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October 26, 2010

Petit (∞,1)-Toposes

Posted by Mike Shulman

I think I’m gradually coming to understand why some people think (,1)(\infty,1)-toposes are the best thing since sliced bread. For me, I think a brief way to describe what’s so amazing about them is that they still have all the wonderful things I love about “good old” 1-toposes, but they also bring in homotopy theory and higher category theory in the “right” way, which fixes a number of problems (or inelegancies) that happen with good old 1-toposes.

In this post I want to describe one aspect of that, at an intuitive level that should hopefully make sense even if you have no idea what an (,1)(\infty,1)-topos is (yet). It’s closely related to stuff that Urs has been talking a lot about recently, but I prefer to think about it in terms of “petit toposes” instead of “gros” ones. From this perspective, the claim is that (,1)(\infty,1)-toposes elucidate the relationship between topological spaces and \infty-groupoids, by providing a context in which both can be embedded (almost) disjointly.

Imagine that you’ve never heard of the homotopy hypothesis, but you know what a topological space is, and you know what a groupoid is, and you have at least some vague idea of what an \infty-groupoid is. And you want to know how they’re related. You might notice first that there’s a particular class of topological spaces that can naturally be identified with a particular class of groupoids: discrete topological spaces are just sets, as are discrete (higher) groupoids (those with no nonidentity kk-morphisms for k>0k\gt 0). Thus, we have two categories which share a common subcategory, so one might naturally ask whether they are in fact both subcategories of a common supercategory. In other words, is there a commutative square: sets (higher) groupoids spaces ??\array{\text{sets} & \to & \text{(higher) groupoids}\\ \downarrow & & \downarrow \\ \text{spaces} & \to & ??}

Before we go any further, let’s replace topological spaces with their open-set lattices. (For suitably nice topological spaces, this doesn’t lose any information.) This is a good idea since lattices are special posets, which are special categories, and groupoids are also special categories. But we can’t just put “(higher) categories” in the ?? box, since continuous maps of spaces correspond, not to arbitrary poset-maps of their open-set lattices, but only to those which preserve finite intersections and arbitrary unions. On the other hand, we can’t put “categories with finite limits and arbitrary colimits” either, since groupoids don’t have those.

But actually, before we jump to that, we should think about how sets sit inside groupoids and spaces. They sit inside groupoids essentially as themselves (with identity morphisms added), but the open-set lattice of the discrete space on a set is not the set itself but its powerset. The powerset of a set is equivalently its free cocompletion qua poset. Thus, we could put something like “categories with finite limits and arbitrary colimits” in our ?? box, as long as we interpret the right-hand vertical arrow not as “inclusion” but as “free cocompletion.”

Of course, it doesn’t make much sense to start with a (higher) groupoid and take its free cocompletion as a poset, since it isn’t a poset. So instead let’s call the right-hand arrow the free cocompletion of a (higher) groupoid as a (higher) category, which of course is just its presheaf category [G op,Grpd][G^{op},\infty Grpd]. Now, however, we need to adjust the bottom horizontal arrow, since while open-set lattices are very well-behaved cocomplete posets, they are not that well-behaved when regarded directly as cocomplete categories. We don’t want to exactly take their free cocompletion, though, since we want to preserve the colimits they already have; thus the bottom arrow is something like “boosting up a cocomplete poset (= (0,1)-category) to a well-behaved cocomplete (higher) category.”

This is essentially the correct picture. One has to be a little more careful with the definition than “categories with finite limits and arbitrary colimits”—the limits have to distribute over the colimits in a suitable way—but once we do that, and explain how that bottom arrow is to be constructed, we’ve arrived at the right notion of (higher) topos. The point of taking this route is that the category of (,1)(\infty,1)-toposes contains topological spaces and \infty-groupoids as full subcategories, whose intersection consists of the discrete spaces. If we think of an (,1)(\infty,1)-topos as a generalized topological space, then we might evocatively say that the right-hand vertical arrow in our square “equips an \infty-groupoid with the discrete topology.”

Next, now that we’ve got spaces and \infty-groupoids in a common framework, we can ask how they’re related. For instance, what is a map from a space XX to an \infty-groupoid YY (both considered as (∞,1)-toposes)? Well, if YY is a set, then we know the answer: it’s a decomposition of XX into connected components labeled by the points of YY. That is, we have to write X= iU iX = \bigcup_i U_i where each U iU_i is open and labeled by some f(U i)Yf(U_i)\in Y, and moreover if U iU jU_i \cap U_j \neq \emptyset, then U iU_i and U jU_j are labeled by the same point of YY. In yet other words, for every nonempty intersection U iU jU_i \cap U_j, we have to specify an equality f(U i)=f(U j)f(U_i) = f(U_j).

That last immediately suggests what the right notion of “map from a space XX into a (higher) groupoid YY is going to be; we should give a decomposition of X= iU iX = \bigcup_i U_i where each U iU_i is labeled by an object f(U i)Yf(U_i)\in Y, together with, for every nonempty intersection U iU jU_i \cap U_j, an isomorphism (or equivalence) f(U i)f(U j)f(U_i) \cong f(U_j), and for every triple intersection, … etc. In other words, a map from a space XX to a groupoid YY is the same as a “YY-bundle over XX” (specified in terms of transition functions). In yet other words, if “space” means “generalized space” in the sense of “(,1)(\infty,1)-topos,” then the “classifying space” of an \infty-groupoid is just itself, regarded as a generalized space “with the discrete topology.”

The really lovely observation, however, is that the full subcategory Grpd(,1)Topos\infty Grpd\hookrightarrow (\infty,1)Topos is (almost) reflective. In other words, for a space XX, there is an \infty-groupoid Π(X)\Pi(X) such that for any \infty-groupoid YY, maps XYX\to Y are equivalent to maps Π(X)Y\Pi(X)\to Y. This \infty-groupoid Π(X)\Pi(X) is just the usual fundamental \infty-groupoid of XX, whose objects are the points of XX, whose 1-morphisms are paths in XX, whose 2-morphisms are homotopies of paths, and so on. In other words, the fundamental \infty-groupoid of a topological space is not an ad hoc algebraic gadget we invent in order to give us information about the space—it has a universal property.

As was just pointed out in another thread, the fact that Π\Pi is a left adjoint can be regarded as the ultimate van Kampen theorem, since any left adjoint preserves all colimits. (The classical van Kampen theorem, of course, says that Π 1\Pi_1 preserves a particular class of pushouts.) Of course, the work is then moved into verifying that a given pushout of topological spaces induces a pushout of (∞,1)-toposes, but I think the conceptual basis of the theorem, at least, is much clearer this way.

Now this does come with a couple of caveats. One is that in order to identify Π(X)\Pi(X) with the fundamental \infty-groupoid as usually defined, we need there to exist lots of maps into XX from the interval. Some spaces have issues with that; for instance, they could be connected but not path-connected. One of my favorite spaces to think about is the “long circle:” take the extended long line and identify its endpoints. The Warsaw circle is another. Both of these clearly have a “loop,” but classical π 1\pi_1 can’t detect it, since no map out of [0,1][0,1] can “make it all the way around.” However, the topos-theoretic Π(X)\Pi(X) does detect this loop. This is generally the way things go: for not-so-nice spaces where the notions disagree, the topos-theoretic version is usually better-behaved. This sort of “covering fundamental groupoid” has been studied in the 1-topos-theoretic literature, using hypercoverings and other model-categorey sort of things, but I think it’s much more intuitively clear what’s going on when we at least state it in the language of higher topoi.

The second caveat is that some spaces may be so badly behaved that Π(X)\Pi(X) doesn’t exist (hence why I said that Grpd\infty Grpd is only “almost” reflective). For Π(X)\Pi(X) to exist, we essentially need XX to be locally contractible. What we do in this case is the same thing that we do when constructing “moduli stacks”—we look at the functor that Π(X)\Pi(X) would represent if it did exist, and study that functor as a stand-in for the nonexistent object. Here that functor is just Hom (,1)Topos(X,):GrpdGrpdHom_{(\infty,1)Topos}(X,-)\colon\infty Grpd \to \infty Grpd. This functor is called the shape of XX. And it’s still true that the map from (∞,1)-toposes to their shapes preserves colimits.

Posted at October 26, 2010 10:22 PM UTC

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27 Comments & 0 Trackbacks

Re: Petit (∞,1)-Toposes

Thanks, Mike.

One could tell this story of forming “Top SetGrpdTop \coprod_{Set} \infty Grpd” also nicely from the gros perspective, leading to the cohesive \infty TopGrpd.

But of course you are right, this does miss objects of non-representable shape. I hope we can see how to put it all together to a big picture that includes the nice aspects of both points of view.

This here might be a way to go: remember the story of Structured Spaces as summarized at Notions of space:

for a given site CC, we want to be looking at the sequence of inclusions of \infty-categories of spaces

CSchemes(C)(,1)Topos C(,1)Sh^(Pro(C)) C \hookrightarrow Schemes(C) \hookrightarrow (\infty,1)Topos^{C} \hookrightarrow (\infty,1)\hat Sh(Pro(C))

where on the far right we have the very large (,1)(\infty,1)-topos on pro-objects in CC, and where (,1)Topos C(\infty,1)Topos^C is supposed to denote here the CC-structured (,1)(\infty,1)-toposes – supposed to be the petit ones! – but equipped with “CC-structure” that makes them be objects in the very large (,1)(\infty,1)-topos themselves.

If we take C=OpenBallsC = OpenBalls or similar then every toplogical space XX naturally gives rise to a petit (,1)Topos C(\infty,1)Topos^C.

Maybe passing to that very large and gros (,1)Sh^(Pro(C))(\infty,1)\hat Sh(Pro(C)) helps to put all aspects into the same perspective.

Posted by: Urs Schreiber on October 26, 2010 11:21 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

One could tell this story of forming “Top SetGrpdTop \coprod_{Set} \infty Grpd” also nicely from the gros perspective, leading to the cohesive TopGrpd\infty TopGrpd.

Well, as I said, tastes differ. But I don’t find the gros approach nearly as compelling. For one thing, it’s not canonical. The picture I sketched above seems to be basically forced upon you once you decide you want to put spaces and \infty-groupoids into one supercategory. But in order to take the gros approach, first you have to invent the notion of (,1)(\infty,1)-topos (instead of being forced into it), and then you have to pick a particular site over which to work.

In particular, what’s special about open balls? They’re an arbitrary choice. Maybe they give you a nice theory for some spaces, but of course there are spaces that aren’t well-probed by open balls. I’m not really happy with calling (concrete) sheaves on the category of open balls something grandiose like “topological ∞-groupoid”—to me, that term should refer to something more basic and general that doesn’t involve any arbitrary choices.

It also seems to me (perhaps unsurprisingly) that the natural way to compare the petit and gros approaches is by embedding the gros in the petit, rather than vice versa. As you know, any object XX of an ((∞,1)-)topos EE gives rise to an ((∞,1)-)topos E/XE/X which contains all the information about XX in EE (such as its fundamental \infty-groupoid), and moreover in this way the ((∞,1)-)category EE is fully embedded in Topos/ETopos/E. This seems to me a very natural construction: if we have a generalized space (= topos) EE which is a sufficiently “fat point” such that even “discrete sets” relative to EE have a lot of topological structure, then we can regard these sets as generalized spaces themselves, over EE. This is again completely canonical and doesn’t require any “universe-hopping”—your Sh^(Pro(C))\hat{Sh}(Pro(C)) is very large indeed!

But, of course, tastes differ.

Posted by: Mike Shulman on October 27, 2010 4:17 AM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

Mike,

tastes don’t differ that much in this case, actually. I do see the point that you are making about (,1)Topos(\infty,1)Topos itself being the right and canonically existing context for generalized spaces in a very nice way.

I am just trying to have that and at the same time not lose too much of the nice structure that we have on a gros cohesive \infty-topos of spaces. We talked about how the axioms of cohesive (,1)(\infty,1)-toposes axiomatize what one may want to demand on a collection of generalized spaces. I happen to know a bunch of good things that do exist in such a context and that I care about.

But, as you point out rightly, these axioms may be nice, but they may also be too strong for some purposes. So I’d like to see if we can lift them to (,1)Topos(\infty,1)Topos in some way. If we can get a quadruple of adjunctions maybe between (,1)Topos(\infty,1)Topos and ProGrpdPro \infty Grpd that mimics the quadruple of adjunctions between a cohesive \infty-topos and Grpd\infty Grpd.

We started talking about that here in the thread on Julia sets, but I guess it is better to continue the discussion here:

so we are looking at the shape functor

Shape:(,1)ToposProGrpd. Shape : (\infty,1)Topos \to Pro \infty Grpd \,.

Two dumb questions:

  1. does this have a right adjoint (and does that have another right adjoint)?

  2. You write (almost) that I can think about this as the composite

    ToposYoneda[Topos,Grpd] lex op[Grpd,Grpd] lex op=ProGrpd. \infty Topos \stackrel{Yoneda}{\to} [\infty Topos, \infty Grpd]_{lex}^{op} \to [\infty Grpd, \infty Grpd]^{op}_{lex} = Pro \infty Grpd \,.

    (I suppose we need to restrict to limit-preserving functors as I have indicated.)

    I guess I should know this, but now I have problems seeing it: why is that composite the shape?

This is looking to me like we might be headed towards considering the large (,1)(\infty,1)-topos

Sh^(Topos) \infty\hat Sh(\infty Topos)

from HTT 6.3.5 and an (essential) geometric morphism to the large Sh^(Grpd)\infty \hat Sh(\infty Grpd) induced from a morphism of large sites Sh:GrpdTopos\infty Sh : \infty Grpd \to \infty Topos.

But just a hunch, I need to think about that.

what’s special about open balls?

It’s true that open balls are not the right test spaces for the most general topological spaces. They are the right test objects for certain sufficiently non-pathological topological spaces.

One way to see how they are special is: they characterize the topos points of Sh(Manifolds)Sh(Manifolds). A stalk of an object in there is the colimit over a sequence of open balls of fixed dimension nn and shrinking radius. (Of course that’s close to being a tautology.)

I’m not really happy with calling (concrete) sheaves on the category of open balls something grandiose like “topological ∞-groupoid”—to me, that term should refer to something more basic and general that doesn’t involve any arbitrary choices.

Agreed, I should try to find a better term. Sorry that it comes across as trying to be grandiose. I was just looking for a descriptive term.

Posted by: Urs Schreiber on October 27, 2010 9:43 AM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

why is that composite the shape?

Sorry, I should have explained—I forgot that I had to do a bit of thinking to convince myself of that. First of all, for an ∞-groupoid GG, we have [G,Gpd]Gpd/G[G,\infty Gpd] \simeq \infty Gpd/G, so that the geometric morphism [G,Gpd]Gpd[G,\infty Gpd] \to \infty Gpd is a local homeomorphism of (∞,1)-toposes.

Now just as for 1-toposes, a geometric morphism EGpd/GE \to \infty Gpd/G into a “discrete” one is equivalent to a global section 1Δ(G)1 \to \Delta(G), where (Δ,Γ):EGpd(\Delta,\Gamma)\colon E \to \infty Gpd is the unique geometric morphism. In the one direction, Gpd/G\infty Gpd/G contains a canonical global section of GG, namely the diagonal 1 GG *G=G×G1_G \to G^\ast G = G\times G, so that for any f:EGpd/Gf\colon E \to \infty Gpd/G we can apply f *f^\ast to get a morphism 1f *(1)f *G *GΔ(G)1 \simeq f^\ast(1) \to f^\ast G^\ast G \simeq \Delta(G). In the other direction, a map 1Δ(G)1 \to \Delta(G) gives us the composite EE/1E/Δ(G)Gpd/GE \simeq E/1 \to E/ \Delta(G) \to \infty Gpd / G.

Therefore, the functor (,1)ToposLex(Gpd,Gpd) op(\infty,1)Topos \to Lex(\infty Gpd,\infty Gpd)^{op} that I described takes a topos EE to the functor GHom E(1,Δ(G))=Γ(Δ(G))G\mapsto Hom_E(1,\Delta(G)) = \Gamma(\Delta(G)), which is by definition the shape of EE.

Posted by: Mike Shulman on October 27, 2010 6:59 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

does this have a right adjoint (and does that have another right adjoint)?

Well, the embedding Gpd(,1)Topos\infty Gpd \to (\infty,1)Topos, at least, sort of has a right adjoint, which is what you’d expect it to be: global sections of a topos. Namely, we first observe that it preserves colimits, since [colim iG i,Gpd]lim i[G i,Gpd][\colim_i G_i,\infty Gpd] \simeq \lim_i [G_i,\infty Gpd] where the transition maps on the right are the inverse image functors of the corresponding diagram in (,1)Topos(\infty,1)Topos, and we already noted that when the limit of such a diagram is a topos, it is the colimit in the category of topoi. Now for an \infty-groupoid GG and (,1)(\infty,1)-topos EE, we have Hom (,1)Topos([G,Gpd],E) =Hom (,1)Topos([colim G*,Gpd],E) =Hom (,1)Topos(colim G,E) =lim GHom (,1)Topos(,E) =Hom Gpd(G,Hom (,1)Topos(,E)) \begin{aligned} Hom_{(\infty,1)Topos}([G,\infty Gpd], E) &= Hom_{(\infty,1)Topos}([\colim_G \ast, \infty Gpd] , E) \\ &= Hom_{(\infty,1)Topos}(\colim_G \star , E) \\ &= \lim_G Hom_{(\infty,1)Topos}(\star , E) \\ &= Hom_{\infty Gpd}(G,Hom_{(\infty,1)Topos}(\star , E)) \end{aligned} so it looks like its right adjoint is the “global points” functor Γ()=Hom (,1)Topos(,)\Gamma(-) = Hom_{(\infty,1)Topos}(\star , -). Here I’m writing \star for the terminal (,1)(\infty,1)-topos, which is of course Gpd\infty Gpd, but the above equations looked much more confusing when I called it that.

There is only one problem: in general Γ(E)\Gamma(E) will not be small. Even an ordinary 1-topos can have a proper class of global points. (Consider, for instance, the classifying topos of objects, for which a global point is just an arbitrary set.) But modulo that consideration, it looks like we do have a string of adjunctions ΠDiscΓ\Pi \dashv Disc \dashv \Gamma at least in the locally \infty-connected case.

One way to prevent a topos from having too many points is to just stipulate a small set of points as its “only” points, analogous to passing from locales back to topological spaces. In other words, we could consider (,1)(\infty,1)-ionads. If we did that, then I think we would also get a further right adjoint CodiscCodisc of Γ\Gamma at least for 1-topoi: the “codiscrete ionad” on a set XX is just the terminal topos equipped with XX as collection of points (all duplicating its unique point). I’m not sure whether an (,1)(\infty,1)-ionad should be equipped with a whole \infty-groupoid of points or just a set of them, though; from this perspective we seem to want the former.

On the other hand, I think in general the Galois toposes that arise as fundamental groupoids of other toposes won’t have enough points, hence cannot be represented as ionads. So the ionad approach doesn’t seem to fit very well with the shape theory. (Once again I am reminded of Steve Vickers’ “topological systems” — maybe we need something in between a topos and an ionad, which has a specified set of points that aren’t required to be enough.)

Posted by: Mike Shulman on October 28, 2010 6:10 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

so it looks like its right adjoint is the “global points” functor Γ()=Hom (,1)Topos(*,)\Gamma(-) = Hom_{(\infty,1)Topos}(*,-) […]

This is (as you just showed!) what a few comments above I had babtized the coshape functor.

Check out the notes I made at here. I have included your observation there, too, now. Please let me know what you think. And…

There is only one problem: in general Γ(E)\Gamma(E) will not be small.

… this is dealt with there by working over Grpd^\infty \hat Grpd.

(I can see why you have reservations against that. But on the other hand, as soon as you write (,1)Topos(\infty,1)Topos at all, this is forced on you: this is after all a “very large” Grpd^\infty \hat Grpd-enriched category. My general attitude is: work in the large context where operations exist naturally and only after having set up the general theory check in special cases whether a construction happens to land in a smaller context.)

But that still just seems to give a (very large) geometric morphism

(ΓCodisc):Sh^(,1)ToposCodiscΓSh^(Grpd). (\Gamma \dashv Codisc) : \hat Sh (\infty,1)Topos \stackrel{\overset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}} \hat Sh(\infty Grpd) \,.

and not the total string (ΠDiscΓCodisc)(\Pi \dashv Disc \dashv \Gamma \dashv Codisc).

Because if now with the above adjunction I’d use your argument to get (DiscΓ)(Disc \dashv \Gamma), I’d have to step up one more universe, it seems…

Posted by: Urs Schreiber on October 28, 2010 9:51 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

what a few comments above I had baptized the coshape functor.

Sorry, I didn’t understand that it was the same thing. Why call it something obscure-sounding like “coshape” instead of a word I understand like “global sections” or “global points”? But also, I was not talking about your huge topos of sheaves on topoi, only about the original topoi themselves – I really don’t like that beast, and not just because it gives me headaches to think about.

My general attitude is: work in the large context where operations exist naturally and only after having set up the general theory check in special cases whether a construction happens to land in a smaller context.

I’m not comfortable with that approach, because it seems to me that usually what makes a difference for whether operations “exist naturally” is the relative size of various things, not the absolute size. So you can’t just “work in a big place” where “everything exists naturally” because there might not be such a place: if you make everything bigger then you still have the same relative sizes.

As you just pointed out, in fact. If you go up one level of \infty-groupoid, so that Γ\Gamma is defined, then if you were just “working in a big place” you’d be consistent by going up a level of (,1)(\infty,1)-topos too, but then Γ\Gamma wouldn’t be defined any more since it would take values at the next level. The point about Γ\Gamma is what it does to relative size.

I don’t have any objection to universes in their own right. What I get worried about is trying to finesse things to avoid having to think about relative sizes.

Posted by: Mike Shulman on October 28, 2010 10:02 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

Why call it something obscure-sounding like “coshape” instead of a word I understand like “global sections” or “global points”?

I did call it “Γ\Gamma”!

It’s the same with “shape”: there we also start out calling it by the obscure sounding term “shape” only to realize then that it’s really just “fundamental \infty-groupoid”.

So we equivalently say

(ShapeDiscCoShapeCoDisc) (Shape \dashv Disc \dashv CoShape \dashv CoDisc)

or

(ΠDiscΓCoDisc). (\Pi \dashv Disc \dashv \Gamma \dashv CoDisc) \,.

Or at least we would say so. If all these adjunctions exited jointly on (,1)Topos(\infty,1)Topos.

Posted by: Urs Schreiber on October 28, 2010 10:09 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

It’s the same with “shape”: there we also start out calling it by the obscure sounding term “shape” only to realize then that it’s really just “fundamental ∞-groupoid”.

Yes, and that’s one reason it’s taken me until now to figure out what the heck this “shape” thing is supposed to be. At least in that case we have the excuse that shape theorists have been calling it that for a long time, so there’s a historical precedent to feel obliged to follow. But here the historical precedent is to call this “the global points of a topos,” so why intentionally use an obscurer-sounding term?

Posted by: Mike Shulman on October 28, 2010 10:45 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

I wrote:

This is looking to me like we might be headed towards considering the large (∞,1)-topos Sh^(Topos)\infty \hat Sh(\infty Topos) from HTT 6.3.5 and an (essential) geometric morphism to the large Sh^(Grpd)\infty \hat Sh(\infty Grpd) […]

By HTT lemma 6.3.5.21 and using remark 6.3.5.18 we have a geometric morphism of very large \infty-toposes

Ind^GrpdSh^(Topos). \hat Ind \infty Grpd \to \infty \hat Sh(\infty Topos) \,.

Maybe that morphism deserves to be called CoshapeCoshape, not sure.

If only there were a few op-s thrown in, this would be going in the direction I’d be looking for.

Posted by: Urs Schreiber on October 27, 2010 10:18 AM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

Hi Mike and Urs.

The (,1)(\infty,1)-category of (,1)(\infty,1)-topoi admits limits and colimits (see Cor. 6.3.1.8 and Prop. 6.3.2.3, and Cor. 6.3.4.7 in HTT; even better, the \infty-bicategory of (,1)(\infty,1)-topoi if cotensored over (,1)-Cat(\infty,1)\text{-}Cat, as it is suggested by Prop. 6.3.4.9 in loc. cit.). Therefore, the inclusion

-Grpd(,1)-Topoi\infty\text{-}Grpd\to (\infty,1)\text{-}Topoi

Extends canonically to a functor

Pro(-Grpd)(,1)-TopoiPro(\infty\text{-}Grpd)\to (\infty,1)\text{-}Topoi

which is fully faithful and is the right adjoint to the ‘shape functor’

Π:(,1)-TopoiPro(-Grpd)\Pi:(\infty,1)\text{-}Topoi\to Pro(\infty\text{-}Grpd)

Posted by: Denis-Charles Cisinski on October 27, 2010 2:00 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

My size-issues warning bell is going off. The (∞,1)-category (∞,1)-Topos has small limits and colimits, so I’ll believe that for any small (∞,1)-category CC, an embedding C(,1)ToposC \hookrightarrow (\infty,1)Topos extends to a functor Pro(C)=Lex(C,Gpd) op(,1)ToposPro(C) = Lex(C,\infty Gpd)^{op} \to (\infty,1)Topos. But Gpd\infty Gpd is not small.

I expect that the real issue here is that Lex(C,Gpd) opLex(C,\infty Gpd)^{op} is not the “right” definition of Pro(C)Pro(C) when CC is a large (∞,1)-category; we should really be using some ∞-version of small presheaves in order to get the actual completion of a large (∞,1)-category under (small) cofiltered limits. With a definition like that, it seems like you should get such a functor—but then we’d need to worry about whether the shape of an (∞,1)-topos is in fact a small presheaf.

Alternately, one could try to give a direct construction of the “classifying topos” of a pro-∞-groupoid. I think one way that people do this in the 1-topos case is to consider a pro-groupoid as a localic groupoid (i.e. regard your inverse system of groupoids as an inverse system of discrete localic groupoids, and then actually take the limit in the category of locales) and then consider “equivariant sheaves” on the localic groupoid. Perhaps one could do something similar with localic ∞-groupoids.

Posted by: Mike Shulman on October 27, 2010 9:46 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

My size-issues warning bell is going off.

It seems to me that my suggestion above takes care of that:

there is a geometric morphism of huge (,1)(\infty,1)-toposes for coshape of an (,1)(\infty,1)-topos

(ΓCodisc):IndGrpd^CodiscΓ(,1)Sh^((,1)Topos) (\Gamma \dashv Codisc) : Ind \infty \hat Grpd \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\to}} (\infty,1)\hat Sh((\infty,1)Topos)

where the coshape of H(,1)Topos(,1)Sh^((,1)Topos)\mathbf{H} \in (\infty,1)Topos \subset (\infty,1)\hat Sh((\infty,1)Topos) is

Γ(H):AHom(PSh(A),H). \Gamma(\mathbf{H}) : A \mapsto Hom(PSh(A), \mathbf{H}) \,.

I tried to write that out at coshape. But have to dash off now. But:

can’t we just op this?

Posted by: Urs Schreiber on October 28, 2010 2:08 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

I’m just really not happy about the idea of taking “big sheaves on the category of topoi”—it seems to be adding a superfluous layer of indirection, not taking advantage of the the advantages of topoi. Plus it needs another universe.

Posted by: Mike Shulman on October 28, 2010 5:23 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

I recorded a few facts at progroup, including the equivalence between surjective progroups and prodiscrete localic groups. There are a lot of different constructions of the “fundamental group(oid) of a topos” in the literature, and I don’t understand the relationship between them all, but at least some of them have the flavor of a “reflection” into the subcategory of “prodiscrete localic toposes,” i.e. toposes that can be presented as equivariant sheaves on a prodiscrete localic groupoid.

So right now my feeling is that we should be looking for a reflection from (∞,1)-toposes into the subcategory of (∞,1)-toposes of equivariant sheaves on prodiscrete localic ∞-groupoids. If the (∞,1)-topos we start with is locally ∞-connected, then this reflection ought to give an (∞,1)-topos of actions of an honestly discrete ∞-groupoid, which is what we called its fundamental ∞-groupoid before. And if all goes well, then in general, the resulting prodiscrete localic ∞-groupoid ought to be the “limit,” in the category of localic ∞-groupoids, of the “shape” of the (∞,1)-topos we started with, considered as a pro-∞-groupoid, i.e. a cofiltered diagram of ∞-groupoids. And this really ought also to be its limit in the category of (∞,1)-toposes as well.

It sounds like I’m saying that Π \Pi_\infty of a general (∞,1)-topos should be thought of as its prodiscrete completion.

Posted by: Mike Shulman on October 28, 2010 5:47 AM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

It seems to me it would be great give a nice characterization of (,1)(\infty,1)-topoi which are equivalent to the classifying topos of a pro-groupoid, in order to get a good understanding of the shape using geometric tools: all this is supposed to be related with the notion of locally constant stack. I also mean that all this story is after all a generalization of Grothendieck’s Galois theory, and one of the things which allows all this to be interpreted as a genuine Galois theory (in the sense of Galois, of course) is that, for any field kk, the category of sheaves on the small étale site of Spec(k)Spec(k) is a Galois topos, that is a topos which is equivalent to the classifying topos of a pro-group GG (the latter being the Galois group of `the’ separable closure of kk). If I try to mimick what we get in usual topos theory, we might consider the following notions.

Let TT be an (,1)(\infty,1)-topos.

A stack XX over TT (i.e. an objet of TT) is locally constant if there exists a covering {V i1} iI\{V_i\to 1\}_{i\in I} of the terminal object of TT such that X×V iX\times V_i is constant in T/V iT/V_i (i.e. in the image of the constant stack functor) for all ii.

Consider now a stack XX on TT, and let G=Aut(X)G=Aut(X) be the \infty-groupoid of its automorphisms in TT, that is the part of Hom T(X,X)Hom_T(X,X) lying above the connected components which are invertible maps in the homotopy category Ho(T)Ho(T)). Denote by X/GX/G the quotient of XX under the action of GG (say, on the left).

Definition: the stack XX is said to be Galois there exists a decomposition of the terminal object of TT as 1=(X/G)⨿Y1=(X/G)\amalg Y (for some YY).

Note that X/GX/G is locally constant: its restriction to X/GX/G is the terminal object of T/X/GT/X/G, while its restriction to YY is empty in T/YT/Y. Therefore, any Galois object is locally constant (being a GG-torsor over a locally constant stack, with GG constant).

Example: If GG is a groupoid, then any representable stack on G^\hat{G} is Galois. And conversely, and Galois stack on G^\hat{G} is representable.

Remark: for any geometric morphism of topoi f:TTf:T'\to T, the inverse image functor f *:TTf^*:T\to T' preserves Galois objects.

I propose the following

Definition. An (,1)(\infty,1)-topos TT is Galois is the class of Galois stacks on TT form a generating family of (the underlying (,1)(\infty,1)-category of) TT.

Example: For any \infty-groupoid GG, the (,1)(\infty,1)-topos G^\hat{G} is Galois. More generally, this is true for the classifying (,1)(\infty,1)-topos of any pro-\infty-groupoid.

Proposition. Let TT be a locally \infty-connected (,1)(\infty,1)-topos. Denote by LC(T)LC(T) the full (,1)(\infty,1)-subcategory of TT which consists of locally constant stacks on TT. Then LC(T)LC(T) is a Galois (,1)(\infty,1)-topos, and the inclusion map LC(T)TLC(T)\subset T is the inverse image functor of a geometric morphism TLC(T)T\to LC(T) which identifies canonically LC(T)LC(T) with the classifying (,1)(\infty,1)-topos of the \infty-groupoid Π(T)\Pi(T).

What I like with the previous proposition is that, although the computation of the shape of TT might be complicated, the geometric meaning of LC(T)LC(T) is very intuitive. So what would be nice is a generalization of the previous proposition to general (,1)(\infty,1)-topoi. I propose the

Conjecture. Let TT be an (,1)(\infty,1)-topos. Denote by LC(T)LC(T) the full (,1)(\infty,1)-subcategory of TT which consists of locally constant stacks on TT, and write CLC(T)CLC(T) for the completion of LC(T)LC(T) by colimits in TT. Then CLC(T)CLC(T) is a Galois (,1)(\infty,1)-topos, and the inclusion map CLC(T)TCLC(T)\subset T is the inverse image functor of a geometric morphism TCLC(T)T\to CLC(T) which identifies canonically CLC(T)CLC(T) with the classifying (,1)(\infty,1)-topos of the pro-\infty-groupoid Π(T)\Pi(T).

Note that, for a Galois (,1)(\infty,1)-topos TT, we obviously have CLC(T)=TCLC(T)=T. Therefore, a trivial consequence of the conjecture above would be that an (,1)(\infty,1)-topos is Galois if anf only if it is the classifying (,1)(\infty,1)-topos of a pro-\infty-groupoid. This would also allow to interpret shape theory by saying that the inclusion of Galois (,1)(\infty,1)-topoi into (,1)(\infty,1)-topos has a left adjoint. Another corollary of this conjecture would be that, for a geometric morphism of (,1)(\infty,1)-topoi f:TTf:T'\to T, the following conditions would be equivalent:

a) the map Π(f):Π(T)Π(T)\Pi(f):\Pi(T')\to\Pi(T) is an equivalence of pro-\infty-groupoids;

b) the functor f *:CLC(T)CLC(T)f^*:CLC(T)\to CLC(T') is an equivalence of (,1)(\infty,1)-categories;

c) the functor f *:LC(T)LC(T)f^*:LC(T)\to LC(T') is an equivalence of (,1)(\infty,1)-categories.

In other words, shape theory should depend only on the basic notion of locally constant stack. Note that all this is in our folklore in a way or another since a long time: this is just a way to express the fact that weak homotopy equivalences are detected by cohomology with coefficients in local systems.

Posted by: Denis-Charles Cisinski on October 28, 2010 3:38 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

That’s a very beautiful picture, Denis-Charles! I think it’s the sort of thing I was driving at, but without enough understanding of how the 1-dimensional picture works to formulate it as cleanly.

Posted by: Mike Shulman on October 28, 2010 5:34 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

Thanks, Dennis-Charles,

You have put flesh on the bones of what I was claiming way back in 1979! (Before even I started hearing about Pursuing Stacks or had moved to Bangor.) I had been working with strong shape theory for a few years and was convinced of the need for a Galois-Poincaré type theory that has finally now begun to appear as the natural way of interpreting strong shape invariants. I was, unfortunately, told by several referees of grant proposals that what I was saying was rubbish, or uninteresting or both. (There was much more to do than any of my proposals would have managed to do even if they had been funded.)

It is looking good. I also had hoped that (strong) shape theory would include ‘generalised spaces’ i.e. toposes and that is now clearly the case.

Posted by: Tim Porter on October 28, 2010 7:01 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

all this story is after all a generalization of Grothendieck’s Galois theory

One reason why I am after having an adjunction (ΠLConst)(\Pi \dashv LConst) is because this makes Galois theory become almost a tautology – or rather: a quadruple application of the \infty-Yoneda lemma:

For FinGrpdGrpdFin \infty Grpd \in \infty Grpd the \infty-groupoid of finite \infty-groupoids we have that morphisms XLConstFinGrpdX \to LConst Fin \infty Grpd are locally constant \infty-stacks on XX That these are co-represented by the fundamental \infty-groupoid of XX is then just the fact that passing to the adjunct Π(X)FinGrpd\Pi(X) \to Fin \infty Grpd is an equivalence.

Tautological as this may look, I think this does in fact capture the higher Galois theory in the literature, such as the articles by Toën, Mike, by you, etc.

Details of what I have in mind here are at geometric homotopy groups in an (,1)(\infty,1)-topos.

Posted by: Urs Schreiber on October 28, 2010 5:02 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

I think that’s the same sort of thing Denis-Charles was getting at when he said

shape theory should depend only on the basic notion of locally constant stack

Posted by: Mike Shulman on October 28, 2010 5:50 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

Dear Urs,

Grothendieck’s Galois theory generalizes Galois theory of fields. In this sense, I don’t see how what one finds at the nLab about pro-homotopy types of topoi gives back Galois theory tautologically. You get a nice picture of the Galois reconstruction theorem for locally \infty-connected topoi (as Toën did for instance - and what I stated as proposition is nothing by a reformulation of this too), but the general picture for arbitrary (,1)(\infty,1)-topoi is not there (as far as I understand): there is nothing in the literature (including the nLab) which reaches something like what I described above (i.e. the explicit link with locally constant stacks for general higher topoi). This means in particular that you don’t get classical Galois theory of fields as a consequence yet. For instance, one of the issues is that locally constant stacks on a non locally \infty-connected (,1)(\infty,1)-topos do not form an (,1)(\infty,1)-topos (for instance an infinite colimit of locally constant stacks might not be locally constant): that is precisely why you get a pro-\infty-groupoid instead of a genuine \infty-groupoid. Therefore, the classifying topos of the pro-\infty-groupoid associated to an (,1)(\infty,1)-topos TT cannot be described in terms of locally constant stacks on TT (I mean, not tautologically!).

Posted by: Denis-Charles Cisinski on October 28, 2010 9:26 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

You get a nice picture of the Galois reconstruction theorem for locally ∞-connected topoi (as Toën did for instance - and what I stated as proposition is nothing by a reformulation of this too), but the general picture for arbitrary (∞,1)-topoi is not there

Yes, exactly. This is why I say:

[the] reason why I am after having an adjunction (ΠConst)(\Pi \dashv Const) is […]

Because once we do have this adjunction (as we do on any locally \infty-connected \infty-topos) Galois theory is “solved” (as for locally constant \infty-stacks on topological space regarded inside for instance the cohesive \infty-topos Sh(OpenBalls)\infty Sh(OpenBalls)).

So therfore it seems interesting to see to which extent we can implement this adjunction on (,1)Topos(\infty,1)Topos. This is what Mike and I have tried to discuss in aspects here.

But I agree with you: since we do not have this adjunction (yet?) on (,1)Topos(\infty,1)Topos, Galois theory with general \infty-toposes is not (yet?) tautologized, yes.

So I am tinking: maybe we need to resort to more “by-hand” constructions. Or maybe we should find a point of view that does support the adjunction after all. Such as maybe after passing from (,1)Topos(\infty,1)Topos to Sh^((,1)Topos)\infty \hat Sh((\infty,1)Topos).

Posted by: Urs Schreiber on October 28, 2010 10:05 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

I think the word “tautologous” is kind of misleading and belittling here. I think the ∞-categorical language is a beautiful conceptual way of packaging up statements of this form, but calling something “tautologous” makes it sound trivial and easy, while it seems likely to me that proving the ∞-categorical version of these statements will essentially incorporate all the same work that goes into the known versions of Grothendieck’s Galois theory.

Posted by: Mike Shulman on October 28, 2010 10:45 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

I think the word “tautologous” is kind of misleading and belittling here. I think the ∞-categorical language is a beautiful conceptual way of packaging up statements of this form, but calling something “tautologous” makes it sound trivial and easy, while it seems likely to me that proving the ∞-categorical version of these statements will essentially incorporate all the same work that goes into the known versions of Grothendieck’s Galois theory.

Certainly, I agree. Realizing/constructing the adjunction (ΠLConst)(\Pi \dashv LConst) is still work that needs to be done. And it may amount to the same trouble as any other formulation. But even then it would still have the advatage of organizing the information neatly.

But also, constructing the adjunction may well be easier than going along another route. I’d dare say that the \simeq 12\frac{1}{2}-page proof that Sh(TopBalls)\infty Sh(TopBalls) is locally \infty-connected including the Galois-theory corollary (by abstract Tannaka duality = pure Yoneda!) that End(x *:LConstStacks(X)FinGrpd)Func(Π x(X),FinGrpd)End(x^* : LConstStacks(X) \to Fin \infty Grpd) \simeq Func(\Pi_x(X), Fin \infty Grpd) for any object XX is simpler than other discussion that arrives at the same conclusion.

Posted by: Urs Schreiber on October 28, 2010 11:03 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

Thanks Mike, thanks Denis-Charles,

I have written this out now at shape of an (,1)(\infty,1)-topos.

I made

Π(H):=(,1)Topos(H,PSh(() op)) \Pi(\mathbf{H}) := (\infty,1)Topos(\mathbf{H}, PSh((-)^{op}))

now the definition and moved the definition that was originally there into the Properties-section.

Posted by: Urs Schreiber on October 27, 2010 8:45 PM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

The topologist’s sine curve is another

Did you mean the Warsaw circle?

Posted by: David Roberts on October 27, 2010 12:35 AM | Permalink | Reply to this

Re: Petit (∞,1)-Toposes

Yes, thank you. For some reason I always say one of those when I mean the other.

Posted by: Mike Shulman on October 27, 2010 4:22 AM | Permalink | Reply to this

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