Dear colleagues, this is to clarify my references to one of George Janelidze's messages in this thread and to add a couple of comments about preservation of finite products by pizero. 2018-02-11 18:38 GMT-03:00 George Janelidze <George.Janelidze@uct.ac.za>:
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 did not know. 2018-02-11 18:38 GMT-03:00 George Janelidze <George.Janelidze@uct.ac.za>:
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)".
Apologies, I inserted my question in the wrong place. What I meant to ask is the following. Let L : Locales ---> Locales be the monad determined by the inclusion 0-Dimensional locales--->Locales. Is it the case that the unit X ---> L X is the first component of a factorization X ---> L X ---> 2^(2^X) of the canonical map to a double dualization, where 2 is a suitable exponentiating object in Locales? 2018-02-11 18:38 GMT-03:00 George Janelidze <George.Janelidze@uct.ac.za>:
Note that Matias speaks of preservation of finite products while the reflection Locales--->0-Dimensional locales does not preserve them.
I did not mean to suggest that a positive answer to my question would entail an answer to George's question. I simply wanted to understand the construction of L : Locales ---> Locales in terms that I find more familiar. (The finite-product preservation result I mentioned is not applicable to Locales.) 2018-02-11 18:38 GMT-03:00 George Janelidze <George.Janelidze@uct.ac.za>:
Note also the big (and well known) difference between semi-left-exactness and preservation of finite products: for every 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)...
Concerning local connectedness, finite-product preservation of pizero and even some homotopy theory, let me add a couple of related remarks. The reason I looked for a finite-product preserving pizero in the Tbilisi paper comes from "Axiom 1" in Lawvere's "Categories of spaces ..." paper, TAC Reprint 9, which postulates an essential geometric morphism Gamma: E ---> S such that the leftmost adjoint Gamma_! : E ---> S preserves finite products. Then Lawvere says: "The axiom is necessary for the naive construction of the homotopic passage from quantity to quality; namely, it insures that (not only Gamma_* but also) Gamma_! is a closed functor thus inducing a second way of associating an S-enriched category to each E-enriched category [...] For example, E itself as an E-enriched category gives rise to a homotopy category in which [E](X, Y )=Gamma_!(Y^X)." Axiomatic Cohesion (TAC 2007) continued the work in "Categories of spaces ..." introducing, among other things, the axiom: (Nullstellensatz) The canonical Gamma_* ---> Gamma_! is epic which captures the intuition that every piece has a point. The above is one source of inspiration for Johnstone's TAC 2011 paper where he studies locally connected geometric morphisms p: E ---> S such that the leftmost adjoint p_! : E ---> S preserves finite products. Among other things he shows that: if p: E ---> S is lc, hyperconnected and local then p_! preserves finite products. It also follows from Johnstone's results that: if p: E ---> S is lc and local then, p is hyperconnected iff the Nullstellensatz holds. A rough conclusion that one may arrive at is that: if pizero does not preserve finite products then there is some connected space without points. Apologies again for the lack of clarity in my previous message. I look forward to see how the different aspects of the 'pizero idea' evolve and I am glad that Steve Vickers' message brought them up. All the best, MatÃas 2018-02-11 18:38 GMT-03:00 George Janelidze <George.Janelidze@uct.ac.za>:
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 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/Commentat MathUnivCarolRetro_42-2001-2_13.pdf
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
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