Re: Topos theory for spaces of connected components
Dear Steve, Before answering your question I have to adjust it in three ways: 1. These two factorization systems cannot be "the same", because one of them is general and the other is special. What we can ask is: Is the pure-entire factorization an example of monotone-light factorization? 2. There are generalizations of the 'classical compact Hausdorff monotone-light' in topology that are not special cases of the categorical monotone-light factorization (by which, here and above I mean what is defined in [CJKP]). 3. Since the category of toposes is 2-dimensional, in order to state the question above we have to define (carefully...) a notion of 2-dimensional monotone-light factorization system. But suppose we ignore item 2 and suppose there a nice 2-dimensional notion of monotone-light factorization system. Then no, the pure-entire factorization system in the category of toposes should not be an example of monotone-light factorization system. It is clear from Lemma C3.4.14 (Page 679) in Elephant, which shows that the pure-entire factorization system is not pullback stable. One might hope, however, to develop a kind of 'relative monotone-light factorization theory' and make the pure-entire factorization system an example of it using what Peter says immediately after that lemma. According to what you say about "tidy", you probably noticed this in some sense, but you continue "maybe tidiness is the correct analogue of local connectedness in this Stone case" and this makes me warn you about a possible confusion of "levels": geometric morphisms of toposes should play the roles of continuous maps of spaces and not of functors between categories of spaces. Or, have I misunderstood EVERYTHING you said? All the best to you, George From: Steve Vickers<mailto:s.j.vickers@cs.bham.ac.uk> Sent: Monday, February 5, 2018 8:55 AM To: George.Janelidze@uct.ac.za<mailto:George.Janelidze@uct.ac.za> Cc: categories@mta.ca<mailto:categories@mta.ca> ; Bunge Marta<mailto:marta.bunge@mcgill.ca> ; Clemens.BERGER@unice.fr<mailto:Clemens.BERGER@unice.fr> Subject: Re: categories: Topos theory for spaces of connected components Dear George, Lots to digest here, but here's one quick question. Is monotone-light the same as the pure-entire factorization of geometric morphisms? I found that when I looked up "factorization" in the Elephant index. "Entire" does seem to mean fibrewise Stone, and then "pure" is orthogonal to "entire". The factorization is stable if the geometric morphism is tidy, so maybe tidiness is the correct analogue of local connectedness in this Stone case. I must check the Moerdijk-Vermeulen monograph to see what it says. All the best, Steve.
On 4 Feb 2018, at 19:11, George.Janelidze@uct.ac.za<mailto:George.Janelidze@uct.ac.za> wrote:
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)
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Dear Colleagues, Let me repeat from my exchange of massages with Steve Vickers:
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...
Almost immediately after writing this I received the following message: "...a sufficient condition for F failing to be semi-left exact is for the coproduct of a connected frame Q and a Boolean frame X to have a complemented element that is not in the image of the inclusion of X. I believe such an example is described in chapter XIII, section 4, pages 260--266 of the book Frames and Locales by Picado and Pultr..." The author is Graham Manuell, a PhD student at the University of Edinburgh who did his MSc in Cate Town. I looked at the book: it will take me a long time (which I don't have now) to check the details, because understanding them will require carefully reading every word of those pages... But if what the book says is correct (I cannot imagine these good authors to be careless of course!), then what Graham says is certainly correct, in spite of the fact that semi-left-exactness is not mentioned in the book. The example, as the authors say, was found by I. Kriz (I apologize for not using proper accents on r, i, and z). Moreover, most of the "weaker questions" I had in mind, are also answered... But I still have a question: Kriz's example is presented as a counter-example to a frame-theoretic counterpart of a purely topological property, but now - thanks to Graham's simple remark - it is also a counter-example to semi-left-exactness, whose topological counterpart also fail (unless we restrict spaces to, say, locally connected, or compact). Is there an easier counter-example? George Janelidze [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Tue, Feb 6, 2018 at 12:01 PM, George Janelidze <George.Janelidze@uct.ac.za> wrote:
Dear Colleagues,
Let me repeat from my exchange of massages with Steve Vickers:
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?
Can Example 1 in https://dml.cz/bitstream/handle/10338.dmlcz/119250/CommentatMathUnivCarolRet... be put to some use to answer the question negatively? It shows that the zero-dimensional reflection in topological spaces does not preserve finite products. The example uses fairly nice subspaces of R and R^2. With kind regards, Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Colleagues, Concerning Steve's messages started with "Topos theory for spaces of connected components" sent on February 4 and comments to them, I am not sure I understand what was the end of the story, but I would like to comment on a part of the story related to my question, in the 'chronological' order: 1. I think on February 6 I have written three messages, the first of which was not posted (which is reasonable since my second message contained its copy). In the third message, whose subject was "Reflection to 0-dimensional locales", I wrote that the answer to my question "Is the reflection Locales--->0-Dimensional locales semi-left-exact?" is NO. I also wrote that I know this from Graham Manuell, who explained to me that this follows from the existence of a counter-example, due to I. Kriz, presented in the book "Frames and Locales" by J. Picado and A. Pultr (Pages 260-266). And I asked if it is possible to construct a simpler counter-example. 2. My question above is mentioned (among many other things) in the message of Matias Menni posted on February 8, although it is not clear to me whether or not Matias already knew then that the answer to it is negative. I also don't understand what exactly does Matias mean by asking whether or not the inclusion functor 0-Dimensional locales--->Locales "is the result of a variant of Bill's construction (using an exponentiating object and a `good' factorization system)". Note that Matias speaks of preservation of finite products while the reflection Locales--->0-Dimensional locales does not preserve them. Note also the big (and well known) difference between semi-left-exactness and preservation of finite products: for every connected locally connected topos E with coproducts, the functor Pizero : E--->Sets is a semi-left-exact reflection - but if it were always finite product preserving, then, say, homotopy theory would not exist (all fundamental groups of 'good' spaces would be trivial)... 3. Andrej Bauer, in his message of February 9, also mentions my question and says:
Can Example 1 in
https://dml.cz/bitstream/handle/10338.dmlcz/119250/CommentatMathUnivCarolRet...
be put to some use to answer the question negatively? It shows that the zero-dimensional reflection in topological spaces does not preserve finite products. The example uses fairly nice subspaces of R and R^2.
I think the topological version does not help; note also that the non-semi-left-exactness there was known for a very long time. Summarizing, I thank again Graham for his help, and Matias and Andrej for their comments, but let me insist: the counter-example of Kriz is so complicated... can someone construct an easier one? George Janelidze [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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Andrej Bauer -
George Janelidze