Severe Strict Monoidal Category Naivete
Ellis D. Cooper <xtalv1 <at> netropolis.net>
2010-12-03 15:51:20 GMT
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 <at> g:X <at> Y->Y <at> Z and g <at> f:Y <at> X->Z <at> 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 <at> 1_Y:X <at> Y->Y <at> Y, and also tensoring 1_Y
with g to get 1_Y <at> g:Y <at> Y->Y <at> Z. So I get the composition f <at> 1_Y followed
by 1_Y <at> g is a morphism from
X <at> Y->Y <at> Z, and you made me realize that f followed by g as a morphism
X->Y cannot possibly equal f <at> 1_Y followed by 1_Y <at> g.
Then again, if the ambient strict monoidal category is symmetric, so
that the latter composition is a morphism X <at> Y->Z <at> 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.
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