From: Andrej Bauer <andrej.bauer@andrej.com>
To: Jocelyn Ireson-Paine <popx@j-paine.org>
Cc: categories <categories@mta.ca>
Subject: Re: The boringness of the dual of exponential
Date: Thu, 10 Nov 2011 02:11:50 +0100 [thread overview]
Message-ID: <E1ROac3-0006hW-JC@mlist.mta.ca> (raw)
In-Reply-To: <E1RN2tu-0005v9-2T@mlist.mta.ca>
>> 1. "forall" goes with "weakening" because it is adjoint to it on the
>> right.
>> 2. "exists" goes with "weakening" because it is adjoint to it on the left.
>>
> What's the definition of "weakening"? I've not seen this word used formally.
Weakening is the map Sub(A) -> Sub(A x B) which takes a subobject M
>-> A to the subobject M x B >-> A x B, i.e., it is pullback along the
projection pi1 : A x B -> A.
The existential quantifier over B is the left adjoint to weakening,
where Sub(A) and Sub(A x B) are viewed as posetal categories. The
universal quantifier is the right adjoint. Working out the details in
Set is instructive.
As far as I know the terminology comes from logic. The rule of
weakening says that a logical derivation may be "wakened" with an
additional (but unneeded) hypothesis:
Gamma |- A
----------------------
Gamma, H |- A
With kind regards,
Andrej
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next prev parent reply other threads:[~2011-11-10 1:11 UTC|newest]
Thread overview: 22+ messages / expand[flat|nested] mbox.gz Atom feed top
2011-11-05 12:52 David Leduc
2011-11-06 20:22 ` FEJ Linton
2011-11-06 21:55 ` Thomas Streicher
2011-11-07 16:32 ` F. William Lawvere
2011-11-06 22:59 ` Ross Street
[not found] ` <F284B070-BBE5-4187-BA3C-E1A3EA560E6A@mq.edu.au>
2011-11-07 12:52 ` David Leduc
2011-11-08 16:20 ` Paul Taylor
2011-11-09 20:57 ` Uwe.Wolter
2011-11-10 9:29 ` Prof. Peter Johnstone
2011-11-11 7:47 ` Vaughan Pratt
2011-11-11 21:08 ` Robert Seely
2011-11-09 11:28 ` Andrej Bauer
2011-11-10 0:45 ` Jocelyn Ireson-Paine
2011-11-13 7:57 ` Vaughan Pratt
2011-11-14 13:36 ` Patrik Eklund
2011-11-15 13:03 ` Robert Dawson
[not found] ` <07D33522-CA8F-4133-A8E8-4B3BF6DFCCB4@cs.ox.ac.uk>
2011-11-16 18:06 ` Robert Dawson
2011-11-10 2:17 ` Peter Selinger
2011-11-07 21:23 ` Michael Shulman
2011-11-10 1:11 ` Andrej Bauer [this message]
2011-11-09 9:19 Reinhard Boerger
2011-11-09 18:58 RJ Wood
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