* Re: Barr's symmetric monoidal closed comma category
@ 1998-09-15 4:42 HASEGAWA Masahito
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From: HASEGAWA Masahito @ 1998-09-15 4:42 UTC (permalink / raw)
To: categories
Related to Mike and Paul's messages to categories, I had the
following observation in my recent draft paper
(http://www.kurims.kyoto-u.ac.jp/~hassei/papers/basic.ps.gz
"Logical predicates for intuitionistic linear logic", Lemma A.1):
Suppose that C and D are symmetric monoidal closed
categories and that G:C->D is a symmetric monoidal
functor. Moreover suppose that D has pullbacks.
Then the comma category (D,G) can be given a symmetric
monoidal closed structure, so that the obvious projection
(D,G)->C is strict symmetric monoidal closed.
I used this (together with the free symmetric monoidal cocompletion)
for deriving "logical predicates (logical relations)" for
intuitionistic linear type theories, thus in a similar way as Lafont's
use of glueing for typed lambda calculi and ccc. In this paper I also
have another lemma for glueing symmetric monoidal adjunctions, for
interpreting the modality !.
I have been wondering if this is a sort of folklore, but never found a
reference.
Best Regards,
Masahito.
Masahito Hasegawa
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502 Japan
MAIL: hassei@kurims.kyoto-u.ac.jp
URL: http://www.kurims.kyoto-u.ac.jp/~hassei
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* Barr's symmetric monoidal closed comma category
@ 1998-09-14 0:55 Paul Taylor
0 siblings, 0 replies; 2+ messages in thread
From: Paul Taylor @ 1998-09-14 0:55 UTC (permalink / raw)
To: categories
Mike Barr asked whether anyone had been aware that the comma category
(F,A) or equivalently (S,U)
is symmetric monoidal closed, where
A
^ |
F| -| |U
| V
S
is a monoidal (tripleable) adjunction over S=Set.
As he said, an object (S,s,A) is an arrow s:FS--> A
(or, by adjointness, an S-tuple of elements of 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.
He said that 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).
I have certainly seen comma categories like (S,U); they are referred to
as gluing, sconing or the Freyd cover. (BTW Peter says that "scone" is
a corruption of Sierpinski cone, so its correct pronunciation is presumably
"shco:ne".) This construction provides an almost magical proof of
strong normalisation, consistency and similar results for various
fragments of symbolic or categorical logic. The algebraic theory that
I have in mind here for Mike's monadic adjunction is that which describes
the fragment of logic in question (the Australians would prefer us to
talk about a 2-monad here).
An early example of such a proof in the unification of these two
traditions was given by Yves Lafont in an appendix to his thesis.
He proved that the embedding of a category in the free CCC that it generates
is full and faithful. In the course of this he described the exponential
in (S,U), which is what Mike's formula gives, though I didn't notice
(and apparently nor did Yves) that the A-object which we have is the
internal version of Mike's Hom-set.
Yves Lafont's result appears as part of a generic account of the gluing
construction in Section 7.7 of my forthcoming book.
Paul
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