoidal Category Naivete David Roberts 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 2010-12-04 14:00 ` Ellis D. Cooper
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