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Put a link to
into weak omega-groupoid… only trouble being that this entry doesn’t exist yet but redirects to infinity-groupoid, which otherwise has no references currently ?!-o . Somebody should take care of editing this a bit. But it won’t be me right now.
I made some modifications to the definition section of root, and added the theorem that finite multiplicative subgroups of a field are cyclic. While I was at it, I added a bit to quaternion.
I have split off a brief entry Zuckerman induction from cohomological induction (since the basic version is not necessarily derived).
In email discussion with somebody I wanted to point to the Lab entry A-infinity space only to notice that there is not much there. I have now spent a minute adding just a tiny little bit more…
Idea section for a new entry cohomological induction and a new stub induced comodule. I have separated corepresentation from comodule&coaction. Sometimes corepresentation is the same as coaction, sometimes there are small differences (defined on dense subspaces etc.) but more important, there is a different notion of corepresentation in Leibniz algebra theory, which will be explained in a separate section later.
A remark at induction.
I could have sworn that we already had the following entry, but it seems we didn’t. Now we do:
As already mentioned in another thread, I have added to infinity-image a brief new section Syntax in homotopy type theory. But please check! And even if correct, it’s still a bit rough.
It’s time to become serious about “higher order” aspects of applications of the the “sharp-modality” in a cohesive (infinity,1)-topos – I am thinking of the construction of moduli -stacks for differential cocycles.
Consider, as usual, the running example Smooth∞Grpd.
Here is the baby example, which below I discuss how to refine:
there is an object called , which is just the good old sheaf of differential 1-forms. Consider also a smooth manifold . On first thought one might want to say that the internal hom object is the “moduli 0-stack of differential 1-forms on ”. But that’s not quite right. For CartSp, the -plots of the latter should be smoothly -parameterized sets of differential 1-forms on , but the -plots of contain a bit more stuff. They are of course 1-forms on and the actual families that we want to see are only those 1-forms on which have “no leg along ”. But one sees easily that the correct moduli stack of 1-forms on is
where is the concretification of .
This above kind of issue persists as we refine differential 1-forms to circle-principal connections: write for the stack of circle-principal connections. Then for a manifold, one might be inclined to say that the mapping stack is the moduli stack of circle-principal connections on . But again it is not quite right: a -plot of is a circle-principal connection on , but it should be one with no form components along , so that we can interpret it as a smoothly -parameterized set of connections on .
The previous example might make one think that this is again fixed by considering . But now that we have a genuine 1-stack and not a 0-stack anymore, this is not good enough: the stack has as -plots the groupoid whose objects are smoothly -parameterized sets of connections on – that’s as it should be – , but whose morphisms are -parameterized sets of gauge transformations between these, where is the underlying discrete set of the test manifold – and that’s of course not how it should be. The reflection fixes the moduli in degree 0 correctly, but it “dustifies” their automorphisms in degree 1.
We can correct this as follows: the correct moduli stack of circle principal connections on some is the homotopy pullback in
where the bottom morphism is induced from the canonical map from circle-principal connections to their underlying circle-principal bundles.
Here the in the bottom takes care of making the 0-cells come out right, whereas the pullback restricts among those dustified -parameterized sets of gauge transformations to those that actually do have a smooth parameterization.
The previous example is controlled by a hidden pattern, which we can bring out by noticing that
where is the 2-image of , hence the factorization by a 0-connected morphism followed by a 0-truncated one. For the 1-truncated object the 2-image doesn’t change anything. Generally we have a tower
Moreover, if we write for the Dold-Kan map from sheaves of chain complexes to sheaves of groupoids (and let stackification be implicit), then
If we pass to circle-principal 2-connections, this becomes
and so on.
And a little reflection show that the correct moduli 2-stack of circle-principal 2-connections on some is the homotopy limit in
This is a “3-stage -reflection” of sorts, which fixes the naive moduli 2-stack first in degree 0 (thereby first completely messing it up in the higher degrees), then fixes it in degree 1, then in degree 2. Then we are done.
I just added a page on the Bondal-Orlov reconstruction theorem. Feel free to edit!
It should have its own announcement: Frankel model of ZFA added to the lab. I should say that in this model there is a map where every fibre has two elements, which has no section (which would be “choosing a sock out of a countable set of pairs”)
Created Abraham Fraenkel, the F in ZFC.
I have splitt off Hamiltonian vector field from symplectic manifold in order to also record the -plectic generalization.
Faà di Bruno formula with redirect Faà di Bruno Hopf algebra
We had discussed here at some length the formalization of formally etale morphisms in a differential cohesive (infinity,1)-topos. But there is an immediate slight reformulation which I never made explicit before, but which is interesting to make explicit:
namely I used to characterize formal étaleness in terms of the canonical morphism between the components of the geometric morphism that defines the differential cohesion – because that formulation made close contact to the way Kontsevich and Rosenberg formulate formal étaleness.
But there is a more suggestive/transparent but equivalent (in fact more general, since it works in all of not just in ) formulation in terms of the -modality, the “fundamental infinitesimal path -groupoid” operator:
a morphism in is formally étale precisely if the canonical diagram
is an -pullback.
(It’s immediate that this is equivalent to the previous definition, using that is fully faithful, by definition.)
This is nice, because it makes the relation to general abstract Galois theory manifest: if we just replace in the above the infinitesimal modality with the finite path -groupoid modality , then the above pullback characterizes the “-closed morphisms” which precisely constitute the total space projections of locally constant -stacks over . Here we now characterize general -stacks over .
And for instance in direct analogy with the corresponding proof for the -modality, one finds for the -modality that, for instance, we have an orthogonal factorization system
I’ll spell out more on this at Differential cohesion – Structures a little later (that’s why this here is under “latest changes”), for the moment more details are in section 3.7.3 of differential cohomology in a cohesive topos (schreiber).
added to disk a brief pointer to Joyal’s combinatorial disks. Needs to be expanded, probably entry should be split and disambiguated. But no time right now.
I have started something in an entry
which has grown out of the the desires expressed in the thread The Wiki history of the universe.
This is tentative. I should have maybe created this instead on my personal web. I hope we can discuss this a bit. If it leads nowhere and/or if the feeling is that it is awkward for one reason or other, I promise to remove it again. But let’s give this a chance. I feel this is finally beginning to converge to something.
non-Hausdorff manifold (just for complenetess, since I was editing the exposition at manifold)
started adding list of references to the page Bill Lawvere
not that I made it very far -- just three items so far :-)
I was really looking for an online copy of "Categorical dynamics" as referenced at synthetic differential geometry and generalized smooth algebra, but haven't found it yet. I was thinking that the "Toposes of laws of motion" that I do reference must be something close. But I don't know.
New, mainly disambiguation, entry affine algebra. Note that affine algebra for most algebraists is not the same as affine Lie algebra. I have corrected a wrong link in Wess-Zumino model which links to affine algebra instead to affine Lie algebra; let us be careful when linking in future. Affine algebras are coordinate rings of affine varieties. I have split affine variety from algebraic variety which also got a redirect algebraic manifold (= smooth algebraic variety). New entry Igor Shafarevich.
New stubs Bargmann-Segal transform and Hall coherent state. Changes to coherent state and coherent state in geometric quantization. We still need Bergman kernel (the coincidence is that the (Segal-)Bargmann kernel is a special case of Bergman kernel from complex analysis :))
started differential Galois theory
expanded at 2-framing
I created Alex Heller at Jim’s suggestion. It is very stubby and could have a lot more added.
New page for A. Ehresmann (and relevant redirects, including Bastiani) and links with Ehresmann, Cahiers.
New entry (!) tangent Lie algebra. Significant changes at invariant differential form with redirect invariant vector field reflecting the vector fields and other tensor cases. Many more related entries listed at and the whole entry reworked extensively at Lie theory. Some changes at Lie’s three theorems and local Lie group. New stubs Chevalley group and Sigurdur Helgason.
By the way, when writing tangent Lie algebra, I had the problem with finding the correct font for the standard symbol of Lie algebra of vector fields on a manifold. Usually one has varchi symbol which looks like Greek chi but with dash through middle. The varchi symbol is not recognized and I put mathcal X which is slanted and script, just alike, but without dash through middle.
By the way, on a real Lie group of dimension , if one expresses the right invariant vector field in terms of left invariant vector fields then at each point there is a -linear operator which sends any frame of left invariant vector fields to the corresponding frame in right invariant vector fields; this gives a -valued real analytic function on (or, in local coordinates, on a neighborhood of the unit element). In other words, if I take a frame in a Lie algebra and interpret it in two ways, as a frame of left invariant vector fields and a frame in right invariant vector fields, then I can take a matrix of real analytic functions on a Lie group and multiply the frame of left invariant vector fields with this matrix to get the correspoding frame of right invariant vector fields. I use in my current research some computations involving this matrix function. Does anybody know of any reference in literature which does any computations involving this matrix valued function on ?
New entry spectral curve.
A stub Massey product and a longer Toda bracket (still plenty gaps of reference, many many unlinked words). No promises w.r.t. spellings.
I now see I’ve missed the convention for capitalization. Will fix that now… done.
Cheers
I am hereby moving the following old Discussion box from interval object to here
Urs Schreiber: this is really old discussion by now. We might want to start putting dates on discussions. In principle it can be seen from the entry history, but readers glancing at this here hardly will. Maybe discussions like this here are better had at the forum after all.
We had originally started discussing the notion of interval objects at homotopy but then moved it to this entry here. The above entry grew out of the following discussion we had, together with discussion at Trimble n-category.
Urs: Let me chat a bit about what I am looking for here. It seems very useful to have a good notion of what it means in a context like a closed category of fibrant objects to say that path objects are compatibly corepresented.
By this should be meant: there exists an object such that
for any other object, is a path object;
and such that has some structure and property which makes it “nice”.
In something I am thinking about the main point of being nice is that it can model compositon: it must be possible to put two intervals end-to-end and get an interval of twice the length. In some private notes here I suggest that:
a “category with interval object” should be
with a compatible structure of a category of fibrant objects
and equipped with an internal co-categoy on for the interval object;
such that co-represents path objects, in that for all objects , is a path object for .
I think there are a bunch of obvious examples: all familiar models of higher groupoids (Kan complexes, -groupoids etc.) with the interval object being the obvious cellular interval .
I also describe one class of applications which I think this is needed/useful for: recall how Kenneth Brown in section 4 of his article on category of fibrant objects (see theorems recalled there and reference given there) describes fiber bundles in the abstract homotopy theory of a pointed category of fibrant objects. This is pretty restrictive. In order to describe things like -vector bundles in an context of enriched homotopy theory one must drop this assumption of the ambient category being pointed. The structure of it being a category with an interval object is just the necessary extra structure to still allow to talk of (principal and associated) fiber bundles in abstract homotopy theory. It seems.
Comments are very welcome.
Todd: The original “Trimblean” definition for weak -categories (I called them “flabby” -categories) crucially used the fact that in a nice category , we have a highly nontrivial -operad where the components have the form , where here denotes the cospan composite of two bipointed spaces (each seen as a cospan from the one-point space to itself), and the hom here is the internal hom between cospans.
My comment is that the only thing that stops one from generalizing this to general (monoidal closed) model categories is that “usually” doesn’t seem to be “nice” in your sense here, and so one doesn’t get an interesting (nontrivial) operad when my machine is applied to the interval object. But I’m generally on the lookout for this sort of thing, and would be very interested in hearing from others if they have interesting examples of this.
to be continued in the next comment
added to representable morphism of stacks the remark that precisely along representable morphisms of stacks over the category of smooth manifolds (i.e. smooth infinity-groupoids) do we have push-forward in generalized cohomology.
But I still need to write out some indication justifying this assertion…
I noticed that the text at loop space didn’t point to smooth loop space and didn’t make clear that such a variant might even exist. So I have now expanded the Idea-section there a little to give a better picture.
I have received a question on the old entry directed object, so I am looking at that now. First of all I’ll clean it up a bit and move old discussion from there to here:
[begin forwarded discussion]
+–{.query}
Eric: I don’t fully “grok” this constructive definition, but I like its flavor. Is it possible to formalize the procedure in a simple catchy phrase? In other words, when you begin with a “category with interval object ”, but whose objects are otherwise undirected (like Top), you construct the “supercategory ” with directed -objected (even though no objects in are directed). I used the term “directed internalization”, but is there a better term?
I just think this concept is important and should have some really slick arrow theoretic description and I’m not having any luck coming up with one myself.
=–
[ continued in next post ]
finally expanded the long-existing table of contents complex geometry - contents and included it as a floating TOC in the relevant entries.
Do we have more entries that need to go here and which I have forgotten?
started a stub n-category object in an (infinity,1)-category, to go in parallel with the existing category object in an (infinity,1)-category.
For the moment, nothing there yet apart from a brief remark that Theta_n spaces are -categories internal to . I hope to expand this entry later.
did I say that I created Theta space?
This is a really nice model. Rezk claims to have shown to get the homotopy hypothesis right for all (n,r)-categories and for both n and r ranging to . If that holds water, it's quite impressive. It seems the only thing missing then is the
- Theta-space of all
-Theta spaces. Does anyone know if there is a proposal for that?
It's also interesting how the result is a mix of globular and simplicial shapes. So in what respect does that build on/improve over Joyal's original proposal?
A query about the new entry on copncurrency theory: Does ‘simultaneously’ make sense if there is no global clock?
If not, then the situation gets a lot more like some models for spacetime and the idea of slices through some evolving state space might be a good model.
Someone, apparently in Berlin, has created a page called www.mfo.de/document/1145/OWR_2011_52.pdf, with just that text (and ’My First Slide’) in the body. The URL points to a report on a logic workshop at Oberwolfach around this time last year. It’s not spam, but what should we do with it?
Someone signing themselves as ‘Joker? at November 3, 2012 08:05:13 from 93.129.88.58’ deleted two lines from sheaf and topos theory. There seemed no reason for this, so I have rolled back to the previous version.
added a bit more to string^c structure, but it’s still stubby
The recent changes to the various modal logic pages have changed the emphasis from the ’many agent’ versions .etc. to a type theoretic one. That would be okay but in so doing they have become a bit garbled so they refer to K(m) but then just describe itself. I am wondering what is planned for these. I originally wrote them with the aim of increasing the nPOV side of the Computer Science entries and to have some brief introduction to modal logs, what should they become?
October 24, 2012 09:26:08 by Anonymous Coward (99.133.144.164) has added a comment questioning the validity of a sentence at reflective subcategory.
wanted to be able to say sum and have a pointer to somewhere.
I made starts on lexicographic order and on compactification. Lexicographic order was defined only for products of well-ordered families of linear orders (probably the most common type of application).
I’m not very happy with the opening of compactification.
I edited the old entry projection a little.
There is no real systematics in common use of “projection” as opposed to “projector”, but I think the following makes good sense:
a projection is a canonical map out of a product;
a projector is an idempotent in a suitably abelian category
and then the relation is: A projector is a projection followed by a subobject inclusion.
That’s how I have now put it in the entry.
New entry enumerative geometry. New stubs Schubert calculus, intersection theory.
By the way (Andrew); the title of this nForum post is not seen but truncated. This happens because of some other stuff is placed into the corner in the same line. It says unimportant info “Bottom of Page” preceded by long space between the truncated title and this info ad. I think it is more important that the titles be spelled entirely.
stub for 5-dimensional supergravity (for the moment I just need the record the reference)
created a table of contents idempotents - contents and included it as a floating TOC into the relevant entries
While writing at k-morphism, I noticed there is no article on globular operad (aka Batanin operad), so I wrote one. Experts please look over, and improve if desired.
While writing the new Idea-section now at Segal condition I felt the need to have a table of contents
So I started one and added it to the relevant entries as a floating TOC.
I was asked by email about the claim at geometry of physics that integration can be axiomatized in cohesive homotopy type theory simply a truncation operation (followed by concretification, for the right cohesive structure). That may sound surprising.
So I have started to work on the section geometry of physics – integration. So far I have there the following introductory text, which is supposed to already indicate at least while the claim is plausible. Eventually maybe I can move parts of this to the entry integration proper.
Here is what it currently has in the intro-paragraph of geometry of physics - integration:
I think we need a floating table of contents categories of categories - contents to connect our entries on related topics. I have started one.
But this needs to be further expanded. also haven’t included it into the relevant entries yet, no time right now.
I noticed that we didn’t really have a general-purpose entry localization of model categories (on top of the detailed Bousfield localization of model categories which we did have). I quickly created something with just some basic pointers.
I have written out in some detail the proof at Grothendieck spectral sequence.
But I still need to go through it and proof-read and polish. Handle with care for the moment. Maybe the whole thing needs to be rearranged, for readability.
spelled out some basics at spectral sequence of a double complex
Key references at Jones polynomial and von Neumann algebra factor.
(Should we have subfactors as a separate entry or put them under factors ?)
I (only) now realize that I pretty much missed the story of familial regularity and exactness. But also it was easy to miss, with the entries that are unified by this not pointing back to it.
To rectify this I have created now a floating TOC and am including it into all the relevant entries:
Please check out that TOC and edit/modify as need be.