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Igor Bakovic created Diaconescu’s theorem
nm.
Nanometer or New Mexico ?
I posted something that was incorrect and quickly edited/deleted it.
Let’s just say I confused Diaconescu and Popescu.
We need to do something to the entry classifying topos now. That focuses on algebraic theories. We need to say something about classifying toposes for torsors. Of course the Diaconesu-page now does precisely that in a way, but it should be expanded on at classifying topos.
Of course, the classifying topos for torsors is a special case of the classifying topos for a geometric theory, and can be identified with the theory of flat functors on a category.
On classifying topos I see a discussion “for groups”, which explains how to construct the essentially algebraic theory (i.e., finite limits theory) of groups. But it doesn’t seem to explain the classifying topos for groups! Is the idea that later someone will come along and explain that taking presheaves on the finite limits theory for groups will give the classifying topos for groups?
This is in fact the only construction of classifying topoi that I thought I understood. I thought Urs was saying there was too much focus on this construction, but now it seems there’s not enough!
Of course, the classifying topos for torsors is a special case of the classifying topos for a geometric theory, and can be identified with the theory of flat functors on a category.
Over a fixed group? How does this work?
$Set^G$ classifies $G$-torsors for a group $G$. This is actually a good illustration of Diaconescu’s theorem proper.
@John: That’s weird… yes, someone should come along and explain that! I didn’t realize that that example left off in the middle that way.
@Urs: What Todd said. The same is true for any category C in place of G, when you define a “C-torsor” to mean a flat functor out of C (which generalizes the notion of torsor for a group), and that is then exactly Diaconescu’s theorem.
Thanks for reassuring me I’m not insane, Mike. I’ll change that entry now while I’m waiting for the shuttle to pick me up and take me to the plane to Singapore.
Welcome to Asia :)
I won’t leave my house for 3 more hours, but thanks!
While I’m waiting, I made a preliminary stab at fixing the discussion “for groups” in classifying topos, and also the previous section, “Background on the theory of theories” - which is really a general overview of 3 of our favorite doctrines. I’m not very happy with it; in particular I find it clunky to distinguish notationally between a “theory” (in some doctrine) and its “classifying category” (which is a category in that doctrine), but I wasn’t feeling brave enough to change this from the way the previous folks had written it.
What is $Set$ the classifying topos for?
What is the category of simplicial sets the classifying topos for?
What is the category of presheaves on $FinSet$ the classifying topos for?
What is the category of presheaves on $FinSet^{op}$ the classifying topos for?
I have known the answer to some of these, at certain times… Someone should put the answers into the $n$Lab!
Let’s see, $FinSet$ is the free category with finite colimits on one object. So $FinSet^{op}$ is the free category with finite limits on one object - that is, the finite limits theory of an object. So presheaves on this should be the classifying topos for an object.
The category $1$ is the free category with finite colimits on nothing - that is, the initial category with finite colimits. So, $1^{op} \cong 1$ is also the free category with finite limits on nothing - that is, the finite limits theory of nothing. So the category of presheaves on this, namely the category $Set$, should be the classifying topos for “nothing”.
Is this right? Just killing time…
Okay, so $SimpSet$ is the classifying topos for interval objects.
What about presheaves on $\Delta^{op}$? What’s that the classifying topos for?
Presheaves on $\Delta^{op}$ classify total orders, essentially because any total order is a filtered colimit of finite total orders aka objects of $\Delta$.
Presheaves on $FinSet$ classify Boolean algebras (allowing the 1-point terminal algebra), because any Boolean algebra is a filtered colimit of finite Boolean algebras aka objects of $FinSet^{op}$.
I think you got the others.
$Set^G$ classifies $G$-torsors for a group $G$. This is actually a good illustration of Diaconescu’s theorem proper.
But $G$ is not a geometric theory, or is it?
And what we want to see is, I thought, torsors in a topos $E$ over a group object $G$ in the topos $E$.
Maybe I still don’t get the the language of “theories”.
For a topological group $G_{top}$ the statement is that the classifying topos for topological $G$-principal bundles is the Deligne topos of sheaves on the simplicial topological space $N G_{top}$.
Is that subsumed in the language that we currently have at classifying topos or not? What would be the geometric theory $T$ for this example?
But G is not a geometric theory, or is it?
No, G is not the syntactic category of a geometric theory. But there is a geometric theory T, called “the theory of G-torsors,” such that a T-model in a topos E is the same as a $\Delta G$-torsor in E. And if $C_T$ denotes the syntactic category of T, then by general nonsense $Sh(C_T)$ is the classifying topos for T, and the statement is that $Sh(C_T) \simeq Set^G$. Hence $Set^G$ also classifies G-torsors, even though G itself is not the syntactic category of a theory.
The same is true for any presheaf category: by Diaconescu’s theorem, $Set^C$ classifies flat functors out of C, but C itself is not the syntactic category of the theory of flat functors out of C. That classifying category is larger than C, but equipped with a topology such that sheaves on it are equivalent to presheaves on C, since both classify the same thing. Make sense?
If G is not a group in sets but a group in a topos E, then you just have to play the whole game relative to E. You can define what you mean by a geometric theory in a topos E, and any such theory T has a classifying E-topos, sometimes denoted $E[T]$, such that for any E-topos F (meaning a topos equipped with a geometric morphism to E, i.e. an object of $Topos/E$), we have
${Topos/E}\;\big(F,E[T]\big) \simeq T Mod(F)$The same argument, relativized to E, shows that if T is the theory of G-torsors for a group G in E, then $E[T] \simeq E^G$. In particular, G-torsors in E are the same as maps $E\to E^G$ of E-toposes, i.e. “points” of $E^G$ considered as an E-topos.
Any statement you can make about classifying topoi must be included in this language in some sense, because every Grothendieck topos is the classifying topos of a geometric theory. Therefore, anything which is classified by some Grothendieck topos E must be equivalent (at least insofar as its models in other Grothendieck topoi go) to some geometric theory, namely the theory classified by E. Of course many different-looking theories can be classified by the same topos and hence be “Morita-equivalent.” I don’t know offhand of a clean geometric presentation of the theory of topological G-principal bundles, but one could obtain a messy theory Morita-equivalent to it by finding a site whose topos of sheaves is equivalent to the Deligne topos of sheaves on $N G_{top}$ and then writing down the theory of flat cover-preserving functors on that site.
Thanks, Mike.
So in a somewhat provocative way I could say: a topos classifies whatever it classifies, namely categories of geometric morphisms into it, and this tautology may be retold as a story about geometric theories.
As you say:
Any statement you can make about classifying topoi must be included in this language in some sense, because every Grothendieck topos is the classifying topos of a geometric theory. Therefore, anything which is classified by some Grothendieck topos E must be equivalent (at least insofar as its models in other Grothendieck topoi go) to some geometric theory, namely the theory classified by E.
I think I can appreciate the purpose of this theory-language, but maybe we could add to the beginning of the entry on classifying toposes some statements along these lines, that make it clear also to non-theory theoreticians such as me that what is discussed there does apply to Diaconescu’s theorem and alike.
I’ll try to write something a little later maybe. You can still check and roll back if I get it wrong.
I have now edited and expanded at classifying topos
the Idea-section
the first 2.5 paragraphs of the Definition-section
a topos classifies whatever it classifies, namely categories of geometric morphisms into it, and this tautology may be retold as a story about geometric theories.
Yes. But you can say the same thing about any representable functor. For example, the classifying space $B G$ of a group $G$ classifies whatever it classifies, namely homotopy classes of maps into it, and this tautology may be retold as a story about principal $G$-bundles. Of course, in both cases the real meat of the story is in the retelling.
Of course, in both cases the real meat of the story is in the retelling.
Yes, but as the above discussion shows, the previous version of the entry didn’t make it clear (to me! :-) that the story can always be retold in terms of geometric theories. That’s what I’d try to amplify now. But please feel free to edit if it sounds bad now.
Same holds for spaces and bundles: while it is not hard, it does require a little bit of discussion that every space is the classifying space of some sort of principal bundle.
By the way, what explcitly is the geometric theory $T$ such that $Sh(C_T) \simeq Set^G$? Can one write this out in a nice useful manner? Or does it become a mess? And for $Sh(N G_{top})$?
Another by the way: it should be useful to write $Set^{G}$ as $Sh(\mathbf{B}G)$. For $G_{top}$ a topological group, that Deligne topos $Sh(N G_{top})$ of sheaves on a simplicial topological space ought to be the 2-colimit
$Sh(N G_{top}) \lim_{\to} ([n] \mapto Sh(G^{\times n} )$in toposes, mimicing the homotopy colimit
$\mathbf{B}G = \lim_{\to }( [n] \mapsto G^{\times n})$in $Top$.
A discussion along these lines might be useful in the entry to make the relation to classifying spaces for principal bundlesmore transparent.
I asked a question at the discussion ’classifying topos’ which I see could have been posed here. In ’The idea’, the classifying topos $B G$ for $G$-torsors is being likened to the classifying space $\mathcal{B} G$. Is there an analogue on the classifying topos side for the total space, $E G$, of the universal $G$-bundle?
Another point, the ’Diaconescu’s theorem’ entry speaks of $C$-principal bundles for a small category $C$, and links to the principal bundle page. However this page only speaks of $G$-principal bundles for a group $G$.
Another point, the ’Diaconescu’s theorem’ entry speaks of C-principal bundles for a small category $C$, and links to the principal bundle page. However this page only speaks of $G$-principal bundles for a group $G$.
I was wondering about this too: from context I gather that a $C$-torsor could be a flat functor out of $C$, but this is just a guess.
I think the only definition I’ve ever seen of “C-torsor” for a category C defines it to mean “flat functor out of C.” I don’t know what I think about generalizing the meaning of “torsor” in that way, or whether it would be better to just say “flat functor.”
A geometric theory T whose models are G-torsors can be described as follows. It has one sort, X, and one unary operation $g:X\to X$ for every element $g\in G$. It has algebraic axioms $\top\vdash_x \;1(x) = x$ and $\top\vdash_x \;g(h(x)) = (g h)(x)$, which make X into a G-set, and geometric axioms $\top \vdash\; \exists x \in X$ (inhabited-ness), $g(x) = x \;\vdash_x \;\bot$ for all $g\neq 1$ (freeness), and $\top\vdash_{x,y}\; \bigvee_{g\in G}\; g(x) = y$ (transitivity).
I have added to Diaconescu’s theorem the general statement of the theorem, a remark on its relevance to classifying toposes, a cautionary remark on the “torsor”-terminology, and a pointer to the Elephant.
The terminology clash that Mike points out was actually a problem in the entry Disconescu’s theorem.
I think that was my fault. I remember how I hastily added in the theorems there a while back, and thought to myself “better write ’flat functor’ instead of the unfortunate ’torsor’”.
In any case, I have tried to quickly fix this by adopting the more careful terminology that Mike suggests, there and at flat functor and site.
But it’s after my bedtime and I shouldn’t be working anymore. Somebody should look over this carefully again.
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