From: rjwood@mathstat.dal.ca (RJ Wood)
To: categories@mta.ca
Subject: Re: The boringness of the dual of exponential
Date: Wed, 9 Nov 2011 14:58:59 -0400 (AST) [thread overview]
Message-ID: <E1ROIlD-0003Qj-9T@mlist.mta.ca> (raw)
Your observation about lack of symmetry in \set is underscored by the
fact that the yoneda functor for \set has a left adjoint which has
a left adjoint which has a left adjoint which has a left adjoint but
the co-yoneda functor for \set has a right adjoint that fails to
preserve even finite sums.
R_j
> When David originally posted his question, I thought it was rather
> a silly one and that it was quite rightly dismissed by various
> people. On the other hand, he now says
>
>> However, I am not yet satisfied. Let me precise my thoughts. In the
>> textbooks and lecture notes on category category that I have read,
>> there are always product and coproduct, pullback and pushout,
>> equalizer and coequalizer, monomorphism and epimorphism, and so on.
>> However exponential is always left alone. That is why I assumed it is
>> boring. If it is not boring, why is it never mentioned in textbooks
>> and lecture notes on category theory?
>
> In other words, these things are "idioms" or "naturally occurring
> things" in mathematics, but there is a gap in the obvious symmetries.
>
> Looking for gaps in symmetries is a good thing to do. For example
> Dirac (whose biography by Graham Farmelo I have just started reading)
> predicted the positron this way.
>
> Actually, if we're looking at the categorical structure of the category
> of sets, it isn't very symmetrical at all. The second edition of
> Paul Cohn's "Universal Algebra" was evidently influenced by Mac Lane's
> famous textbook, but illustrates how categorists had way overemphasised
> duality.
>
> For example the terminal object yields the classical notion of element
> or point, whereas the initial object is strict and boring.
>
> Products and coproducts of sets are very different.
>
> I explored this kind of thing in my book. For example, the section
> on coproducts shows how different they are in sets/spaces and algebras.
>
> So David's question becomes a good one that deserves an answer if
> we read it as one about the phenomenology of mathematics rather than
> its technicalities.
>
> Paul Taylor
>
> PS There is a boring technical answer that I don't think anyone has
> mentioned, namely copowers, especially of modules.
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next reply other threads:[~2011-11-09 18:58 UTC|newest]
Thread overview: 22+ messages / expand[flat|nested] mbox.gz Atom feed top
2011-11-09 18:58 RJ Wood [this message]
-- strict thread matches above, loose matches on Subject: below --
2011-11-09 9:19 Reinhard Boerger
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
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