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I keep working on the entry now titled homotopy groups in an (infinity,1)-topos (used to be "homotopy groups of an oo-stack").
The following subsections I added
Geometric Pi_oo of the terminal object in a locally connected oo-topos
Geometric Pi_0 of a genral object in a locally connected topos
Examples (meaning: general examples, neither purely categorical nor geometrical, currently a discussion of simple examples that distinguish the two notions)
This all needs more work. But I'll stop for a moment and instead start now an entry on locally n-connected (oo,1)-toposes, which I need for further discussion here.
continued from here
my proposal:
Connes fusion is used to define fusion of positive energy representations of the loop group in * Antony Wassermann, Operator algebras and conformal field theory III (arXiv) and to define elliptic cohomology in * Stephan Stolz and Peter Teichner, What is an elliptic object? (link)
and removing the query box.
New entry to support the discussion with Urs about infinitesimally thickened topos.
Some of you may remember that a while ago I had started wondering how one could characterize geometric morphisms of toposes that would exhibit as an “infinitesimal thickening” of .
Instead of coming to a defnite conclusion on this one, I worked with a concrete example that should be an example of this situation: that of the Gorthendieck toposes on the sites CartSp and ThCartSp of cartesian spaces and infinitesimally thickened cartesian spaces.
But now I went through my proofs for that situation and tried to extract which abstract properties of these sites they actually depend on. Unless I am mixed up, it seems to me now that the essential property is is a coreflective subcategory of and that in the respective adjunction
buth functors preserve covers.
So maybe it makes sense to take this as a definition: a geometric morphism of Grothendieck toposes is an infinitesimal thickening if it comes from such a coreflective embedding of sites.
Details of this, with more comments on the meaning of it all and detailed proofs, I have now typed into my page on path oo-functors in the section Infinitesimal path oo-groupoids.
added to differential bimodule the basic example, thanks to Zoran
I added a disambiguation note to conjunction, since most of the links to that page actually wanted something else. Then I changed those links to something else: logical conjunction (not yet extant).
An Internet and dictionary search suggests that there is no analogous danger for disjunction (also not yet extant).
Wrote two-sided bar construction. There is a lot to add, but I added a query box under the subsection “Delooping machines” which I’m hoping someone like Mike could answer.
Moved the subsection on Hilbert space quantum mechanics from quantum information to quantum mechanics as had been suggested by someone in a query box.
Tim van Beek has written about unbounded posets at partial order.
Where is this used?
In another thread I came up with a definition of a local isomorphism in a site, working from the definition of a local homeomorphism/diffeomorphism in Top/Diff respectively (with the open cover pretopology in both cases). Then I find that there is a page local isomorphism talking about maps in presheaf categories: such a map is a local isomorphism if becomes an isomorphism on applying the sheafification functor . To quote my definition again
Definition: Let (C,J) be a site (J a pretopology). A map is a J-local isomorphism if there are covering families and such that for each the restriction is an isomorphism onto some .
I don’t claim, in the time I have available, to understand the implications of the definition at local isomorphism. I just wonder how it relates to concrete notions like local homeomorphisms (let us work with Top and open covers as covering families). Is a local homeomorphism, after applying Yoneda, a local isomorphism? Does a local isomorphism in the image of Yoneda come from a local homeomorphism? I suspect the answer is yes. Now for the biggie: can a local isomorphism be characterised in terms as basic as my definition as quoted? With my definition one avoids dealing with functor categories (and so size issues, to some extent: is very big), so if they are equivalent, I’d like to put this somewhere.
Obviously we can take the site in my definition to be a presheaf category with the canonical pretopology or something, and potentially recover the definition at local isomorphism, but for the ease of connecting with geometric ideas, I prefer something simpler.
Any thoughts?
Created small site and large site of an object in a site, as a spin off from discussion around petit topos. The latter is so named because large site is taken for sites that happen to be large. The content of this page, however, looks as though it could go somewhere discussing sheaves.
Urs has erased the sentence explanining the purpose of the entry. Why ??
"In fact not only that it is a good survey but it has a nice bibliography. The main plan of this entry is to build a hyperlinked bibliography of the above article!"
Started thinking about smooth paths.
(Incidentally, David, do you want query boxes added to your web? And would you like to change the CSS for off-web links from those boxes to some nice colour?)
I felt the need to write down what it means for a subspace to have the Baire property, so I did.
A discussion of the cartesian closed monoidal structure on an (oo,1)-topos is currently missing on the nLab.
I started making a first step in the direction of including it:
at model structure on simplicial presheaves I added a section Closed monoidal structure with a pointer to Toen’s lectures (where the following is an exercise) and a statement and proof of how is a monoidal model category by the Cartesian product.
as a lemma for that I added to Quillen bifunctor the statement that on cofib generated model cats a Quillen bifunctor property is checked already on generating cofibrations (here).
More later…
Based on recent discussions here primarily with Harry and Urs, I added a proof at co-Yoneda lemma in terms of extranaturality, and some didactic material over at adjunction bridging hom-functors to units/counits, involving some but hopefully not too much overlap with related material Urs recently added at adjoint functor. Still need to work in some links.
Created paracategory and Kleene equality.
I created cylinder on a presheaf and will fill it in more as I read through Ast308. I plan on adding more stuff as I get to it (things about test categories and localisers, etc.).
This is similar but not the same as cylinder object, since even though it is specialized to presheaf categories, we don’t require any notion of a weak equivalence a priori.
I have quietly submitted the beginning of an article on "surface diagrams" on my web. There is still quite a lot left to write up, and it needs to be formatted more prettily, but I thought I'd throw what I have (so far) out there.
I moved the characterization of pointwise kan extensions as those preserved by representable functors to the top (of the section on pointwise kan extensions) and made it the definition (since there was no unified definition before). This is for aesthetic reasons. Since being pointwise is a property, I like that this property has a definition independent of the computational model we’re using.
Are there size issues that I might be glossing over?
I added a paragraph about to countable choice
I think the definition of the Grothendieck construction was wrong. The explicit definition was right, but the description in terms of a generalized universal bundle didn’t work out to that, if by “the category of pointed categories” was meant for the functors to preserve the points, which is the usual meaning of a category of pointed objects. I corrected this by using the lax slice. Since while I was writing it I got confused with all the op’s, I decided that the reader might have similar trouble, so I changed it to do the covariant version first and then the contravariant.
I expanded the Examples-section at petit topos and included a reference to Lawvere’s “Axiomatic cohesion”, which contains some discussion of some aspects of a characterization of “gros” vs “petit” (which I wouldn’t have noticed were it not for a talk by Peter Johnstone).
I am thinking that it should be possible to give more and more formal discussion here, using Lawvere’s article and potentially other articles. But that’s it from me for the time being.
Swapped the order of the propositions that small limits commute with small limits and that limits commute with right adjoints, which allowed me to give a proof that small limits commute with small limits by citing the result on right adjoints and the characterization of the limit as right adjoint to the constant diagram functor.
Started a stub at family of sets. This should also explain concepts like a family of subsets of a given set or a family of groups. And how to formalise them all in material and structural set theories, predicative foundations, internally in indexed categories, etc.
An anonymous coward put something blank (or possibly some spam that somebody else blanked within half an hour) at Hausdorff dimension, so I put in a stub.
I moved the proof of the claim that the Segal-Brylinski “differetiable Lie group cohomology” is that computed in the (oo,1)-topos of oo-Lie groupoids from the entry group cohomology to the entry Lie infinity-groupoid and expanded the details of the proof considerably.
See this new section.
Towards the end I could expand still a bit more, but I am not allowed to work anymore today… :-)
I’ve added a bit about these to free monoid. (These are the computer scientists’ stacks, not the geometers’ stacks!) There is a query about queues too; I’ve forgotten something and can’t reconstruct it.
Changed a page title from topological topos to Johnstone’s topological topos. Urs said I should call for help when making a name change, so that someone can clear the cache to get the change to propagate properly.
started a disambiguation page basis
I just started nonabelian homological algebra.
Regarding that the nlabizens have discussed so much various generalizations of Grothendieck topology, maybe somebody knows which terminology is convenient for the setup of covers of abelian categories by finite conservative families of flat localizations functors, or more generally by finite conservative families of flat (additive) functors. Namely the localizations functors do not mutually commute so the descent data are more complicated but if you produce the comonad from a cover then the descent data are nothing but the comodules over the comonad on the product of the categories which cover. In noncommutative geometry we often deal with stacks in this generalization of topology and use ad hoc language, say for cocycles, but the thing is essentially very simple and the language barier should be overcome. There are more general and ore elaborate theories of nc stacks, but this picture is the simplest possible.
stub for crystalline cohomology
There are notes by Jacob Lurie on crystals, but I forget where to find them. Does anyone have the link?
I got the book “Counterexamples in Topological Vector Spaces” out of our library, and just the sheer number of them made me realise that my goal of getting the poset of properties to be a lattice would produce a horrendous diagram. So I’ve gone for a more modest aim, that of trying to convey a little more information than the original diagram.
Unfortunately, the nLab isn’t displaying the current diagram, though the original one displays just fine and on my own instiki installation then it also displays just fine so I’m not sure what’s going on there. Until I figure that out, you can see it here. The source code is in the nLab: second lctvs diagram dot source.
A little explanation of the design:
(It seems that Instiki’s SVG support has … temporarily … broken. I’ll email Jacques.)
created basis for a topology and linked to it with comments from coverage and, of course, Grothendieck topology
added a still somewhat stubby section on tensoring over ooGrpd to limits in a quasi-category
polished and expanded (infinity,1)-category of (infinity,1)-sheaves
In particular I spelled out the proof that the full subcategory of (oo,1)-presheaves on (infinity,1)-sheaves is a left exact reflective sub-(oo,1)-category.
started at infinity-Lie groupoid a section The (oo,1)-topos on CartSp.
Currently this gives statement and proof of the assertion that for a smooth manifold regarded as an object of the Cech nerve of a good open cover provides a cofibrant replacement.
The discussion of topological localization and that at (infinity,1)-category of (infinity,1)-sheaves for obtaining (oo,1)-sheaf toposes focuses on Grothendieck topologies. In the rest of the nLab, though, we exhibit a certain moral preference for coverages.
I therefore started a section Localization at a coverage at model structure on simplicial presheaves, where I state and prove a handful of statements that are useful for understanding this.
There is more to be said here, but that’s it from me for the moment.
Wrote about Poincare sphere, which led to perfect group. Also added a subsection “Metrizable spaces” to metric space.
Added Manifold Atlas Project to Online Resources.
I added a section on triangulable spaces and PL structures to simplicial complex, but this is the type of thing which gets beyond my ken pretty quickly. My real motivation is to convince myself that a space is homeomorphic to the realization of a simplicial complex (in short, is triangulable) if and only if it is homeomorphic to the realization of a simplicial set – perhaps this seems intuitively obvious, but it should be given a careful proof, and I want such a proof to have a home in the Lab. (Tim Porter said in a related discussion that there was a relevant article by Curtis in some early issue of Adv. Math., but I am not near a university library to investigate this.)
I’ll put down some preliminary discussion here. Let denote the poset of finite nonempty subsets of . A simplicial complex consists of a set and a down-closed subset such that every singleton belongs to . Thus is itself a poset, and we can take its nerve as a simplicial set. The first claim is that the realization of this nerve is homeomorphic to the realization of the simplicial complex. This I believe is or should be a basic result in the technique of subdivision. Hence realizations of simplicial sets subsume triangulable spaces.
For the other (harder) direction, showing that realizations of simplicial sets are triangulable, I want a lemma: that the realization of a nerve of a poset is triangulable. Basically the idea is that we use the simplicial complex whose vertices are elements of the poset and whose simplices are subsets for which we have a strictly increasing chain . Then, the next step would use the following construction: given a simplicial set , construct the poset whose elements are nondegenerate simplices (elements) of , ordered if is some face of . The claim would be that the realization of is homeomorphic to the realization of the nerve of this poset.
All of this could very well be completely standard, but it’s hard for me to find an account of this in one place. Alternatively, my intuitions might be wrong here. Or, perhaps I’m going about it in a clumsy way.
I started adding some illustrations to my personal web related to Vistoli’s paper on descent. If you like them or have suggestions to improve them, I can maybe migrate some to nLab pages:
Notes on Grothendieck Topologies, Fibered Categories and Descent Theory (ericforgy)
Todd Trimble requested currying (on the Sandbox, of all places), and I wrote it (also linking to it from closed monoidal category).
So, I have some pending changes on operad that I made in the sandbox and am waiting for a go-ahead to post from the interested parties, but I was also wondering if someone would be willing to write up a follow-up to the very nice definition of an operad as a monoid in the blah blah monoidal category. That is, it seems like this should give us a very nice way to define an algebra, but I don't know how one would actually go about doing it.
rearranged a bit and expanded category theory - contents. In particular I added a list with central theorems of category theory.
added Eric’s illustrations to the Idea-section at representable presheaf. Also added a stub-section on Definition in higher category theory.
Created faithful morphism, conservative morphism, pseudomonic morphism, and discrete morphism, and added to fully faithful morphism.
Added complete topological vector space including various variants (quasi-complete, sequentially complete, and some others). Hopefully got all the redirects right!
I only have Schaefer’s book at home so couldn’t check “locally complete” - I know that Jarchow deals with this in his book. Kriegl and Michor naturally only consider it in the context of smootheology so I’m not sure what the “best” characterisation is. There’s also a notational conflict with “convenient” versus “locally complete”. As Greg Kuperberg pointed out, in some places “convenient” means “locally complete and bornological” whereas in others it means just “locally complete” (in the contexts where convenient is used the distinction is immaterial as the topology is not considered an integral part of the structure).
I added these whilst working on the expansion of the TVS relationships diagram. That brought up a question on terminology. In the diagram, we have entries “Banach space” and “Hilbert space” (and “normed space” and “inner product space”). These don’t quite work, though, as for a topological vector space the correct notion of a normed space should be normable space as the actual choice of norm is immaterial for the TVS properties. I’m wondering whether or not this is something to worry about. Here’s an example of where it may be an issue: a nuclear Banach space is automatically finite dimensional. That implies that its topology can be given by a Hilbert structure. However, the Hilbertian norm may not be the one that was first thought of. But that’s a subtlety that’s tricky to convey on a simple diagram. So I’d rather have “normable” than “normed”. Does anyone else have an opinion on this?
If “normable” is fine, then the important question is: what’s a better way of saying “Hilbertisable”, or “Banachable”? Length doesn’t matter here, as I’m putting the expanded names in tooltips and only using abbreviations in the diagram.
started rational homotopy theory in an (infinity,1)-topos
With just slightly more it could also be called "Lie theory in an oo,1-topos" I suppose.
if you looked at this yesterday, as it was under construction, maybe have another look: I believe I could clarify the global story a bit better.
Looking at the entry Banach spaces, I found the following in the introduction:
So every -dimensional real Banach space may be described (up to linear isometry, the usual sort of isomorphism) as the Cartesian space equipped with the -norm for
which seems to imply that every norm on a finite dimensional Banach space is a -norm for some . That feels to me like a load of dingo’s kidneys. To define a norm on some I just need a nice convex set, and there’s lots more of these than the balls of -norms, surely.
Am I missing something?
Moonshine, intentionally with capital M as most people do follow this convention for the Monster and (Monstrous) Moonshine VOA.
created (infinity,1)-site
created finite limit (this was previously a redirect to finitely complete category)
It got announced in another category, but here it is in Latest Changes:
Todd began (and then I edited) simple group.
I wrote a quick entry conformal group, just from memory. Somebody could check and expand. In fact it would not be bad to have also a separate entry on conformal and on quasiconformal mappings.
chiral algebra and improvements to vertex operator algebra
somehow I missed that there already is a page compact operator and created compact operators. The plural is another error :-) the unsatisfied link that I used to create the page was “compact operators”. When I tried to rename it to the singular term it failed, of course. Now the page compact operators is simply superfluous, but as a non-administrator I cannot delete it…