sheaves on localic groupoids
Dear Categorists - Joyal and Tierney proved that any Grothendieck topos is equivalent to the category of sheaves on a localic groupoid. I gather that we can take this localic groupoid to have a single object iff the Grothendieck topos is connected, atomic, and has a point. In this case the topos can also be seen as the category of continuous actions of a localic group on (discrete) sets. I'm curious about how these three conditions combine to get the job done. So suppose G is a localic groupoid. Under which conditions is the category of sheaves on G a connected Grothendieck topos? Under which conditions is the category of sheaves on G an atomic Grothendieck topos? Under which conditions is the category of sheaves on G a Grothendieck topos with a point? (Maybe we should interpret "with a point" as an extra structure on G rather than a mere extra property; I don't know how much this matters.) Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
A time ago when I was working on the subject I was also very curious about the same (or related) questions of Johon Baez. More concretely, if you take a pointless connected atomic topos (we know they are): How or which is the localic point that Joyal-Tierney using change of base take out of the hat ? How or which is the localic groupoid of Joyal-Tierney ? On 17/10/18 23:12, John Baez wrote:
Dear Categorists -
Joyal and Tierney proved that any Grothendieck topos is equivalent to the category of sheaves on a localic groupoid. I gather that we can take this localic groupoid to have a single object iff the Grothendieck topos is connected, atomic, and has a point. In this case the topos can also be seen as the category of continuous actions of a localic group on (discrete) sets.
I'm curious about how these three conditions combine to get the job done. So suppose G is a localic groupoid.
Under which conditions is the category of sheaves on G a connected Grothendieck topos?
Under which conditions is the category of sheaves on G an atomic Grothendieck topos?
Under which conditions is the category of sheaves on G a Grothendieck topos with a point?
(Maybe we should interpret "with a point" as an extra structure on G rather than a mere extra property; I don't know how much this matters.)
Best, jb
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi Eduardo & all If F is a Grothendieck topos then there is a bounded geometric morphism p:F->Set. Let B be a bound for p, then there is a locale of surjections [N->>B], where N is the natural numbers. This is a locale over F that can be mapped to a locale over Set (apply the direct image of p to the frame of opens of [N->>B]). You get the object locale of the localic groupoid that ‘represents’ F via Joyal and Tierney. The morphism locale is the image of [N->>B]x[N->>B]. Not sure if that sheds much light on what the localic groupoid really is, but it is one way of constructing it. I’d like to add that any geometric morphism p:F->E, we know, gives rise to a ‘localic’ adjunction between locales over F and locales over E; the right adjoint being effectively pullback in the category of toposes. If you can find a locale W over F such that W->1 is an effective descent morphism and the slice of this localic adjunction at W is an equivalence then in fact the adjunction is a connected components adjunction of a localic groupoid in E; this can be shown by application of Janelidze’s categorical Galois Theorem (the result holds at the generality of cartesian categories - it’s quite straightforward). If you know that the localic adjunction is a connected components adjunction it is easy to get the Joyal and Tierney result by restricting to local homeomorphisms. But of course you should complain that it must be hard to show that any localic adjunction, sliced at W, is an equivalence. In fact even at the general level of cartesian categories it is not that hard once we recall that the localic adjunction is stably Frobenius (something that is in the original Joyal and Tierney paper but not dwelt on). For a Frobenius adjunction to be an equivalence it only needs to have its left adjoint preserve 1 and its unit to be a regular monomorphism. At the slice, 1 is automatically preserved, so we are just left checking that the (sliced) unit is a regular monomorphism. It turns out that this is so for W=[N->>B] precisely when B is a bound, effectively completing a proof of the Joyal and Tierney representation theorem. I hope it is OK that I’ve used this thread to effectively advertise some work(*) that I did a few years ago. Kind regards, Christopher (*) A localic proof of the localic groupoid representation of Grothendieck toposes Proc. Amer. Math. Soc. 142 (2014), 859-866
19 Oct 2018, at 19:57, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:
A time ago when I was working on the subject I was also very curious about the same (or related) questions of Johon Baez. More concretely, if you take a pointless connected atomic topos (we know they are):
How or which is the localic point that Joyal-Tierney using change of base take out of the hat ?
How or which is the localic groupoid of Joyal-Tierney ?
On 17/10/18 23:12, John Baez wrote: Dear Categorists -
Joyal and Tierney proved that any Grothendieck topos is equivalent to the category of sheaves on a localic groupoid. I gather that we can take this localic groupoid to have a single object iff the Grothendieck topos is connected, atomic, and has a point. In this case the topos can also be seen as the category of continuous actions of a localic group on (discrete) sets.
I'm curious about how these three conditions combine to get the job done. So suppose G is a localic groupoid.
Under which conditions is the category of sheaves on G a connected Grothendieck topos?
Under which conditions is the category of sheaves on G an atomic Grothendieck topos?
Under which conditions is the category of sheaves on G a Grothendieck topos with a point?
(Maybe we should interpret "with a point" as an extra structure on G rather than a mere extra property; I don't know how much this matters.)
Best, jb
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear John, A topos has a point if and only if it can be represented by a (?tale complete open) groupoid whose space of objects has a point. But it is also easy to construct pointless groupoids representing the topos of sets: for any non zero locale X, the trivial groupoid on $X$ represents the topos of sets. An explicit description of points of the topos attached to an ?tale complete open localic groupoid as some sort of G-torsor can be extracted from Ieke Moerdijk's paper the classyfing topos of a continuous groupoid" (I and/or II), I don't think you can get anything better than that. For the other property: If $T$ has a groupoid representation $G$ then the coequalizer of $G$ in the category of locales is the localic reflection of T. This follows from the fact that sheaves over groupoids are colimits in the category of toposes and localic reflection preserves colimits. So any property of a topos that can be tested on its localic reflection can be tested on this quotient. I believe this applies to connectedness as the inverse image of the localic reflection geometric morphisms T -> L is always fully faithful. "Atomic" and "Connected atomic" have nice characterization as soon as one restricts to open ?tale complete localic groupoids. It is proved in sketches of an elephant C.3.5.14 that a topos T is atomic if and only if both the geometric morphisms T -> 1 and T->T \times T are open. It also follows from combining several results of section C.3.5 that T is atomic connected if and only if these two maps are open surjections. Now if T is represented by an open ?tale complete localic groupoid G, it means that you have an open surjection G_0 -> T and that G_1 = G_0 \times_T G_0 In particular by manipulating a little all the pullback square involved, and using the fact that pullback along an open surjection preserve and detect open map and open surjection (C5.1.7 in sketches) one can see that T->1 is open (resp. an open surjection) if and only if G_0 ->1 is, and T -> T \times T is open (resp. an open surjection) if and only if G_1 -> G_0 \times G_0 is. Best wishes, Simon Henry
Dear Categorists -
Joyal and Tierney proved that any Grothendieck topos is equivalent to the category of sheaves on a localic groupoid. I gather that we can take this localic groupoid to have a single object iff the Grothendieck topos is connected, atomic, and has a point. In this case the topos can also be seen as the category of continuous actions of a localic group on (discrete) sets.
I'm curious about how these three conditions combine to get the job done. So suppose G is a localic groupoid.
Under which conditions is the category of sheaves on G a connected Grothendieck topos?
Under which conditions is the category of sheaves on G an atomic Grothendieck topos?
Under which conditions is the category of sheaves on G a Grothendieck topos with a point?
(Maybe we should interpret "with a point" as an extra structure on G rather than a mere extra property; I don't know how much this matters.)
Best, jb
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear John, Eduardo and Simon, A few comments and questions. If G is a connected localic group, then every continuous action of G on a set is trivial. Hence the topos of G-sets contains no information about G. This is annoying. Of course, we could replace G-sets by G-locales (= actions of G on locales). But the category of G-locales is not a topos in general. Should we consider the gross-topos of sheaves on the category of G-locales ? What should be the Grothendieck topology? What are the applications ? (the applications may guide the developement of a theory). I do not have a satisfactory answer to these questions . Remark: There are plenty of connected Lie groups. Equivariant homotopy theory is an important branch of topology. Best wishes, André You are considering the topos T of equivariant G-sheaves on a localic groupoid G. In the case where G is a group, T is the topos of continuous G-sets. Under which conditions is the category of sheaves on G a connected Grothendieck topos? ________________________________________ From: John Baez [baez@math.ucr.edu] Sent: Wednesday, October 17, 2018 10:12 PM To: categories Subject: categories: sheaves on localic groupoids Dear Categorists - Joyal and Tierney proved that any Grothendieck topos is equivalent to the category of sheaves on a localic groupoid. I gather that we can take this localic groupoid to have a single object iff the Grothendieck topos is connected, atomic, and has a point. In this case the topos can also be seen as the category of continuous actions of a localic group on (discrete) sets. I'm curious about how these three conditions combine to get the job done. So suppose G is a localic groupoid. Under which conditions is the category of sheaves on G a connected Grothendieck topos? Under which conditions is the category of sheaves on G an atomic Grothendieck topos? Under which conditions is the category of sheaves on G a Grothendieck topos with a point? (Maybe we should interpret "with a point" as an extra structure on G rather than a mere extra property; I don't know how much this matters.) Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Christopher Townsend -
Eduardo J. Dubuc -
henry＠phare.normalesup.org -
John Baez -
Joyal, André