From: John Duskin <duskin@math.buffalo.edu>
To: categories@mta.ca
Subject: Re: Michael Healy's question on math and AI
Date: Tue, 30 Jan 2001 14:54:53 -0500 [thread overview]
Message-ID: <p05001902b69cc3f60487@[208.28.188.38]> (raw)
At the risk of putting my head on a platter (as Mike Barr
said).....If you take the point of view of ``representable functors",
an object P, together with an orderded pair of arrows (pr_A:P --->A,
pr_B:P---->B) is a product of A with B iff for all objects T, the
mapping Hom(T,P)--->Hom(T,A)xHom(T,B) defined by
f|---->(f.pr_A,f.pr_B) is a bijection. If you compose this map with
the bijection
(a,b)|--->(b,a): Hom(T,A)xHom(T,B)--->Hom(T,B)xHom(T,A), you get that
P, together with the ordered pair of arrows ( pr_B:P---->B, pr_A:P
--->A) represents a "product of B with A". and this is different even
if we are talking about a product of A with itself. In other words,
the "product we usually are thinking about" is universal for ordered
pairs of arrows (a:T--->A,b:T--->B). If this means that "ordered
pair" needs to be made a primitive notion within the underlying logic
(as the Bourbakists did), so be it, because the simple translation of
the above statement out of "set theoretic" terms" into "category
theoretic" terms (no Hom-sets allowed) needs the notion of "ordered
pair" in order to state it independently of any overarching "theory
of sets".
next reply other threads:[~2001-01-30 19:54 UTC|newest]
Thread overview: 9+ messages / expand[flat|nested] mbox.gz Atom feed top
2001-01-30 19:54 John Duskin [this message]
-- strict thread matches above, loose matches on Subject: below --
2001-01-31 0:21 zdiskin
2001-01-29 15:21 S.J.Vickers
[not found] <F37M5o1gXXX3kRC9QnC00001cc1@hotmail.com>
2001-01-28 0:07 ` Michael Barr
2001-01-27 10:45 Colin McLarty
2001-01-26 21:37 Peter McBurney
2001-01-24 22:26 F. William Lawvere
2001-01-26 3:08 ` Todd Wilson
2001-01-26 18:14 ` Michael Barr
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