The n-Category Café

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

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 $\Gamma(E)$ will not be small.

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

(I can see why you have reservations against that. But on the other hand, as soon as you write $(\infty,1)Topos$ at all, this is forced on you: this is after all a “very large” $\infty \hat Grpd$-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

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

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

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

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 $\Gamma$ is defined, then if you were just “working in a big place” you’d be consistent by going up a level of $(\infty,1)$-topos too, but then $\Gamma$ wouldn’t be defined any more since it would take values at the next level. The point about $\Gamma$ 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.

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

I did call it “$\Gamma$”!

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

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

or

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

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

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?

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 $(\infty,1)$-toposes for coshape of an $(\infty,1)$-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 $\mathbf{H} \in (\infty,1)Topos \subset (\infty,1)\hat Sh((\infty,1)Topos)$ is

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

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

can’t we just op this?

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.

It seems to me it would be great give a nice characterization of $(\infty,1)$-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 $k$, the category of sheaves on the small étale site of $Spec(k)$ is a Galois topos, that is a topos which is equivalent to the classifying topos of a pro-group $G$ (the latter being the Galois group of `the’ separable closure of $k$). If I try to mimick what we get in usual topos theory, we might consider the following notions.

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

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

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

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

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

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

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

I propose the following

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

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

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

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

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

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

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

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

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

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

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

For $Fin \infty Grpd \in \infty Grpd$ the $\infty$-groupoid of finite $\infty$-groupoids we have that morphisms $X \to LConst Fin \infty Grpd$ are locally constant $\infty$-stacks on $X$ That these are co-represented by the fundamental $\infty$-groupoid of $X$ is then just the fact that passing to the adjunct $\Pi(X) \to Fin \infty Grpd$ 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 $(\infty,1)$-topos.