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From: Mamuka Jibladze <jib@rmi.acnet.ge>
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Subject: Re: Chew on this
Date: Fri, 25 Sep 1998 15:55:51 +0300 (EET DST)
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One more comment on that fascinating ugly monoidal structure. Many
years ago D. Pataraia as a student was asked to realise tensor product of
vector spaces (V and W over k) as a colimit. He then came up with a
diagram (sorry for still more ugly notation)

                                 k_{v,w}
                                 /    \
                                /      \
                              |_        _|
                            V_w          W_v

That is, vertices of the diagram consist of U(W) copies of V, U(V) copies
of W, and U(V)xU(W) copies of k (U is the forgetful functor to sets). And
the maps... well, you guess.

The reason this is relevant is that in the Barr's monoidal category, the
product of (S->U(A)) and (T->U(B)) is (SxT->U(C)) where C is the colimit,
in the category of algebras, of

                                 F(1)_{s,t}
                                 /    \
                                /      \
                              |_        _|
                            A_t          B_s

It does not look so ugly after all, does it?

:),
Mamuka