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Michael Barr | 5 Dec 2001 14:59

Re: Sketches and Platonic Ideas

There are a number of definitions of sketch around, some of which require
it to be a category with finite products.  In one of Ehresmann's (and
Bastiani's, I believe) there is mentioned the possibility of its being
what they called a quasicategory (or some such substructure term) in which
composition is a partly defined multi-ary operation (in other words, fgh
could be defined without fg or gh being defined).  Charles and I realized
that this was equivalent to what we called a graph with diagrams, which
seemed a more useable notion.  So what we called a sketch was a graph with
diagrams as well as certain cones and cocones that were singled out to be
taken to limits and colimits, resp.  Peter Johnstone criticized us for
doing the equivalent of replacing groups by generators and relations,
which is correct, but it was a conscious decision and there were reasons
for it.  I had never heard the term "idea" in this connection or we might
have used it.  But anyway, "sketch" is used in different ways and I guess
Charles and I contributed to this, but didn't create it. 

On Mon, 3 Dec 2001 baez <at> math.ucr.edu wrote:

> Toby Bartels writes:
> 
> > There could be multiple ideas that generate the same sketch;
> > how do we decide which is the correct idea among equivalent ones?
> > OTOH, if we take equivalence classes of ideas, then we're taking sketches.
> > For example, one could define the idea of multiplication in a monoid
> > as a binary operation and a nullary operation
> > or alternatively as an operation on finite tuples.
> > The former is more common, but I prefer the latter;
> > who has the right idea?
> 
> I'm confused: in my understanding, a sketch basically amounts to
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Tom Leinster | 4 Dec 2001 03:29
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Re: the walking adjunction and biadjunction


Toby Bartels wrote:
> 
> For example, one could define the idea of multiplication in a monoid
> as a binary operation and a nullary operation
> or alternatively as an operation on finite tuples.
> The former is more common, but I prefer the latter;
> who has the right idea?

An interesting question in itself.  I don't think either idea is "right", but
I (presumably) share with you the feeling that often the latter is more
appropriate.  However, if you resolve wholeheartedly never to use a binary +
nullary presentation of a monoid-like structure then you actually find
yourself in quite an extreme position.  For instance, a monoid would be
defined as a set M together with an n-fold operation

(m_1, ..., m_n) |---> [m_1 ... m_n]

on M for each natural n, subject to axioms.  This is as expected so far, but
we've disallowed ourselves from using what would probably be the natural
choice of axioms,

[[m_1^1 ... m_1^{k_1}] ... [m_n^1 ... m_n^{k_n}]] = [m_1^1 ... m_n^{k_n}],
m = [m],

since this is a binary + nullary presentation.  So instead we should derive
from the n-fold multiplications a k-ary operation o_T on M for each (finite,
planar) k-leafed tree T; and the axioms then become that o_T = o_U for any
two k-leafed trees T and U.
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F. William Lawvere | 5 Dec 2001 05:36
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Re: Sketches and Platonic Ideas

Certainly I did not mean to suggest that either John or Andree were
supporting platonism as a philosophy of mathematics. In fact I had
momentarily even forgotten that John had used the term. In my 1972
Perugia Notes I had made an attempt to characterize the relation between
these sorts of mathematical considerations and philosophy by saying that
while platonism is wrong on the relation between Thinking and Being,
something analogous is correct WITHIN the realm of Thinking. The relevant
dialectic there is between abstract general and concrete
general.
Not concrete particular ("concrete" here does not mean
"real").There is another crucial dialectic making particulars
(neither abstract nor concrete) give rise to an abstract
general; since experiments do not mechanically give rise to theory, it is
harder to give a purely mathematical outline of how that dialectic
works, though it certainly does work. A mathematical model of it can be
based on the hypothesis that a given set of particulars is somehow itself
a category (or graph), i.e., that the appropriate ways of comparing the
particulars are given but that their essence is not. Then their
"natural structure" (analogous to cohomology operations) is an
abstract general and the corresponding concrete general receives a
Fourier-Gelfand-Dirac functor from the original particulars. That
functor is usually not full because the real particulars are infinitely
deep and the natural structure is computed with respect to some
limited doctrine; the doctrine can be varied, or "screwed up or down" as
James Clerk Maxwell put it, in order to see various
phenomena.

From: baez <at> math.ucr.edu 
>To: categories <at> mta.ca (categories) 
>Subject: categories: Sketches and Platonic Ideas
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Charles Wells | 5 Dec 2001 21:52

Sketches

This is in reply to Toby Bartels, quoted below.  I don't believe that those 
of us who have written about "ideas" in Ehresmann's sense ever conceived 
that each theory (sketch) was based on one right idea.  There is no 
"correct" idea for a given sketch.

I want to add, for those new to the subject, that the word "sketch" has 
been used with at least three meanings.  Ehresmann and his students use it 
for a structure which is a weakening of the concept of category (the 
composite may not be defined for all composable pairs) plus specified cones 
and/or cocones.  Many others have used the word sketch to refer to a 
category with specified cones and/or cocones.  Michael Barr and I in our 
two books used "sketch" to mean a graph with specified cones and/or cocones 
plus some commutativity conditions on paths; that is in the same spirit as 
Ehresmann's "idea".

--Charles Wells

>Andree Ehresmann wrote in part:
>
> >He thought
> >first of calling a sketch an idea, but then reserved the word "idea" for
> >the smallest part which helps reconstruct the sketch; for instance for a
> >category, the arrows which 'represent' the domain and codomain maps and the
> >composition law.
>
>There could be multiple ideas that generate the same sketch;
>how do we decide which is the correct idea among equivalent ones?
>OTOH, if we take equivalence classes of ideas, then we're taking sketches.
>For example, one could define the idea of multiplication in a monoid
>as a binary operation and a nullary operation
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Toby Bartels | 6 Dec 2001 09:23
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Sketches and ideas (Was: the walking adjunction and biadjunction)

Tom Leinster wrote in part:

>Toby Bartels wrote:

>>For example, one could define the idea of multiplication in a monoid
>>as a binary operation and a nullary operation
>>or alternatively as an operation on finite tuples.
>>The former is more common, but I prefer the latter;
>>who has the right idea?

>An interesting question in itself.  I don't think either idea is "right",

That was supposed to be my point.
Just as a group can be described many ways by generators and relations,
so a sketch (if we define a sketch to be an entire category;
apparently that varies) can be described many ways by ideas.
It's the category that truly characterises what a monoid is
(in the given doctrine), so it better deserves the name "idea",
if we're trying to hark back to Plato-n (even just to be cute).
(Whether or not it's too late to change, I can't say.)

>we've disallowed ourselves from using what would probably be the natural
>choice of axioms,

>[[m_1^1 ... m_1^{k_1}] ... [m_n^1 ... m_n^{k_n}]] = [m_1^1 ... m_n^{k_n}],
>m = [m],

>since this is a binary + nullary presentation.

2 indices and 0 indices.  *Gulp*  You're right!
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Oswald Wyler | 6 Dec 2001 15:00

Reference wanted

I'm sure that the following is known, but I've never seen it in print.
Does someone have a reference for it?

Proposition.  Let U be a monadic functor, in the sense of Mac Lane's CWM.
If U factors U=HG with H faithful and amnestic, and G having a left adjoint,
then G is monadic.
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Colin McLarty | 6 Dec 2001 09:28
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Two Days

	Working on the history of category theory I find that Mahlon Marsh Day
hired several category theorists at the University of Illinois
Champaign-Urbana in the 1960s. I would like to know whatever people can
tell me about his connections to category theory--perhaps through Eilenberg?

	Also, does anyone here know whether Mahlon Michael Day was Mahlon Marsh
Day's son? Mahlon Michael Day got a PhD at Chicago in 1967 with Kaplansky.

Thanks, Colin
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Charles Wells | 6 Dec 2001 21:58

Defining monoids


In talking about defining monoids, Toby Bartels wrote:

"We could also go straight to trees and define them as the basic operations,
then requiring as axiom that grafting of trees produces the same result
as composing the operations."

This is the mu operation of the corresponding monad.  Every single-sorted 
"idea" in the sense of the recent discussion generates a monad in sets with 
a mu like this.  For each set S there is a set of possible computations TS, 
a mu:TTS to TS, and a "OneIdentity" operation in the sense of Mathematica 
that says the computation consisting of a single node results in that node; 
these subject to the monad laws.  In other words, the phenomenon you noted 
is an instance of a general result.

--Charles Wells

Charles Wells,
Emeritus Professor of Mathematics, Case Western Reserve University
Affiliate Scholar, Oberlin College
Send all mail to:
105 South Cedar St., Oberlin, Ohio 44074, USA.
email: charles <at> freude.com.
home phone: 440 774 1926.
professional website: http://www.cwru.edu/artsci/math/wells/home.html
personal website: http://www.oberlin.net/~cwells/index.html
genealogical website: 
http://familytreemaker.genealogy.com/users/w/e/l/Charles-Wells/
NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm
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Martin Escardo | 7 Dec 2001 16:20
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CFP: Workshop Domains VI

                 Call for abstracts

                     Domains VI

           Birmingham, 16-19 September 2002.

    The Workshop on Domains is aimed at computer scientists and
mathematicians alike who share an interest in the mathematical
foundations of computation. The workshop will focus on domains, their
applications and related topics.  Previous meetings were held in
Darmstadt (94,99), Braunschweig (96), Munich (97) and Siegen (98). The
emphasis is on the exchange of ideas between participants similar in
style to Dagstuhl seminars.

INVITED SPEAKERS

    Ulrich Berger         University of Wales Swansea
    Thierry Coquand       Goeteborg University
    Jimmie Lawson         Louisiana State University
    John Longley          University of Edinburgh
*   Dag Normann           University of Oslo
    Prakash Panangaden    McGill University
    Uday Reddy            University of Birmingham
    Thomas Streicher      Darmstadt University

* to be confirmed

SCOPE

    Domain theory has had applications to programming language
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Marco Grandis | 7 Dec 2001 11:43
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Preprint: Directed homotopy theory, II. Homotopy constructs,

The following preprint is available:

M. Grandis,
Directed homotopy theory, II. Homotopy constructs,
Dip. Mat. Univ. Genova, Preprint 446 (Dec 2001).
(19 p.)

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Abstract.

Directed Algebraic Topology studies phenomena where privileged directions
appear, derived from the analysis of concurrency, traffic networks,
space-time models, etc.

   This is the sequel of a paper, 'Directed homotopy theory, I. The
fundamental category', where we introduced directed spaces, their non
reversible homotopies and their fundamental category. Here we study some
basic constructs of homotopy, like homotopy pushouts and pullbacks, mapping
cones and homotopy fibres, suspensions and loops, cofibre and fibre
sequences.

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Part I and II are available, in ps:

ftp://www.dima.unige.it/Home/grandis/public/Dht1.ps
ftp://www.dima.unige.it/Home/grandis/public/Dht2.ps

___
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Gmane