Topos theory for spaces of connected components
Topos theory gives a solid account of local connectedness, where each open - indeed, each sheaf - has a set (discrete space) of connected components. The definition of locally connected geometric morphism covers not only individual spaces but also bundles, considered fibrewise. It also covers generalized spaces as well as ungeneralized. Is there an analogous theory for where the space of connected components is Stone? ("Connected" is now defined by orthogonality with respect to Stone spaces instead of discrete spaces.) The obvious example is any Stone space X, for instance, Cantor space, where X is its own space of connected components. We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to notice the Stone space aspects in the usual examples based on real analysis, since they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components. (By the way, if you wonder what brought me to this, it was from pondering the symmetric monad M on Grothendieck toposes. Bunge and Funk proved that for ungeneralized spaces its localic reflection is the lower powerlocale, which raises the question of whether there is a missing topos construction whose localic reflection is the upper powerlocale. On the other hand, the symmetric monad is related to local connectedness. Points of MX are cosheaves on X, and X is locally connected if there is a terminal cosheaf in a strong sense, with that cosheaf providing the sets of connected components. Perhaps understanding the Stone space view of connected components would cast light on this missing construction.) All the best, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Steve, I have nothing to say about your Stone spaces question in general, except for your remarks in the second part of your message about the symmetric monad M, where you suggest that the Stone locale view of connected components would perhaps cast light on the missing construction of a topos version N of the upper power locale P_U, just as the symmetric monad M is a topos version of the lower power locale P_L. In my paper “Pitts monads and a lax descent theorem” (2015), (Remark 7.6), I leave it as an open question (more or less) the construction of such an N. [ The name “Pitts monad” I gave on account on a condition which first appears in a theorem of A.M. Pitts whereby, in a lax pullback with bottom map an S-essential geometric morphisms, the top map is locally connected. The S-essential geometric morphisms are precisely the M-maps, and for the lower power locale monad P_L, the P_L-maps are the open maps. ] However, toposes are more complicated than locales and a perfect analogue may not be what one should seek Indeed, one can view the symmetric monad M (classifier of distributions on toposes X, or equivalently of complete spreads over X with a locally connected domain) as a topos version of the lower power locale P_L. There is however another such candidate, which is the bagdomain monad B_L (classifier of bags of points, or equivalently of branched coverings over X, namely of those complete spreads that are purely locally equivalent to a locally constant cover). See M. Bunge and J. Funk, Singular Coverings of Toposes (2006), (Def. 9.32). In the same source SCT ( 8.3) there is a diagram which shows that there are two factorizations of the unit X—> M(X), namely one through the unit X—> B_L(X) and the other through the unit X—> T(X) where T (classifier of probability distributions, that is of distributions on X which preserve the terminal object, equivalently of complete spreads over X whose domains are locally connected and have totally connected components, the latter meaning that the connected components functor preserves pullbacks). In particular, M(X) is equivalent to B_L(T(X)). It is therefore of interest (to me at least) to find, not just the N that I mentioned above, but also a monad B_U, as both would presumably be topos versions of the upper power locale monad P_U. In addition, it is of interest (to me at least) to find versions of a "single universe”, by which I mean an analogue to the double power locale monad P, which as you and C. Townsend have shown, is such that P(X) for X a locale, can be viewed either as a composite in either direction of P_L and P_U applied to X, or as equivalent to the double exponentiation O^O^X (even if X not necessarily exponentiable) where O is the Sierpinski locael. For O = the objects classifier in Top_S, the double exponential is in fact relevant already in my first (Algebra Universalis 1995) paper where I construct the symmetric topos by forcing methods, in that distributions on X can be seen as carved out of O^O^X (suitably interpreted via points). Similarly, an “upper” version N of M can be constructed as the classifier N of local homomorphisms over toposes. The question then in my view is now how to deal with the “upper” version B_U of B_L. The analogues semiopen-open versus perfect-proper (or tidy-relatively tidy) are of course relevant to this and constitutes work in progress. Best regards, Marta ________________________________ From: Steve Vickers <s.j.vickers@cs.bham.ac.uk> Sent: February 4, 2018 5:52 AM To: categories@mta.ca Subject: categories: Topos theory for spaces of connected components Topos theory gives a solid account of local connectedness, where each open - indeed, each sheaf - has a set (discrete space) of connected components. The definition of locally connected geometric morphism covers not only individual spaces but also bundles, considered fibrewise. It also covers generalized spaces as well as ungeneralized. Is there an analogous theory for where the space of connected components is Stone? ("Connected" is now defined by orthogonality with respect to Stone spaces instead of discrete spaces.) The obvious example is any Stone space X, for instance, Cantor space, where X is its own space of connected components. We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to notice the Stone space aspects in the usual examples based on real analysis, since they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components. (By the way, if you wonder what brought me to this, it was from pondering the symmetric monad M on Grothendieck toposes. Bunge and Funk proved that for ungeneralized spaces its localic reflection is the lower powerlocale, which raises the question of whether there is a missing topos construction whose localic reflection is the upper powerlocale. On the other hand, the symmetric monad is related to local connectedness. Points of MX are cosheaves on X, and X is locally connected if there is a terminal cosheaf in a strong sense, with that cosheaf providing the sets of connected components. Perhaps understanding the Stone space view of connected components would cast light on this missing construction.) All the best, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Steve, Let me use this opportunity to ask a question 'at a lower level', referring to papers listed at the end of this message. How seriously it is related to your question? I don't know, but since I was going to ask it one day anyway, let me ask it now: As you know, taking connected components gives reflections: (a) Locally connected spaces--->Sets, (b) Compact Hausdorff spaces--->Stone spaces, and although it is easy to put them together to involve all topological spaces, there is no NICE such reflection. But what is "nice"? To me, inspired by Galois theory, "nice" would mean "Grothendieck fibration", or, equivalently in this case, it means "semi-left-exact" in the sense of [CHK]. The fact that (a) is semi-left-exact is used in Galois theory in my several papers with and without co-authors, but I would rather call it a folklore result (probably very old, and, for example, hidden in a sense in [BD]). The fact that (b) is semi-left-exact and even has stable units in the sense of [CHK], which is also easy, is explicitly stated and used in [CJKP], to define the (compact) monotone-light factorization categorically; various analogous results (but in different categories) are obtained by J. J. Xarez. A more general story, but with weaker results (still sufficient for something in Galois theory) are in [CJ]. Another kind of developments, very interesting and involving toposes, are in several papers of M. Bunge, some with J. Funk - I am not listing them since Marta can obviously do it better. My question is a 'localic question' (this is what I mean by "lower level"), but it might indeed be related to your 'topos-theoretic question': As you know, a locale is called 0-dimensional if all its elements are joins of complemented ones. By a morphism L--->L' of locales I shall mean a map L'--->L that preserves all joins and finite meets (as usually). The inclusion functor 0-Dimensional locales--->Locales has a left adjoint F, for which F(L)={x in L | x is a join of complementary elements}. Question: Is F semi-left-exact? I mentioned this question several times in past to several people... I am very interested to know the answer, no matter whether it is YES or NO; if NO, then I have weaker questions... Best regards, George References: [BD] M. Barr and R. Diaconescu, On locally simply connected toposes and their fundamental groups, Cahiers de Topologie et Geométrie Différentielle Catégoriques 22-3, 1980, 301-314 [CHK] C. Cassidy, M. Hébert, and G. M. Kelly, Reflective subcategories, localizations, and factorization systems, Journal of Australian Mathematical Society (Series A), 1985, 287-329 [CJKP] A. Carboni, G. Janelidze, G. M. Kelly, and R. Paré, On localization and stabilization of factorization systems, Applied Categorical Structures 5, 1997, 1-58 [CJ] A. Carboni and G. Janelidze, Boolean Galois theories, Georgian Mathematical Journal 9, 4, 2002, 645-658 (Also available as Preprint 15/2002, Dept. Math. Instituto Superior Téchnico, Lisbon 2002) -------------------------------------------------- From: "Steve Vickers" <s.j.vickers@cs.bham.ac.uk> Sent: Sunday, February 4, 2018 12:52 PM To: <categories@mta.ca> Subject: categories: Topos theory for spaces of connected components
Topos theory gives a solid account of local connectedness, where each open - indeed, each sheaf - has a set (discrete space) of connected components. The definition of locally connected geometric morphism covers not only individual spaces but also bundles, considered fibrewise. It also covers generalized spaces as well as ungeneralized.
Is there an analogous theory for where the space of connected components is Stone? ("Connected" is now defined by orthogonality with respect to Stone spaces instead of discrete spaces.)
The obvious example is any Stone space X, for instance, Cantor space, where X is its own space of connected components. We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to notice the Stone space aspects in the usual examples based on real analysis, since they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components.
(By the way, if you wonder what brought me to this, it was from pondering the symmetric monad M on Grothendieck toposes. Bunge and Funk proved that for ungeneralized spaces its localic reflection is the lower powerlocale, which raises the question of whether there is a missing topos construction whose localic reflection is the upper powerlocale. On the other hand, the symmetric monad is related to local connectedness. Points of MX are cosheaves on X, and X is locally connected if there is a terminal cosheaf in a strong sense, with that cosheaf providing the sets of connected components. Perhaps understanding the Stone space view of connected components would cast light on this missing construction.)
All the best,
Steve.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to notice the Stone space aspects in the usual examples
Steve Vickers wrote: based on real analysis, since they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components. Digressing a bit, this reminds me of some results David Roberts recently pointed out. However, they concern path-connected components rather than connected components. The set of path-connected components of a space X is a quotient set of X, so we can give it the quotient topology. What can the resulting space be like? Anything! For every topological space X, there is a paracompact Hausdorff space whose space of path-connected components is homeomorphic to X. D. Harris, Every space is a path-component space, Pacific J. Math. 91 no. 1 (1980) 95-104. http://dx.doi.org/10.2140/pjm.1980.91.95 There is more here: https://mathoverflow.net/questions/291443/paths-in-path-component-spaces Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear John, For point-set results like this it can be a bit delicate working out how the point-free topos treatment goes. Moerdijk has proved that for a connected, locally connected topos X, the map ends: X^I -> XxX is an open surjection. (Here I = [0,1] is the closed real interval, and if p: I -> X then ends(p) = (p(0), p(1)).) This is interpreted as the appropriate point-free way to say that X is path-connected, so connected, locally connected => path connected - which goes against the classical account. Part of the issue is that a point-free surjection is not necessarily surjective on points. Hence even for locally connected spaces, which are supposed to be the well behaved ones, the path-connected components got from the topos theory (which, by Moerdijk's result, agree with the connected components) may be different from the ones got from point-set topology. All the best, Steve. On 04/02/2018 20:57, John Baez wrote:
Steve Vickers wrote:
We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to notice the Stone space aspects in the usual examples based on real analysis, since they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components.
Digressing a bit, this reminds me of some results David Roberts recently pointed out. However, they concern path-connected components rather than connected components. The set of path-connected components of a space X is a quotient set of X, so we can give it the quotient topology. What can the resulting space be like?
Anything! For every topological space X, there is a paracompact Hausdorff space whose space of path-connected components is homeomorphic to X.
D. Harris, Every space is a path-component space, Pacific J. Math. 91 no. 1 (1980) 95-104. http://dx.doi.org/10.2140/pjm.1980.91.95
There is more here:
https://mathoverflow.net/questions/291443/paths-in-path-component-spaces
Best, jb
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Steve, This is response to your message reproduced below. I am aware of Johnstone’s results on the lower bagdomain. However, both the symmetric monad M and the lower bagdomain monad B_L on BTop_S are “on the same side” as the lower power locale monad P_L on Loc_S, and the latter is the localic reflection for both. The upper power locale monad P_U on Loc_S is “on the other side”, in a sense that is explained in my ‘Pitts monads paper”.In it I deduce effective lax descent theorems in a general setting of what I call "Pitts KZ-monads" and "Pitts co-KZ-monads" on a “2-category of spaces”. In the case of M on BTop_S, it is the S-essential surjective geometric morphisms that are shown to be of lax effective descent (a result originally due to Andy Pitts). In the case of P_L on Loc-S it is the open surjections of locales that are shown to be of lax effective descent (a result originally due to Joyal-Tierney). Both M and P_L are instances of "Pitts KZ-monads". Now, P_U on Loc_S is instead an instance of a" Pitts co-KZ-monad" and the result recovered from my general setting is that proper surjections of locales are of effective lax descent (a result originally due to Jaapie Vermeulen). What I seek is a co-KZ-monad N (or perhaps B_U) on BTop_S for which my general theorem would give me that relatively tidy surjections of toposes are of effective lax descent (a result due to I. Moerdijk and J.C.C.Vermeulen). In my Pitts paper there is another consequence of the general theorem proved therein and it is that coherent surjections between coherent toposes are of effective lax descent (a result proven by different methods and by several people, such asM. Zawadowsky 1995, D.Ballard and W.Boshuck 1998, and I.Moerdijk and J.C.C.Vermeulen 1994,thus establishing a conjecture of Pitts 1985 (in the Cambridge Conference whose slides you have requested to Andy). It is of interest for what we are discussing to point out that the “coherent monad C” that I use therein to deduce the latter from my general theorem is a Pitts co-KZ-monad, hence on the “same side” as P_U for Loc_S. For a coherent topos E, the coherent monad C(E) applied to it classifies pretopos morphisms E_{coh} —> S. where E_{coh} is the full subcategory of E of coherent objects with the topology of finite coverings. This theorem is perhaps all I can get in my setting when searching for the still elusive N or B_U but I have not given up yet. Also in my 2015 Pitts paper there are characterizations of the algebras for a Pitts KZ-monad M (dually for a Pitts co-KZ-monad N) as the "stably M-complete objects" ("stably N-complete objects"), where the former is stated in terms of pointwise left Kan extensions along M-maps, and the latter in terms of pointwise right Kan extensions along N-maps. These notions owe much to the work of M, Escardo, in particular to his 1998 "Properly injective spaces and function spaces”. I will say more when i know more myself. Thanks very much for your pointers. I will most certainly look into them even if I do not at the moment think they are what I need. Best regards, Marta ________________________________________________ From: Steve Vickers <s.j.vickers@cs.bham.ac.uk> Sent: February 5, 2018 9:03 AM To: martabunge@hotmail.com Cc: categories@mta.ca Subject: Re: categories: Topos theory for spaces of connected components Dear Marta, Johnstone showed that B_L(X) is a partial product of X against the "generic local homeomorphism", a geometric morphism p from the classifier of pointed objects to the object classifier. A point of B_L(X) is a family of points of X, indexed by elements of a set. He also proposed other partial products, for example those against the generic entire map, which goes to the classifier for Boolean algebras from the classifier of Boolean algebras equipped with prime filter. Wouldn't that be your B_U? A point would be a family of points of X, indexed by points of a Stone space. Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
For any topos in SGA4 SLN 169 IV Exercice 8.7 it is established that the constant sheaf functor has a proadjoint. Thus the "connected components" of any topos form a proset, which in the locally connected case is an actual set. I do not remember a characterization of the category Pro(Set), but I do remember that the category Pro(finiteSet) is the category of stone spaces (this means that the inverse limit set with the product topology wholy characterize the proset). Thus, the topos with an Stone space of connected component are those in which the proset of connected components is a proset of finite sets. This are exactly the quasi-compact Topos (all covers of 1 admits a finite subcover). Obvious question is if this can be extended to the general case, that is taking the inverse limit of the proset with the product topology (that is, totally disconnected topological spaces). We know this can not be the case since the inverse limit may be empty, but may be the inverse limit in the category of locales is worth to investigate. Best e.d. On 04/02/18 07:52, Steve Vickers wrote:
Topos theory gives a solid account of local connectedness, where each open - indeed, each sheaf - has a set (discrete space) of connected components. The definition of locally connected geometric morphism covers not only individual spaces but also bundles, considered fibrewise. It also covers generalized spaces as well as ungeneralized.
Is there an analogous theory for where the space of connected components is Stone? ("Connected" is now defined by orthogonality with respect to Stone spaces instead of discrete spaces.)
The obvious example is any Stone space X, for instance, Cantor space, where X is its own space of connected components. We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to notice the Stone space aspects in the usual examples based on real analysis, since they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components.
(By the way, if you wonder what brought me to this, it was from pondering the symmetric monad M on Grothendieck toposes. Bunge and Funk proved that for ungeneralized spaces its localic reflection is the lower powerlocale, which raises the question of whether there is a missing topos construction whose localic reflection is the upper powerlocale. On the other hand, the symmetric monad is related to local connectedness. Points of MX are cosheaves on X, and X is locally connected if there is a terminal cosheaf in a strong sense, with that cosheaf providing the sets of connected components. Perhaps understanding the Stone space view of connected components would cast light on this missing construction.)
All the best,
Steve.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Steve, You wrote:
Topos theory gives a solid account of local connectedness, where each open - indeed, each sheaf - has a set (discrete space) of connected components. [...]
Is there an analogous theory for where the space of connected components is Stone? ("Connected" is now defined by orthogonality with respect to Stone spaces instead of discrete spaces.)
I have only partial answers to your question. Consider F a bdd S-topos, not necessarily locally connected. There are two instances of non-discrete localic generalizations of the discrete \Pi_0(F) of connected components that may be relevant. They were both reported in my lecture "On two non-discrete localic generalizations of \pi_0” at the Colloque Internationale ‘Charles Ehresmann : 100 ans”, Amiens, 2005. An abstract is included in Cahiers de Top.et Geo.Diff.Cat 46-3 (2005). A fuller account of my lecture can be found in my Research Gate page. It consists of two unrelated parts. The first part (otherwise unpublished) reports my construction of the totally (paths) disconnected topos P_0(F) of path components of F by collapsing paths to a point. It was also the subject matter of a lecture that I gave at UNIGE Seminar in 2003, and of another that I gave at the Workshop on the Ramifications of category Theory, Firenze, 2003. The second part (in collaboration with J. Funk, published as “Quasicomponents in topos theory : the hyperpure-complete spread factorization”, Math.. Proc. Camb. Phil. Soc 142. 2007 ) contains a construction of the zero-dimensional topos \P_0(F) of quasicomponents of F. Both reduce to the usual (discrete) in the case of a locally connected topos F. With best regards, Marta ----- Original Message ----- From: "Steve Vickers" <s.j.vickers@cs.bham.ac.uk> To: categories@mta.ca Sent: Sunday, February 4, 2018 5:52:14 AM Subject: categories: Topos theory for spaces of connected components Topos theory gives a solid account of local connectedness, where each open - indeed, each sheaf - has a set (discrete space) of connected components. The definition of locally connected geometric morphism covers not only individual spaces but also bundles, considered fibrewise. It also covers generalized spaces as well as ungeneralized. Is there an analogous theory for where the space of connected components is Stone? ("Connected" is now defined by orthogonality with respect to Stone spaces instead of discrete spaces.) The obvious example is any Stone space X, for instance, Cantor space, where X is its own space of connected components. We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to notice the Stone space aspects in the usual examples based on real analysis, since they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components. (By the way, if you wonder what brought me to this, it was from pondering the symmetric monad M on Grothendieck toposes. Bunge and Funk proved that for ungeneralized spaces its localic reflection is the lower powerlocale, which raises the question of whether there is a missing topos construction whose localic reflection is the upper powerlocale. On the other hand, the symmetric monad is related to local connectedness. Points of MX are cosheaves on X, and X is locally connected if there is a terminal cosheaf in a strong sense, with that cosheaf providing the sets of connected components. Perhaps understanding the Stone space view of connected components would cast light on this missing construction.) All the best, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (6)
-
Eduardo J. Dubuc -
George Janelidze -
John Baez -
Marta Bunge -
Marta Bunge -
Steve Vickers