Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 PSh(S)Sh(S,J)PSh(S) \to Sh(S,J). To quote my definition again

      Definition: Let (C,J) be a site (J a pretopology). A map f:abf:a \to b is a J-local isomorphism if there are covering families (v ib)(v_i \to b) and (u ja)(u_j \to a) such that for each u ju_j the restriction f|u jf|u_j is an isomorphism onto some v iv_i.

      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: [Top op,Set][Top^{op},Set] 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?

    • 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!"

      Geometric and topological structures related to M-branes

    • 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 [C op,sSet] proj[C^{op},sSet]_{proj} 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.

    • 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 have started an entry on pre-Lie algebras, which are much more interesting than you might think at first. My friend Bill Schmitt, the combinatorist, is visiting and telling me amazing things about combinatorics and operads.... this is a little bit of the story.
    • 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 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 the article dependent choice, and did some editing at COSHEP to make clearer to myself the argument that COSHEP + (1 is projective) implies dependent choice. It’s not clear to me that the projectivity of 1 is removable in that argument; maybe it is.

    • 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.

    • started a disambiguation page basis

    • 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:

      1. Abbreviate all the nodes to make the diagram more compact (with a key by the side, and tooltips to display the proper title).
      2. Added some properties: LF spaces, LB spaces, Ptak spaces, B rB_r spaces
      3. Taken out some properties: I took out those that seemed “merely” topological in flavour: paracompactness, separable, normal. I’m pondering taking out completeness and sequential completeness as well.
      4. Tried to classify the different properties. I picked three main categories: Size, Completeness, Duality. By “Size”, I mean “How close to a Banach space?”.

      (It seems that Instiki’s SVG support has … temporarily … broken. I’ll email Jacques.)

    • 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 sPSh(CartSp) proj,covsPSh(CartSp)_{proj,cov} the Cech nerve of a good open cover provides a cofibrant replacement.

    • 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 P fin(X)P_{fin}(X) denote the poset of finite nonempty subsets of XX. A simplicial complex consists of a set VV and a down-closed subset ΣP fin(V)\Sigma \subseteq P_{fin}(V) such that every singleton {v}\{v\} belongs to Σ\Sigma. Thus Σ\Sigma 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 {x 1,x 2,,x n}\{x_1, x_2, \ldots, x_n\} for which we have a strictly increasing chain x 1<x 2<<x nx_1 \lt x_2 \lt \ldots \lt x_n. Then, the next step would use the following construction: given a simplicial set XX, construct the poset whose elements are nondegenerate simplices (elements) of XX, ordered x<yx \lt y if xx is some face of yy. The claim would be that the realization of XX 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.

    • 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.

    • 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 nn-dimensional real Banach space may be described (up to linear isometry, the usual sort of isomorphism) as the Cartesian space n\mathbb{R}^n equipped with the pp-norm for 1p1 \leq p \leq \infty

      which seems to imply that every norm on a finite dimensional Banach space is a pp-norm for some pp. That feels to me like a load of dingo’s kidneys. To define a norm on some n\mathbb{R}^n I just need a nice convex set, and there’s lots more of these than the balls of pp-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.

    • somebody signing as “Anonymous Coward” had created special relativity and typed in a confused paragraph (the smallest confusion being that the paragraph concerned not special but general relativity).

      I removed that paragraph and quickly wrote a brief “Idea”-section . But have no time to do this justice now.

    • 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.

    • 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…

    • Created sequential compactness, should probably link to all these compactness variations from compact space. Not sure if I got the “iff” bit right in the relationship with compactness itself; will check it myself if no-one fixes it in the meantime.

      I decided that this was the key property in manifolds of mapping spaces and to stop trying to figure out a Froelicher version of sequentially compact for the time-being.

    • the term “twist” or “twisted” is one of those insanely ambiguous terms in math. Trying to follow our recent agreement on how to deal with ambiguous page names, I tried to indicate this at twist .

    • Created lax-idempotent 2-monad, with some definitions from Kelly–Lack. I think Kock has a couple of others. I’ll add more, like proofs of the equivalence of the definitions, and more on the cocompletion example, later (next week, probably).

    • The final copy of my thesis is up on the lab. Available from Fundamental Bigroupoids and 2-Covering Spaces. I’ve fixed the typo in definition 5.1 that made it into the print copy ;)

      Thus I’ve updated the links at David Roberts, the above linked page, and on my private web home page. If anyone knows of any other places it is linked, please let me know, or update the link to point to DMRthesis_final.pdf, instead of DMR_thesis.pdf.

      Now to all the other projects that are on the back burner, time permitting…

    • started a floating toc for topos theory. See at the right of topos.

      Please feel encouraged to expand and improve the structure.