26 May 2012 00:09

## The Idea of Structure as Data and Conditions

In the 1952 document at
http://mathdoc.emath.fr/archives-bourbaki/PDF/nbt_029.pdf the only
mathematician
"pr\'{e}sent" referenced by first name only is Sammy.

I was permitted to audit a graduate course on category theory guided
by Sammy at Columbia University in the early 1960s.
I recall his insistence that mathematical structure is given by data
and conditions. Is that idea
implicit or explicit in Bourbaki?  Has that idea been superceded? How
does it relate to the
development of  algebraic theories as understood by Lawvere, Linton,
Barr-Wells, the Elephant, and so on?

Ellis D. Cooper

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]


27 May 2012 01:48

### Re: The Idea of Structure as Data and Conditions

Let me point out that not every structure comes with an obvious notion of
morphism.  For example, if I just gave the bare-bones definition of
topological space, the obvious definition of morphism would be open
mappings.  On complete lattices, we can have complete homomorphisms,
complete sup homomorphisms and, needless to say, complete inf
homomorphisms.  And I have recently helped characterize the injectives in
the category of partially-ordered monoids and marphisms that satisfy
f(x)f(y) =< f(xy).  There are Heyting algebras.  Isomorphisms are always
the same, so that is safe.

I never understood why the founding paper in category theory was called
"The general theory of natural equivalences", when they do consider more
general natural transformations.

Michael

On Fri, 25 May 2012, Ellis D. Cooper wrote:

> In the 1952 document at
> http://mathdoc.emath.fr/archives-bourbaki/PDF/nbt_029.pdf the only
> mathematician
> "pr\'{e}sent" referenced by first name only is Sammy.
>
> I was permitted to audit a graduate course on category theory guided
> by Sammy at Columbia University in the early 1960s.
> I recall his insistence that mathematical structure is given by data
> and conditions. Is that idea
> implicit or explicit in Bourbaki?  Has that idea been superceded? How
> does it relate to the
> development of  algebraic theories as understood by Lawvere, Linton,


27 May 2012 20:16

### Re: The Idea of Structure as Data and Conditions

On 26/05/12 20:48, Michael Barr wrote:

> I never understood why the founding paper in category theory was called
> "The general theory of natural equivalences", when they do consider more
> general natural transformations.

Well, I always understood that title as meaning that they GENERALIZE
the notion of natural equivalences that had as particular examples.

e.d.

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]


28 May 2012 08:00

### Re: The Idea of Structure as Data and Conditions

On Sat, 26 May 2012 19:48:26 -0400 (EDT), Michael Barr wrote:

> Let me point out that not every structure comes with an obvious notion of
> morphism.  --- [examples snipped] ---

The most common example: sets. The structure is based on membership.
Virtually no one ever wants to restrict attention to functions that respect
(preserve or reflect) membership (other than "preserve" between ordinals).

Cheers, -- Fred

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]


30 May 2012 08:26

### Re: The Idea of Structure as Data and Conditions

On 5/27/2012 11:00 PM, FEJ Linton wrote:
> Virtually no one ever wants to restrict attention to functions that respect
> (preserve or reflect) membership (other than "preserve" between ordinals).

Wouldn't that depend on the context in which membership arises?
Certainly group homomorphisms aren't expected to respect membership of
group elements in groups, but neither are they expected to respect the
composition of group homomorphisms equipping the category Grp.

The latter kind of respect is accorded categories by functors between
them.  By the same token the former kind is accorded elementary models
of set theory by elementary homomorphisms between them, the appropriate
counterpart of functors in that context.  Elementary homomorphisms
preserve elementary structure, which in the case of models of set theory
has membership as a basic part.  ZFC structure is very different (at the
bottom few layers) from the algebraic-in-Grph structure of the category
Set, which has function composition as a basic part.

For those who prefer algebra to logic, Joyal, Moerdijk and Awodey offer
Algebraic Set Theory, AST, as a middle ground here.  This replaces
membership by (set-sized) unions and singleton a |--> {a}.
Homomorphisms then have their usual algebraic meaning, which is arguably
less fiddly than for elementary maps.  Union allows the subset relation
to be defined as X <= Y  iff  X U Y = Y, from which one can then define
membership X e Y as {X} <= Y.  Both relations are preserved by the
homomorphisms of AST.  For a crash course see Awodey's

http://www.andrew.cmu.edu/user/awodey/preprints/astIntroFinal.pdf

ZFC, Set, and AST differ only at the bottom few layers, above which


27 May 2012 15:02

### Re: The Idea of Structure as Data and Conditions

Data and conditions constitute a presentation.  The graph, diagrams, cones
& cocones of a sketch are a presentation.  This idea has not been
superseded, not at all, but it has been completed (in two senses) by the
concept of theory, which is the  object generated by the presentation: The
theory of a sketch, the classifiying topos, the algebraic theory in the
sense of Lawvere, and so on.  This object contains all the information

That idea is in some way the other face of, or the complementary point of
view about, data and conditions.

Charles

On Fri, May 25, 2012 at 6:09 PM, Ellis D. Cooper <xtalv1 <at> netropolis.net>wrote:

> In the 1952 document at
>
http://mathdoc.emath.fr/**archives-bourbaki/PDF/nbt_029.**pdf<http://mathdoc.emath.fr/archives-bourbaki/PDF/nbt_029.pdf>the only
> mathematician
> "pr\'{e}sent" referenced by first name only is Sammy.
>
> I was permitted to audit a graduate course on category theory guided
> by Sammy at Columbia University in the early 1960s.
> I recall his insistence that mathematical structure is given by data
> and conditions. Is that idea
> implicit or explicit in Bourbaki?  Has that idea been superceded? How
> does it relate to the
> development of  algebraic theories as understood by Lawvere, Linton,
> Barr-Wells, the Elephant, and so on?
>


27 May 2012 15:06

### Re: The Idea of Structure as Data and Conditions

The sentence "contains all the info about any model" is ambiguous. I mean
the theory contains everything you can say that is correct about every
model. --C

On Sun, May 27, 2012 at 9:02 AM, Charles Wells <charles <at> abstractmath.org>wrote:

> Data and conditions constitute a presentation.  The graph, diagrams, cones
> & cocones of a sketch are a presentation.  This idea has not been
> superseded, not at all, but it has been completed (in two senses) by the
> concept of theory, which is the  object generated by the presentation: The
> theory of a sketch, the classifiying topos, the algebraic theory in the
> sense of Lawvere, and so on.  This object contains all the information
> about any model.
>
> That idea is in some way the other face of, or the complementary point of
> view about, data and conditions.
>
> Charles
>
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]


28 May 2012 01:25

### Re: The Idea of Structure as Data and Conditions

The proposition that "mathematical structure is given by data and
conditions" is so broad as to be vacuous from a foundational standpoint.
While there may be disagreement between mathematical camps over
whether algebraic frameworks rest on logical or vice versa, common to
both is the idea that one starts with a (non-logical) language and
equips it with a theory.  I don't see how "data and conditions" can be
interpreted as giving undue weight to either the equational or
first-order starting points.    The sentiment could just as validly
preface a graduate course on first order model theory.

Vaughan Pratt

On 5/25/2012 3:09 PM, Ellis D. Cooper wrote:
> I was permitted to audit a graduate course on category theory guided
> by Sammy at Columbia University in the early 1960s.
> I recall his insistence that mathematical structure is given by data
> and conditions. Is that idea
> implicit or explicit in Bourbaki? Has that idea been superceded? How
> does it relate to the
> development of algebraic theories as understood by Lawvere, Linton,
> Barr-Wells, the Elephant, and so on?

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]


29 May 2012 05:15

### Re: The Idea of Structure as Data and Conditions

On Mon, 28 May 2012 07:25:19 PM EDT, Florian Lengyel
<florian.lengyel <at> gmail.com>, protesting my assertion that

>> Virtually no one ever wants to restrict attention to functions that
respect
>> (preserve or reflect) membership (other than "preserve" between ordinals).

remonstrated that

> Within set theories that satisfy the axiom of regularity, one's attention
> is restricted to functions that both preserve and reflect self-membership.
>
> f(x) \in f(x) iff x\in x

Hereto, I in turn ask: Why only self-membership? why not membership outright

-- f(x) \in f(y) if (and/or only if) x \in y -- ?

And how often, really, do we actually impose either of those restrictions? (Or

did FL inadvertently omit a "sometimes" between "is" and "restricted"   ?)

Cheers, -- Fred

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]


29 May 2012 15:28

### Re: The Idea of Structure as Data and Conditions

Something has gone entirely wrong here.

The axiom of regularity forbids x\in x.

colin

On Mon, May 28, 2012 at 11:15 PM, Fred E.J. Linton <fejlinton <at> usa.net> wrote:
> On Mon, 28 May 2012 07:25:19 PM EDT, Florian Lengyel
> <florian.lengyel <at> gmail.com>, protesting my assertion that
>
>>> Virtually no one ever wants to restrict attention to functions that
> respect
>>> (preserve or reflect) membership (other than "preserve" between ordinals).
>
> remonstrated that
>
>> Within set theories that satisfy the axiom of regularity, one's attention
>> is restricted to functions that both preserve and reflect self-membership.
>>
>> f(x) \in f(x) iff x\in x
>
> Hereto, I in turn ask: Why only self-membership? why not membership outright
>
> -- f(x) \in f(y) if (and/or only if) x \in y -- ?
>
> And how often, really, do we actually impose either of those restrictions? (Or
>
> did FL inadvertently omit a "sometimes" between "is" and "restricted"    ?)
>
> Cheers, -- Fred


29 May 2012 15:28

### Re: The Idea of Structure as Data and Conditions

I am not certain that is a good example, actually. The membership relation is certainly what gives the
structure of a model of ZFC, so it may be relevant to morphisms between such models. However, it does not
have to have anything to do with the structure of individual sets.

The structure of a single set is, I believe, most fruitfully thought of as just consisting of the identity
relation on its elements. That way, the morphisms come out as functions (i.e. identity-preserving total
relations), just as we expect them to.

Best wishes,
Staffan

________________________________________
Från: FEJ Linton [FLinton <at> Wesleyan.edu]
Skickat: den 28 maj 2012 08:00
Till: categories <at> mta.ca
Ämne: categories: Re: The Idea of Structure as Data and Conditions

On Sat, 26 May 2012 19:48:26 -0400 (EDT), Michael Barr wrote:

> Let me point out that not every structure comes with an obvious notion of
> morphism.  --- [examples snipped] ---

The most common example: sets. The structure is based on membership.
Virtually no one ever wants to restrict attention to functions that respect
(preserve or reflect) membership (other than "preserve" between ordinals).

Cheers, -- Fred

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]