* Chew on this
@ 1998-09-08 17:58 Michael Barr
1998-09-25 12:55 ` Mamuka Jibladze
0 siblings, 1 reply; 2+ messages in thread
From: Michael Barr @ 1998-09-08 17:58 UTC (permalink / raw)
To: Categories list
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
^ permalink raw reply [flat|nested] 2+ messages in thread
* Re: Chew on this
1998-09-08 17:58 Chew on this Michael Barr
@ 1998-09-25 12:55 ` Mamuka Jibladze
0 siblings, 0 replies; 2+ messages in thread
From: Mamuka Jibladze @ 1998-09-25 12:55 UTC (permalink / raw)
To: Categories list
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
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1998-09-08 17:58 Chew on this Michael Barr
1998-09-25 12:55 ` Mamuka Jibladze
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