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Have created a table of contents descent and locality - contents.
Am including this now as a “floating TOC” into the relevant entries.
I am a bit confused with the abstract langauge used. So first of all we sharply distinguish descent data and descent objects (let me know if anybody disagrees!). Descent object is NOT a generalization of the descent datum but rather of the entire category of descent data, i.e. it is a generalization when the 2-category of categories is replaced by a more general category. Second, the descent is with respect to a pseudofunctor or with respect to a fibred category. Now the entry descent object talks about a single category (or higher category) and the main example for presheaves of sets about a SET of matching families (it should be a category of descent data in the case of presheaves). Thus the categorical level does not agree in the main example and also the descent seems to be just the one internal to a topos and not for a general fibered category. Probably it is ot so, I just do not understand very abstract formulation in the entry. Is this in agreement with what I just said and I do not yet understand the language or there is a terminological or other disagreement ?
Second, the descent is with respect to a pseudofunctor or with respect to a fibred category.
That’s for 2-presheaves / stacks only. For 1-presheaves / sheaves descent is for functors. For $n$-presheaves it is for $n$-functors.
it should be a category of descent data in the case of presheaves
Not sure what you mean.
I understand that Street and you want to force some formalism for all n. But the basic case is n=2. I am able to look for n bigger than 2, and totally unable to look at n=1. It is beyond my understanding.
I give up my atempt to understand the entry. I trust you now that it is correct but going to n=1 is beyond my intuition (n-descent is in n+1 categories, so here you talk 0-descent!). It would be good to have descent datum :) (I can not contribute now, sorry)
I understand that Street and you want to force some formalism for all n.
Long before Street, the case $n \leq (\infty,1)$ was considered and understood. The one who originally wanted this formalism was Grothendieck, when he began to pursue higher stacks. But today they have been successfully pursued, and the theory is sort of classical by now.
I am able to look for n bigger than 2, and totally unable to look at n=1. It is beyond my understanding.
I am a bit confused by what you are saying here.
I understand that Street and you want to force some formalism for all n.
Here the emphasis is on the 0-dimensional descent. People before Street looked at descent to generalize the descent from 1-descent to include higher cells, this n=1 , 2, 3 etc. Here the confusing thing is to get to 0 and take it as a prime example. What confused me is that this entry is written in terms with emphasis on extension to the case of sets what was not comprehensible to me. It would be nice to eventually have an entry on descent datum, the notion which is much more used in mathematics. I would write it but I am not in a position now to work here extensively.
In my opinion, the category of descent data (the descent object in the case of 1-descent in Cat) is not only the pseudoequalizer, it is a particular way to realize the pseudoequalizer. The distinction is important in my opinion.
I think that I am getting to understand n=0 (though it still confuses me a bit),
The case $n = 0$ is just the age-old case of sheaves and the sheaf condition. This is what the concept of stack and descent for stacks was originally motivated from, by categorification.
Toby Bartels negative thinking -2 categories and -1 categories are also known objects from centuries ago, but the point of view that they are sensibly viewed as “higher” categories is a new discovery. The descent object is a newer name for an abstract generalization of the category of descent data for other 2-categories. The objects of the category of the descent data are the descent data, i.e. the pairs, (object in the category + additional isomorphism needed to descend and in higher dimension plus higher cells). Now to get to decategorify to the descent 0-category (descent set) from that concept is a new task in translation. Now the object in the category is just an element in the set, and the iso drops out by decategorification, and to descend indeed means the matching condition at the pullback. Great, after the chain of translations, we got to the agreement.
“Descent object” is not that new a name — I think it’s been around in 2-category theory circles for decades.
I know. It is new in the sense that it came later after the internalization of the “category of descent data”, due Grothendieck. My own thinking is still largely tied to the Grothendieck’s original setup and its literal higher generalizations (with my drift in last year or so gradually away from the category theory it is likely to stay there).
A description of the descent category in style of Streets definition has the clarifying advantage that the ”objects of descent data” (the objects of the descent object which in this case happen to be categories) occur explicitly and are described intrinsically.
This is true also for the older theory of descent for presheaves with values in higher categories modeled simplicially. In fact that can be described much more explicitly as one does not need to pass through the construction of orientals.
For what it’s worth, you can find examples spelled out in full pedestrian detail for descent with values in $n$-groupoids in low $n$ in section 1.3 here.
More general discussion is in section 2.1.5 “oo-Sheaves and descent”, where I recall how to derive the explicit “descent-ends” that Street postulated. Then in section 2.1.6 “oo-Sheaves with values in chain complexes and strict oo-groupoids” is discussed the relation to abelian sheaf cohomology and Street’s proposal.
For what it’s worth, you can find examples spelled out in full pedestrian detail for descent with values in -groupoids in low in section 1.3 here.
As I understand it, in this introductory chapter you consider descent with values in n-groups (defined by n-principal G-bundles).
More general discussion is in section 2.1.5 “oo-Sheaves and descent”, where I recall how to derive the explicit “descent-ends” that Street postulated.
Then in the chapter on descent this is generalized to the case of descent with values in sSet (in particular to the many-objects-groupoid case). But when you define principal ∞-bundles (in 2.3.4) we are in the one object-case again. So I am unsure: Do you mean by ”principal ∞-bundle” a principal bundle for an ∞groupoid or do you always restrict this notion to ∞groups? Are there conceptual reasons (for example the statement that every groupoid object in an ∞-topos is effective or some other reasoning concerning delooping) to restrict it?
The restriction to connected coefficient objects is just a restriction in view of some appliction, not of principle. All the diagrams in 1.3 have immediate generalizations to the many-object case.
Currently in the document the many-object case is discussed abstractly in section 2.3.6 “$\infty$-group representations and associated $\infty$-bundles” and then in applications in section 3.3.7 “Smooth cohesion – Twisted cohomology”.
If you let me know what kind of examples or applications or theory you would like to see spelled out further, I’ll add whatever it takes to the $n$Lab.
Technical note @ Stephan: If you want to quote a comment or post on the Forum that contains math, then you should click ‘Source’ above the comment to be quoted and copy and paste from the source. (Also be sure to use the same markup format as the comment that you’re quoting, but you’re already using the same format as Urs, which is indeed the only sensible one for anybody to use here.)
Hi Urs,
thank you for your kind offering. Well, my agenda is to define ∞-orbifolds and as I see it, the sensible way to do so is by saying it is an ∞-stack presented by a proper étale Lie ∞-groupoid where
A Lie ∞-groupoid is a smooth ∞-groupoid which is concrete (in the sense of //nlab.mathforge.org/nlab/show/concrete+object (http))
”Étale” shall mean that all source- and target maps (in the simplicial model) are étale (in the sense of 2.4.1.3 of your ªHabilitationsschrift”)
”Proper” shall mean that all source- and target maps are proper. For the notion of proper it would be intuitive to take some appropriate cover of the source- and targets and stipulating for all inverse images of the induced maps of the covers to be compact objects (in the sense of //nlab.mathforge.org/nlab/show/compact+object (http)) whenever the elements of the cover were compact objects themselves. Now this not per se the correct notion of compact. There are hopefully ways to get around this but I don’ t know them yet (the entry on the nlab mentions only some fixing in the case of a topological spaces) so some hints in this direction would be a great help to me. I think that this is the (maybe only) part of my definition which goes not through verbatim.
So far my ideas on how to do it. There may of course arise problems, occurring in details which you are invited to comment on, too. Also ideas in term of leading questions to ∞-orbifolds which I could think about are very welcome (since my current advisor is no category theorist…).
I know that this post somehow drifted away from topic of ”descent object” even when it is obviously circulating around material having to do with it and which is line with ”differential cohomology in a cohesive ∞-topos” so you, Urs or anyone who wishes contribute to this, could reply to stephanspahn1@me.com, too. ps. Thanks Toby, I am reliantly coming to terms with the technics.
my agenda is to define ∞-orbifolds
Ah, I see. This is certainly something that is in the air and should be written out by someone. Glad to hear that you are working on this!
Let me mention that there is at least one existing definition already, even though no details have been spelled out yet (as far as I am aware).
Namely, as you probably know, what in the smooth context is an orbifold, in the algebraic context is a Deligne-Mumford stack. The joint higher generalizations of these, for any kind of geometry $\mathcal{G}$ (algebraic, smooth, or whatever) is what the article Structured Spaces by Jacob Lurie is all about. He calls then $\mathcal{G}$-schemes (see structured (infinity,1)-topos).
A main result of the article is that just as the “0-localic” algebraic $\infty$-schemes are just ordinary schemes, the 1-localic algebraic $\infty$-schemes are precisely the Deligne-Mumford stacks, hence the “algebraic orbifolds”.
I’d expect that when going through this argument for smooth geometry (see the very, very last paragraph of Structured Spaces for a brief remark on this) one recovers smooth orbifolds (but I have not even remotely tried to check that) and their higher analogs.
Of course, all this is from the petit topos perspective on orbifolds. What you say above is coming from the gros topos perspective. Certainly that remains to be worked out. But it should eventually be cross checked with the petit perspective.
My local expert on étale groupoids and related stuff used to be Dave Carchedi, who is currently at the MPI in Bonn (lucky him). With Dave I have occasionally chatted about the desire for higher smooth orbifolds. But we never got to the point of doing some actual research there. I am just mentioning this in case you feel like wanting to talk to somebody who has all the requisite expertise and might be eager to collaborate on something in this direction.
Thanks for your elating response. I am indeed longing for an expert to discuss my ideas with and will contact Dave Carchedi. (To be with Ieke Moerdijk does not sound bad either).
I did not find any definition or rough idea of ∞-orbifolds in the literature but would willingly take it into account.
Indeed I know the article on structured spaces by Jacob Lurie and its motivating truncations ”algebraic orbifolds” but let me indicate that there are some terminological inconsistencies to them in the literature: As I remember Eugene Lerman in ”orbifolds as stacks?” does not distinguish clearly between the notions of geometric stack, Artin stack and Deligne-Mumford stack (on manifolds).
Up to now I concentrated on the big topos perspective but I am planning to think about the other, too; of course a sensible treatment requires both.
Hi Stephan,
did you contact David Carchedi meanwhile? I just heard him speak about this, and he seems to have made some substantial progress on étale $\infty$-stacks. I have just posted a short note on his talk here.
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