From: David Leduc <david.leduc6@googlemail.com>
To: categories@mta.ca
Subject: Re: Equality as an adjunction
Date: Sat, 4 Sep 2010 04:03:02 +0000 [thread overview]
Message-ID: <E1OrtAr-0008CX-5B@mlist.mta.ca> (raw)
In-Reply-To: <E1Orfgs-0007j0-UP@mlist.mta.ca>
Thank you all for your replies. If I understood you well, the
adjunction is as follows:
[C,2](P o Delta, Q) =~ [CxC,2](P, Ran_Delta Q)
where Ran_Delta Q is the right Kan extension if Q along Delta.
(I write [C,D] for the functor category, and 2 for the category with
only two objects True and False and identities). If T is the constant
functor that sends every object to True, then Ran_Delta T : CxC -> 2
is the equality predicate, right?
Now I write this adjunction in logical form:
P(x,x) |- Q(x)
==============
P(x,y) |- Q(x) /\ x=y
But in Pitts it was reversed:
Q(x) |- P(x,x)
==============
Q(x) /\ x=y |- P(x,y)
Where is my mistake?
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next prev parent reply other threads:[~2010-09-04 4:03 UTC|newest]
Thread overview: 8+ messages / expand[flat|nested] mbox.gz Atom feed top
2010-09-02 1:40 David Leduc
2010-09-02 15:22 ` Robert Seely
2010-09-02 17:56 ` Michael Shulman
2010-09-02 22:54 ` Andrej Bauer
2010-09-03 7:06 ` Vaughan Pratt
2010-09-04 4:03 ` David Leduc [this message]
[not found] ` <AANLkTikUtD3SbB+4OBRpniqgLRBg0szKYkMSwiWx+ycr@mail.gmail.com>
2010-09-04 7:21 ` David Leduc
[not found] ` <Pine.LNX.4.64.1009041105300.2080@prism.math.mcgill.ca>
2010-09-05 2:26 ` David Leduc
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