Chew on this

Chew on this

[Michael Barr]

Has anybody seen the following symmetric closed monoidal category?  Let
#A# be a category tripleable over Set.  Let #V# be the comma category
(F,#V#), F being the free functor.  So an object (S,s,A) is an arrow s:FS
--> A and a map (S,s,A) to (T,t,B) is a pair f: S --> T and g: A --> B
making the obvious square commute.  The closed structure (S,s,A) --o
(T,t,B) is a certain arrow of the form (Hom((S,s,A),(T,t,B),?,B^S).  The
monoidal structure is fairly ugly, but it exists. 

Of course, an object (S,s,A) can also be thought of as an S-tuple of
elements of A, by adjointness.

Michael

Re: Chew on this

[Mamuka Jibladze]

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