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 Has that idea been superceded?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/ ]
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 Has that idea been superceded?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, Barr-Wells, the Elephant, and so on?
Ellis D. Cooper
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
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 On Fri, May 25, 2012 at 6:09 PM, Ellis D. Cooper <xtalv1@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?
Ellis D. Cooper
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
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/ ]
On Mon, 28 May 2012 07:25:19 PM EDT, Florian Lengyel <florian.lengyel@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/ ]
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@usa.net> wrote:
On Mon, 28 May 2012 07:25:19 PM EDT, Florian Lengyel <florian.lengyel@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/ ]
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@Wesleyan.edu] Skickat: den 28 maj 2012 08:00 Till: categories@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/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 foundational variations are tied to more fundamental issues involving Choice vs. Determinacy etc. One might compare the differences at the bottom with the wave-particle dichotomy in quantum mechanics or the event-state and time-information dichotomies in concurrency that I spoke on at Physics & Computation 1992 and 1994, see http://boole.stanford.edu/pub/ph94.pdf and the earlier (1992) http://boole.stanford.edu/pub/ql.pdf Or at least that's how it all looks to this outsider. Happy to be corrected on details I've got wrong. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (9)
-
Charles Wells -
Colin McLarty -
Eduardo J. Dubuc -
Ellis D. Cooper -
FEJ Linton -
Fred E.J. Linton -
Michael Barr -
Staffan Angere -
Vaughan Pratt