Dear Christopher, What I, personally, mean by structure is not the point. This word is used, very often, in mathematical texts. Sometimes giving the impression that it has a precise meaning on which the community of mathematicians agree. And I was sure there was at least one definition on which the majority of users did agree Then I received 3 answers all referring to: The joy of Cats, but different: For Carsten Führman, only faithfulness is required, which obviously is not enough Jiri Adamek adds: an isomorphism in S is an identity if its image is. I agree with this; but again not enough. Thomas Streicher adds a third condition, with which I would probably agree if was sure of the precise meaning of isofibration. Could you please, even at the risk of being pedantic say what you mean by that Many thanks to all
â€ژHi Jean - I don't quite understand this question but would like to. What do you mean by 'structure'? Thanks
Sent from my BlackBerry 10 smartphone on the O2 network. Original Message From: Jean Benabou Sent: Wednesday, 8 February 2017 16:18 To: Categories Reply To: Jean Benabou Subject: categories: Terminology
Dear all,
I'm sure the following question has been answered to. Could anyone give me a precise answer and references to this answer. Many thanks.
QUESTION Let p: S --> X be a functor. What conditions should satisfy p to be called a structure functor, i.e. such that every object s of S can be thought of as a structure on the object p(s).
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Jean, My own understanding (superficial and possibly wrong) of the history is that since Bourbaki there have been definitions of "structure" with the aim of reconciling the algebraic examples (where the homomorphisms preserve structure) with the topological spaces (where the continuous maps have inverse images that preserve structure). Certainly if you look at Joy of Cats, the prime classes of examples are those of topological and algebraic categories. But, as we know from topos theory, it is not foundationally robust to treat topological spaces as "sets with structure", i.e. point-set topology. In general we have to work point-free, at least if we want to save important parts of topology from going down the drain. If such an important source of examples, the point-set topological spaces, turned out to be misleading, then, in retrospect, any "precise meaning [of structure] on which the community of mathematicians agree", was probably misguided. It's like looking for a definition of "fish", but on the understanding that it has to include whales. All the best, Steve.
On 9 Feb 2017, at 16:38, Jean Benabou <jean.benabou@wanadoo.fr> wrote:
Dear Christopher, What I, personally, mean by structure is not the point. This word is used, very often, in mathematical texts. Sometimes giving the impression that it has a precise meaning on which the community of mathematicians agree. And I was sure there was at least one definition on which the majority of users did agree
Then I received 3 answers all referring to: The joy of Cats, but different: For Carsten Führman, only faithfulness is required, which obviously is not enough Jiri Adamek adds: an isomorphism in S is an identity if its image is. I agree with this; but again not enough. Thomas Streicher adds a third condition, with which I would probably agree if was sure of the precise meaning of isofibration. Could you please, even at the risk of being pedantic say what you mean by that
Many thanks to all
â€ژHi Jean - I don't quite understand this question but would like to. What do you mean by 'structure'? Thanks
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Steve, et al., If you want
a definition of "fish", but on the understanding that it has to include whales
let me offer: "legless marine vertebrates" :-) . Cheers, -- tlvp [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Andree, I admire very much the pioneer work of Charles Ehresmann on structures, but I think faithful+amnestic is too weak. if S is contained in X the inclusion S --> X satisfies these two properties . Let X se the category of finite sets and S be the full subcategory of finite ordinals. Do you know any structure on finite sets such that 1 has that structure and some other singleton doesn't? Thus if an inclusion is a structure S should be a replete subcategory of X. Dear Steve, My question was NOT what is a structure on an object of a category, but what is a structure FUNCTOR p: S --> X. Topological spaces are the objects of many DIFFERENT categories S, e.g. with morphisms continuous, open, closed, proper maps, each of these categories is equipped with a functor with codomain Sets. All these functors coincide on the objects, but their properties AS FUNCTORS are completely different. The situation is the same with toposes with geometric, open, closed, and proper morphisms and the structure functorS with codomain in CAT. (As you see neither points nor whales are involved; but they could be with Fred's help ). Dear Thomas, I totally agree with your definition. which is confirmed by George Janelidze. I asked the question because I have seen the word structure used with various meanings; and sometimes without any meaning at all.In particular in the nLab. I hope the category community agrees with this definition. I shall wait a few days to see if there are any more comments, and then write a new mail to show how rich this definition is, and how much can be done with it. Meanwhile let me add a definition, and make a few remarks. DEFINITION.. A structure functor p: S --> X is algebraic if p reflects isos. REMARKS ON TERMINOLOGY Let me first introduce some notations adapted to the typographical requirements of this list. If p; S --> X is a functor. I denote by Iso(S) and Iso(X) the groupoids of or isos of S and X and by p/Iso the restriction of p to these groupoids. For each object x of X I denote by p*(x) the fiber of p over x . As I do not want to run into set-class distinctions, which are meaningless here, I shall assume S and X small. I think we should call transportable what is called uniquely transportable in the book on Algebraic Theories (AT), and weakly transportable what they call transportable. This would fit with Bourbaki's definition, given even before the birth of Category Theory,. It means that p/Iso is not merely a fibration but a DISCRETE fibration Thus we would say that p is a structure (resp. a weak structure) if it is faithful and transportable (resp. weakly transportable). In presence of faithfulness the weak case amounts to saying that p/Iso is not merely a fibration but what I call an elementary fibration, i.e. equivalent to a discrete fibration, which is then unique up to a unique isomorphism of fiibrations. What about amnesticity? I think it should be kept as a very secondary notion. It says nothing global about the functor p but only about its fibers; which isn't much unless p is a fibration. Let me make an analogy. If p is faithful all the fibers are posets. The converse is of course not true. Would anyone consider functors such that all the fibers are posets without any cohesion between these posets? Or again; p is amnestic iff its restriction to the subcategory Vert(p) of vertical maps of p is amnestic. It is only in presence of other strong axioms on p that amnesticity makes sense. (Look also at my answer to Andree). I'd of course appreciate any comments or objections, in particular about the last definition and the terminology I propose in the last section. Many thanks., Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Fred, A good answer, but my point was that it was a bad question. You see this once you start pressing at the details. Are seals and turtles fish? No, but on your definition it depends on whether flippers count as legs or not. What about sea snakes? Obviously not - they're snakes, that just happen to live in the sea. But then eels do seem a bit more fishy. A meticulous zoologist would start piling on the subclauses to pin it down more precisely, but we know that that does not actually refine our understanding of zoology. It just amplifies the misconceptions underlying the original question. I'm saying the same can happen in mathematics. All the best, Steve.
On 11 Feb 2017, at 20:42, Fred E.J. Linton <fejlinton@usa.net> wrote:
Steve, et al.,
If you want
a definition of "fish", but on the understanding that it has to include whales
let me offer: "legless marine vertebrates" :-) .
Cheers, -- tlvp
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Steve, I totally agree with you. Let me apply your zoological criteria to Category Theory. You begin with the very simple notions of category, functor and natural transformation. But then you start piling in subclauses such as categories with finite limits, or regular, or abelian, or the glorious toposes. For functors you refine the notion to fully faithful ones or those who have an adjoint, or are flat, or are fibrations. I could give hundreds of examples, and even a meticulous zoologist would say:To much is to much! Obviously Category Theory is very bad and the very idea of putting in a same bag groups, topological spaces, locales, and the glorious toposes is a misconception. Serious mathematicians agreed with this. You are too young to remember the time when these mathematicians called, with zoological justification, this theory : Abstract general nonsense. All the best, Jean Le 14 févr. 17 à 09:48, Steve Vickers a écrit :
Dear Fred,
A good answer, but my point was that it was a bad question.
You see this once you start pressing at the details. Are seals and turtles fish? No, but on your definition it depends on whether flippers count as legs or not. What about sea snakes? Obviously not - they're snakes, that just happen to live in the sea. But then eels do seem a bit more fishy.
A meticulous zoologist would start piling on the subclauses to pin it down more precisely, but we know that that does not actually refine our understanding of zoology. It just amplifies the misconceptions underlying the original question.
I'm saying the same can happen in mathematics.
All the best,
Steve.
On 11 Feb 2017, at 20:42, Fred E.J. Linton <fejlinton@usa.net> wrote:
Steve, et al.,
If you want
a definition of "fish", but on the understanding that it has to include whales
let me offer: "legless marine vertebrates" :-) .
Cheers, -- tlvp
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
-
Fred E.J. Linton -
Jean Benabou -
Steve Vickers