Of course, thank you for calling our attention to Carchedi’s paper ! Pedro was mainly working in localic context and the 2-categorical and Morita aspects were studied more recently than his main known results on the subject. He has also some very nice works in preparation.

]]>His result which you quote is quite similar to main results of a series of important works of Pedro Resende on relations between etale groupoids and inverse semigroups.

]]>I have added to *etale groupoid* some of the relevant facts proven in Carchedi 12.

Added a brief remark on “noncommutative Gelfand duality for etale Lie groupoids”.

]]>Added to the Definition-section at *étale groupoid* some paragraphs on the Morita/stacky-invariant formulation via “foliation groupoids” and the characterization theorem by Crainic-Moerdijk.

I have added more standard references to *étale groupoid*.

- $G^\ad_1:=$ morphisms effectively covered by Pi-closed morphism
Yes, indeed. They are the right class in a factorization system.

I am trying to figure out if factorization systems are the right notion to describe ”étale morphisms”: The $(\Pi_\eq\dashv\Pi_\cl)$ orthogonal factorization system is dual to the factorization system $(\Pi_cl^\op,\Pi_\eq^\op)$ on $H^\op$. Since we are interested in diagrams of type

$\array{A&\to&X\\\downarrow&&\downarrow\\B&\to&Y}$with the left arrow (a reversed) $\Pi$-closed and the horizontal morphisms being effective epimorphisms, we could consider ”$E$-orthogonal factorization systems” where the filling diagonal is required only for squares where the horizontal morphisms are in the class $E$ of effective epimorphisms. This idea occurred to me since in the motivating example of a coverage (by a covering space) we had indeed a unique diagonal.

Another question -now, that we have two (different) canonical admissible structures on a cohesive topos - is, what they have to do with each other: I tried to compare them via the factorization of the global section geometric morphism of the infinitesimal cohesive neighbourhood of $H$ through $H$.

]]>Yes, indeed. They are the right class in a factorization system.

That’s fine! Then they give an admissible structure, too. In fact the right class of every orthogonal factorization system gives an admissible structure.

]]>Formally étale morphisms satisfy 2-out-of-3

I must correct this. If we consider the triple $\{f,g,f\circ g\}$ the implication ”$f$ and $g\circ f$ formally étale entails $g$ is formally étale” is not shown. Nevertheless formally étale maps satisfy the admissibility axioms.

There was also a typo in proposition 3 in formally etale morphism which I corrected

]]>A necessary condition for 1. to be admissible is that $\Pi$-closed morphisms are themselves closed under pullback and I guess Pi-closed morphisms are closed under pullback:

Yes, indeed. They are the right class in a factorization system.

I just posted it here as a kind of status report on what I’m currently thinking of, so there is no need that you comment it instantly.

Sure, and thanks for this. But I did want to comment instantly, because you made a good comment! So I just wanted to warn that even though I am commenting, my comment is not to be read as the result of intesive thinking. ;-)

]]>I don’t have the head free for this right now

I just posted it here as a kind of status report on what I’m currently thinking of, so there is no need that you comment it instantly.

]]>A necessary condition for 1. to be admissible is that $\Pi$-closed morphisms are themselves closed under pullback and I guess Pi-closed morphisms are closed under pullback:

If $f:X\to Y$ is Pi-closed and $f^':X^'\to Y^'$ is a pullback of $f$ there is a pasted an (∞,1)-pullback diagram

$\array{X^'&\to&X&\to& \Pi X\\\downarrow&&\downarrow&&\downarrow\\Y^'&\to&Y&\to &\Pi Y}$By naturality and universality of the $(\Pi\dashv\Disc)$-unit this diagram is equivalent (I have to check this again tomorrow) to

$\array{X^'&\to&\Pi X^'&\to& \Pi X\\\downarrow&&\downarrow&&\downarrow\\Y^'&\to&\Pi Y^'&\to &\Pi Y}$where the left pullback shows that $f^'$ is $\Pi$-closed.

]]>In fact, according to this proposition, the cohesively formally étale morphisms between smooth manifolds are ordinary étale maps.

Right, very good point, I should have made that connection earlier. It’s pretty compelling, now that I think about it.

Also, it makes good sense in view of the notion of “geometry”: the extra information of “admissibility” is somehow encoded in the extra choice of infinitesimal cohesion. Makes perfect sense, now that you made me think this way.

]]>I don’t have the head free for this right now, but you have a very good point by focusing attention back to the formall étale maps. We should push that, since here we really know a good deal of what’s going on.

Formally étale morphisms satisfy 2-out-of-3 and are stable under pullback if the full and faithful embedding into a cohesive neighbourhood of $\mathbf{H}$ preserves pullbacks.

Right. Hm, do we have this for our preferred models? I’d need to think about that. Right, that would be good…

]]>does a cohesive higher topos induce its own canonical geometry

I am looking for this now. In the above discussion I see three candidates for an admissible structure on a (∞,1)-category $G$ which hopefully lead to a canonical geometry $G^'$ such that $O:G^'\to H$ is left exact and ”preserves admissible covers” and hence gives a $G$ structure on a cohesive (∞,1)-topos $H$ - namely: $G^\ad_0:=H_0$ and

$G^\ad_1:=$ morphisms effectively covered by Pi-closed morphism, or

$G^\ad_1:=\Etale_X\hookrightarrow \Disc\Fib_X \hookrightarrow H/X$ where the reflector of the composite inclusion preserves finite limits, or

$G^\ad_1:=$ formally étale morphisms

As it stands none of these can be a geometry since $H$ is not essentially small. But this may be fixed by sorting out or by starting with a $G$ which is.

Formally étale morphisms satisfy 2-out-of-3, are stable under retracts and are stable under pullback if the full and faithful embedding $u^*:H\to H_\th$ into a cohesive neighbourhood of $H$ preserves pullbacks so in case 3 we have an admissible structure.

]]>But then how do we define fibres of local isomorphisms for a general higher topos?

First a nitpicky remark, repeating comments that I made before:

Where you say “local isomorphism” I do hope you mean “local homeomorphism”. Despite the similar terminology, these notions have nothing to do with each other! They are sort of dual to each other: a local isomorphism is, for typical situations that you may have in mind and roughly: something that becomes an iso in the vicinity of points in the codomain. A local homeomorphism becomes an iso in the vicinity of points in the domain.

So then let’s look at this question:

But then how do we define fibres of

local homeomorphismfor a general higher topos?

This has an obvious answer: we look at homotopy fibers. :-)

]]>does a cohesive higher topos induce its own canonical geometry

this I think is the key question. Why? Because the toy example Urs started with used covering spaces, which trivialise over, you guessed it, open covers. Just saying all the fibres are isomorphic seems weaker to me (but I’m happy to be proved wrong). But then how do we define fibres of local isomorphisms for a general higher topos? I might have missed this somewhere in the flurry of discussion above.

]]>But we meant to find an ”intrinsic” description of open and étale maps and and mentioning an underlying geometry is not intrinsic as I understand this term.

Right, that was exactly the idea. That in a cohesive (higher) topos there should be a canonical notion of étale map, because there is already a canonical notion of covering étale map (an étale map whose fiber is the same everywhere).

That may or may not turn out to be true in the end. But even if it should turn out not to be true, it would still be interesting to see to which extent it is true. For instance the $\infty$-category of “discretely fibered” maps that I mentioned above may already have its use.

From another persective, one could ask: does a cohesive higher topos induce its own canonical *geometry* (in the sense you mentioned)?

But if these questions start be more of a hinderance than an inspiration, one should start to try other approaches.

]]>now we have to disambiguate

At least english grammar in many cases doesn’t force me to distinguish between singular and plural when I write ”you” :-)

I only skimmed …

…Toens masters course since I guess the material presented there is covered and further developed by Luries Structured Spaces V.

There are analogies with Zariski maps of rings and the etale topology on Sch.

To return to your remark on the étale topology: I begin to get a more complete idea of the topic now. The étale site defined by the étale topology is a pregeometry and its enveloping geometry is one motivating example for a geometry (for structured (infinity,1)-toposes) David (:-) mentioned above. This notion reduces to what Toen and Ieke Moedijk do in giving some distinguished class of ”prototypical” open resp. étale maps.

But we meant to find an ”intrinsic” description of open and étale maps and and mentioning an underlying geometry is not intrinsic as I understand this term.

]]>The definition of open maps given here is less general than the one given by the (A1)-(A9).

ah, ok. I didn’t compare. Also I only skimmed this discussion, so may not have followed all your details.

Thanks David and Urs

Well, now we have to disambiguate: me and David C :)

]]>you

Thanks David and Urs - of course anyone is addressed and invited to contribute to this discussion; I addressed to Urs in first place just because we started this discussion.

Recall also definition 2.3 (local homeomorphism) here: http://ens.math.univ-montp2.fr/~toen/cours1.pdf (Toen’s Master course)

The definition of open maps given here is less general than the one given by the (A1)-(A9).

As for ’geometries’, see section 1 here: http://ens.math.univ-montp2.fr/~toen/cours2.pdf

Here is the given the characterization of open maps via the ”archetypical open maps” you mentioned It is interesting here that he uses representable presheaves for his definition of open maps since above we tried it with the other inclusion of the base site in its sheaf category: locally constant sheaves.

There are analogies with Zariski maps of rings and the etale topology on Sch.

Yes, I have to look at that more closely - in principle this is a special case of that of localic presheaves…

]]>Did you know this paper?

Yes. I had added that material to *open map* in revision 4.

Admittedly, though, I hadn’t thought of it in the present discussion.

What do you [think]?

Yes, I think this is well worth looking into. I’ll try to get back to you later. Need to be dealing with some other things right now.

]]>One thing to consider is how the open maps in $Top$ are generated by the class of maps $\coprod U_i \to X$, the archetypal open maps. Quite possibly we can generate, given the other axioms, from the inclusions $U_i \hookrightarrow X$. There are analogies with Zariski maps of rings and the etale topology on $Sch$.

Recall also definition 2.3 (local homeomorphism) here: http://ens.math.univ-montp2.fr/~toen/cours1.pdf (Toen’s Master course)

As for ’geometries’, see section 1 here: http://ens.math.univ-montp2.fr/~toen/cours2.pdf

]]>Hi Urs,

I think a quick example shows that what is missing is some open-ness condition.

yes, I think this is true:-) In the (very enlightening) paper (Ieke Moerdijk and Andre Joyal: *A completeness theorem for open maps* doi
) are given axiomatizations of the notions of (classes of) open and of étale maps in toposes (and under additional asuumptions in any category). These axioms are (given in open map -but since the collection axiom needed for the completeness theorem below lacks there, I insert them here for convenience):

Let $\array{Z&\to Y\to^p&X\\\downarrow^g&&\downarrow^f\\B&\to^h}&H$ be a topos.

A class $R$ of $H$-morphisms is called **class of open maps** if (A1)-(A6) are satisfied, $R$ is called **class of étale maps** if (A1)-(A5) and (A7)-(A8) are satisfied. The axiom (A9) is called **collection axiom**.

(A1) any isomorphism belongs to $R$ and $R$ is closed under composition.

(A2)(stability) arbitrary pullbacks of a morphism in $R$ is in $R$

(A3)(descent) if in the pullback square in $H$ the left arrow is in $R$ and the bottom arrow is epi then the right arrow is in $R$

$\array{X&\to&Y\\\downarrow&&\downarrow\\X^'&\to& Y^'}$(A4) for any set $I$ the morphism $\coprod_{i\in I}1\to 1$ is in $R$

(A5) for any family of arrows $(Y_i\to X_i)_{i\in I}$ the morphism $\coprod_{i\in I}Y_i\to \coprod_{i\in I}X_i$

(A6)(quotient) for any factorization $f\circ p=q$ such that $p$ is dpi and $g$ is in $R$ then $f$ is in $R$.

$\array{ Y &&\stackrel{p}{\to}&& X \\ & {}_{\mathllap{g}}\searrow && \swarrow_{\mathrlap{f}} \\ && B }$(A7) if $f:Y\to X$ is in $R$ then its diagonal $\Delta_f:Y\to Y\times_X Y$ is in $R$

(A8) if in the pullback square the left arrow and the bottom arrow are in $R$ and the bottom arrow is epi then the right arrow is in $R$.

(A9)(collection) if $p$ is epi and $f$ is in $R$ there exists a quasi pullback with $h$ epi and $g$ is in $R$.

The collection axiom appears in

Let $R$ be a class of open maps satisfying the collection axiom (A9) in a topos $H$.

There is a topos $T$ and a geometric morphism $F:\Arr(T)\to H$ such that $R$ is induced by the canonical class of open maps in $\Arr(T)$;

i.e. $f\in H_1$ is open iff $F^* f$ is a quasi pullback square in $T$.

and

Let $H$ be a topoos satisfying the collection axiom (A9), let $X\in H$ be an object.

Then there is a coreflective geometric morphism $(i^*\dashv i_*):H/X\to\Et/X$

Apparently one can write down these axioms in higher dimension, too. So I guess my worthwhile agenda is now to consider how exactly they categorify, which statements of this paper hold in higher dimension, and see how they simplify if we assume the (higher) topos to be cohesive (and accordingly what this has to do with discreteness). What do you mean? Did you know this paper?

]]>