From: Dusko Pavlovic <D.Pavlovic@doc.ic.ac.uk>
To: u-reddy@cs.uiuc.edu (Uday S. Reddy)
Cc: categories@mta.ca
Subject: Re: Quantifiers for monoids
Date: Mon, 2 Mar 1998 11:16:12 +0000 (GMT) [thread overview]
Message-ID: <199803021116.LAA05245@beauty.doc> (raw)
In-Reply-To: <199803010524.XAA25141@reddy.cs.uiuc.edu> from "Uday S. Reddy" at Feb 28, 98 11:24:32 pm
[Note from moderator: apologies to Dusko for prepending nonsense to his
post, regards, Bob]
Uday S. Reddy:
> Consider a monoid <M,*,1> in a CCC. The operations of interest are
> natural transformations E_A : [A => M] -> M that satisfy the following
> equations (in the internal language of the CCC):
>
>1) E_A(\lambda x. 1) = 1
>2) E_A(\lambda x. a * g`x) = a * E_A(g)
>3) E_A(\lambda x. g`x * a) = E_A(g) * a
>4) E_A(\lambda x. E_B(\lambda y. h`x`y)) =
> E_B(\lambda y. E_A(\lambda x. h`x`y))
The naturality of E_A in A seems to be a very strong requirement
(provided that am not misuncerstanding anything, ofcourse).
Let T the terminal object (since 1 already denotes the unit of
M). Equations (1) and (2) imply
E_T(\lambda x. a) = a,
so E_T is iso. The naturality, on the other hand, implies that for
every a,b : T --> A holds
a=>M ; E_T = E_A = b=>M ; E_T
Such E_A, I think, shouldn't be thought of as a quantifier: modulo
E_T, it actually boils down to the *evaluation* at (ie substitution
of) an arbitrary global point of A:
E_A(g) = g`a = g`b
Instanciating A = M and g = \lambda x. x yields a = 1, ie 1: T --> M
is the only global point of M. If T generates, M is T.
On the other hand, if there is an initial object 0 in the CCC, the
naturality in A implies that all E_A are constantly 1...
Without the naturality, conditions (1--4) seem to be rather easy to
satisfy. For a fixed A, conditions (2) and (3) are a kind of
naturality on E_A itself. To make this precise, define on A=>M the
pointwise monoid structure
g * h = \lambda x. g`x * h`x
Thinking of M and A=>M as categories, each with one object *, we have
the Hom-functors from M^op x M and from A=>M^op x A=>M to the CCC
where they live.
On the other hand, there is the monoid morphism
I : M ---> A=>M
m |--> \lambda x. m
which can be construed as a functor. Precomposing Hom_{A=>M} with I^op
and I, we get a functor from M^op x M. Conditions (2) and (3) are now
exactly the naturality of
E_A : Hom_{A=>M} (I*, I*) ---> Hom_M (*,*)
in each of the arguments. Hom_M thus appears as a retract of the
functor Hom_{A=>M} o (I^op x I)... All together, E_A thus appear as a
weak kind of *abstraction operators* (like in
www.cogs.susx.ac.uk/users/duskop/papers/CLNA.ps.gz).
All this may not be enlightening at all, but it does seem to help pin
down the models: eg, evaluations/substitutions of the arbitrary points
will work again, as well as, when M is a complete lattice, the infima
and the suprema, corresponding to the quantifiers... (No naturality in
A, ofcourse.)
Apologies if I got carried away a bit. It's a nice question. I hope
others will tell more.
Regards,
-- Dusko Pavlovic
prev parent reply other threads:[~1998-03-02 11:16 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
1998-03-01 5:24 Uday S. Reddy
1998-03-02 11:16 ` Dusko Pavlovic [this message]
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