From: Vaughan Pratt <pratt@cs.stanford.edu>
To: categories@mta.ca
Subject: Re: Equality as an adjunction
Date: Fri, 03 Sep 2010 00:06:18 -0700 [thread overview]
Message-ID: <E1Orfgs-0007j0-UP@mlist.mta.ca> (raw)
In-Reply-To: <E1Or9Uc-00034j-JW@mlist.mta.ca>
On 9/1/2010 6:40 PM, David Leduc wrote:
> In "Categorical Logic", Pitts explains that equality can be defined as
> an adjunction. See Fig. 8, page 55
> (http://www.cl.cam.ac.uk/~amp12/papers/catl/catl.ps.gz)
>
> He writes the definition as a bidirectional typing rule for the
> internal language of a suitable category.
>
> Phi |- P(x,x) [x:A]
> ===================
> Phi, x=x' |- P(x,x') [x x':A]
>
> What are the left and right adjoint functors here?
There's a certain tension between logic and algebra here. The
algebraist's response to a question of this nature should be, adjunction
is third-order equality, where isomorphism is second-order and
commutative diagrams are first-order.
The adjunction case is best understood 3-categorically, representable
quite easily with origami. I can't find the origami I constructed to
demonstrate this, I should build it again and post a photo of it at some
point.
Using adjunctions to explain equality is fine, just as using a hammer
two sizes too large to crack a peanut works fine when wielded gently, as
Andy has done here.
Vaughan
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next prev parent reply other threads:[~2010-09-03 7:06 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 [this message]
2010-09-04 4:03 ` David Leduc
[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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