Re: property_vs_structure
Dear Eduardo,>In a previous message of mine and in response to your questions >
What do we mean by structure ?, and, what do we mean by property ?
I wrote>
On a given data, a structure is additional data on it such that, if it exists, it is not necessarily unique up to isomorphism. >>A property is additional data such that, if it exists, it is unique up to isomorphism (in the model theoretic sense).> to which you replied> Well, here it is necessary first to establish what do we mean by "isomorphism". To do this we need a way to compare the structures, that is, we have to define morphism of structures (see below). Without this, the above is meaningless.> Notice that I specified "in the model theoretic sense". In that context, the notions of a structure and of a morphism of structures are defined and it is that sense that I meant them. They are perfectly meaningful.
You added>
People discussing structure vs property were giving examples where all this was clear and straightforward (invertibility in a monoid, neutral element in a semigroup, etc). My original purpose when I wrote my first mail was to consider a less trivial example testing the following "definitions", that I see you subscribe above at least in what it concerns "structure" and "property":
Michael Shulman wrote:
(**) property = forgetful functor is full and faithful structure = forgetful functor is faithful property-like structure = forgetful functor is pseudomonic
I do not "subscribe" to these notions. I simply use the well-known notions from first-order logic and model theory as I said earlier. I am aware of the difference between the locally connected and the general case concerning covering projections and it is not the mathematics that I was disputing. But I still have trouble following your analysis of this situation as was your purpose. The difference is that, whereas you consider the notion of a covering projection to be a property of a continuous map p from X to B which, in the general (non locally connected case) is a "hidden structure", I view the notion of a covering projection to be a structure in the first place and, in the specific locally connected case, one that may be equivalently reduced to a property. Every property is a structure, but not every structure is reducible to a property.
All the best, Marta [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear All, Apologizing to those who heard this many times, I would like to say - since we are talking about "property_vs_structure" for covering maps: What I called "Galois structures" more than 20 years ago were exactly the structures needed to define covering morphisms in general categories. It is well known (although it is not clear what it really means!) that Poincare's first ideas about covering maps were inspired by Galois theory, and so algebra was "there" even before topology. Having also in mind Grothendieck's work, topos-theoretic developments, and Magid's work, I am certainly not original in saying that covering maps should belong to category theory rather than to topology. A Galois structure (say, on a category C) essentially consists of three ingredients (although various modifications are possible): (i) A functor I : C ---> X. It should better have a right adjoint, and it is wonderful if it is semi-left-exact or, which is almost the same in a sense, if it is a fibration. But relative versions of these conditions involving F below are also good. (ii) A class F of morphisms in C. All covering morphisms we are going to define will be inside F. (iii) Another class E of morphisms in C. Although usually I do not mention it separately because I prefer to define it as the class of effective F-descent morphisms. According to the terminology, Marta and Eduardo used in their messages, E should now be called a "trivialization structure". For a given Galois structure, the covering morphisms are defined as in my papers, and I would repeat the question about "property vs structure" for covering morphisms as follows: Given a category C, where the concept of a covering seems to be important, do we want to fix "the best" Galois structure, or we should consider several/many such structures? It is certainly a matter of taste, but to feel the taste one surely needs examples. In my opinion the ones listed below are especially important; they were investigated together with several people you know, whom I was very honoured to work with. Example 1. C = the opposite category of commutative rings. Here we have a very good candidate, which is: X = the category of Stone spaces; I : C ---> X = the Boolean spectrum functor; F = the class of all morphisms in C; E = the class of all effective descent morphisms in C. In this case covering morphisms are the same what A. R. Magid called componentially locally strongly separable algebras (considering an R-algebra A as a morphism A ---> R in C), and are THE most general algebras for which he developed his "separable Galois theory" presented in [A. R. Magid, The separable Galois theory of commutative rings, Marcel Dekker, 1974]. Example 2. C = the category of locally connected topological spaces. Here again, we seem to have "the best candidate". It is: X = the category of sets; I : C ---> X = the functor sending spaces to the sets of their connected components; F = the class of local homeomorphisms of locally connected spaces; F = the class of surjective maps from E. The covering morphisms are then the same the covering morphisms of locally connected spaces in the usual sense. Example 3. C is a locally connected topos. This essentially generalizes the previous example and everything happens as there (although here F is the class of all morphisms and E the class of all epimorphisms in C of course). Marta knows much more than I do about this example and its connections with other topos-theoretic constructions; my only contribution is the short paper [G. Janelidze, A note on Barr-Diaconescu covering theory, Contemporary Mathematics 131, 3, 1992, 121-124]. Example 4. C = Fam(A) (or FiniteFam(A)), where A is an arbitrary category with terminal object and "multi-pullbacks" (which simply means that C has pullbacks). This is a further generalization of the same thing, and everything can be repeated, but instead of "epimorphism" we should say "effective descent morphisms" (which is the same thing in the case of a topos). There are many non-topos-theoretic important special cases. For instance if C is the category of all (small) categories, then the covering morphisms are as they should be, that is functors that are discrete fibrations and discrete opfibrations at the same time (this observation is due to Steve Lack, although Steve never published it). If C is the category of all (small) groupoids, then this becomes even nicer since the discrete fibrations of groupoids are the same as discrete opfibrations, are Ronnie Brown often tells us how nicely can they be used in homotopy theory... Example 5. C = the category of compact Hausdorff spaces. Here "the best candidate" seems to be: X = the category of Stone spaces; I : C ---> X sending compact Hausdorff spaces to the Stone spaces of their connected components; F = the class of all morphisms in C; E = the class of all morphisms in C that are surjections. As shown in [A. Carboni, G. Janelidze, G. M. Kelly, and R. Paré, On localization and stabilization of factorization systems, Applied Categorical Structures 5, 1997, 1-58], the covering morphisms here are the same as light maps in the sense of Eilenberg and Whyburn. Example 6. C = the category of simplicial sets. Since it is a category of the form Fam(A), Example 4 can be used. However, [R. Brown and G. Janelidze, Galois theory of second order covering maps of simplicial sets, Journal of Pure and Applied Algebra 135, 1999, 23-31] gives a Galois structure that produces a larger (new) class of covering morphisms. In that structure X is the category of groupoids; I : C ---> X the fundamental groupoid functor, F the class of Kan fibrations, and E the class of surjective Kan fibrations. Another "simplicial Galois theory" is presented in [M. Grandis and G. Janelidze, Galois theory of simplicial complexes, Topology and its Applications 132, 3, 2003, 281-289]. Example 7. C = the category of groups. The "most classical" candidate would be: X = the category of abelian groups; I : C ---> X = the abelianization functor; E = F = the class of group epimorphisms. In this case the covering morphisms are the same as central extensions. This was my first example of a "very-non-Grothendieck Galois theory". A bit later I realized that C can be replaced with any variety of groups with multiple operators in the sense of [P. J. Higgins, Groups with multiple operators, Proc. London Math. Soc. (3)6, 1956, 366-416] and X with any subvariety in C - and then we get central extensions relative to a subvariety in the sense of A. Fröhlich's school (see e. g. [A. Fröhlich, Baer-invariants of algebras, Trans. AMS 109, 1963, 221-244], [A. S.-T. Lue, Baer-invariants and extensions relative to a variety, Proc. Cambridge Philos. Soc. 63, 1967, 569-578], [J. Furtado-Coelho, Varieties of W-groups and associated functors, Ph.D. Thesis, University of London, 1972]). The next step was to get rid of groups completely and Max Kelly and I found out that the crucial property that helps to work with generalized central extensions is congruence modularity, and we wrote [G. Janelidze and G. M. Kelly, Galois theory and a general notion of a central extension, Journal of Pure and Applied Algebra 97, 1994, 135-161]. Many further results were obtained by Marino Gran, partly in collaboration with Dominique Bourn (see [M. Gran, Applications of categorical Galois theory in universal algebra, Fields Institute Communications 43, 2004, 243-280] and references there for what was done until 2002/3), Tomas Everaert and Tim Van der Linden, and by Tim and Tomas separately and together. Writing this I feel now bad not to say more about their brilliant results, also involving higher central extensions (see in particular [T. Everaert, M. Gran, and T. Van der Linden, Higher Hopf formulae for homology via Galois theory, Advances in Mathematics 217, 2008, 2231-2267]). I shall gladly say more at another occasion. Examples 8-11?. Let C be one of the following three categories: (a) topological spaces; (b) locales; (c) toposes; (d) schemes of algebraic geometry (although (c) is 2-dimensional). I still do not know anything like "the best candidates"... And finally there are trivial examples: (a) C = X, with I the identity functor; and (b) X = 1. Taking (in both of them) E to be the class of all morphisms in C (and suitable F) we will have: all morphisms in C are covering morphisms in the situation (a), and only isomorphisms are covering morphisms in the situation (b). In particular, since "the largest" Galois structure is trivial, I would conclude: there is no "the best" Galois structure, and one should rather consider several/many such structures. George [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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George Janelidze -
Marta Bunge