Not signed in (Sign In)

Start a new discussion

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-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory 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 limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural 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 string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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).
    • We don’t use the term “range” much here, and I explained why.

    • added to filtered (infinity,1)-category the statement that these are precisely those shapes of diagrams such that \infty-colimits over them commute with finite \infty-limits.

    • algebrad and additions at Nikolai Durov. The movie starts slow and boring but gets very interesting after a while when the topic develops.

    • I have edited posite

      In particular I tried to work the query box into the text. Mike and David R. please check if you agree with the result.

    • New entry thread needed at newly expanded pro-object. Adequate changes at filtered limit.

      While threads are elements in the cofiltered limit of sets or spaces, germs are classes of equivalence which appear in the treatment of filtered colimits. However the entry germ is taking germs just in the special case of colimits forming stalks of an etale space. In my practice the notion of germ could be used more generally for directed colimits or even filtered colimits in groups, sets, and alike. Is it only me using the terminology in this extended sense ? I would like to know the opinions.

    • added to infinity-cohesive site statement and proof that if all objects of the \infty-cohesive site have points, then the cohesive \infty-topos over it satisfies the axiomm pieces have points .

      (Easy proof using the previous results and Dugger’s cofibrant replacement theorem for [C op,sSet] proj,loc[C^{op}, sSet]_{proj,loc}).

    • added to over-(infinity,1)-category the statement that the hom-spaces are computes by homotopy fibers of hom-spaces in the underlying \infty-category in direct analogy to the 1-categorical case

    • added at local (infinity,1)-topos

      • statement and proof that over Grpd\infty Grpd the \infty-connectivity condition is redundant;

      • statement and proof that for any \infty-topos H\mathbf{H} over Grpd\infty Grpd and XHX \in \mathbf{H} any small-projective object, the slice H/X\mathbf{H}/X is local.

    • because I needed the link I have created a stub (2,1)-site. I define it there as an (infinity,1)-site whose underlying category is a (2,1)-category. Don’t have the leisure right now to check that this is also a 2-site whose underlying 2-category is a (2,1)(2,1)-category, as hopefully it is

    • have a quick suggestion for a definition at embedding,

      This was the first idea that came to mind when reading Toby’s initial remark there, haven’t really thought much about it.

    • have created infinity-connected site

      • moved over from infinity-cohesive site the proof that the \infty-topos over a locally and globally \infty-connected site is locally and globally \infty-connected; (and had occasion to polish and streamline it a bit more)
    • created strongly infinity-connected site with the analogous definition, analogous proposition and analogous proof as at strongly connected site.

      Only difference is that I define it as a cosifted \infty-connected site instead of as a cosifted locally \infty-connected site, because I am currently not quite sure about the definition of the latter because my model-category theoretic proofs rely on the existence of a terminal object, without that my standard Quillen adjunction model for the terminal \infty-geometric morphism fails and I’d need to think harder.

      But all these entries of higher connected and local sites currently have the issue that they give sufficient conditions and don’t prove necessary conditions, so I think it’s okay, but we should keep in mind that there might be refinements of these definitions.

    • created connected site and declared it to be a locally connected site with a terminal object.

      That’s sufficient for its sheaf topos to be connected, I don’t now if it is necessary: If we find a weaker sufficient condition we should refine the entry.

    • I plan to write few foundations/set theory stubs including Skolem paradox. It will wait for a bit as the nnLab seems to be down at the moment.

      The entry forcing has phrase downward Löwenheim-Skolem theorem. What does it mean downward in this phrase ? Is it a modifier at all ?

    • I added a number of facts and proofs to connected topos, and shortened the example since it is a special case of the general fact about connected locally connected sites.

    • I rewrote the Idea-section at n-localic (infinity,1)-topos (trying to make it more to the point) and added propositions in the Properties- and the Examples-section.

    • added the full statement to (infinity,1)-Yoneda lemma

      It's an easy consequence of the sSet-enriched Yoneda lemma, using the theorem that oo-presheaves are presented by fibrant-cofibrant sSet enriched presheaves.

      Notice that this appears without stating the name "Yoneda lemma" as Lemma 5.5.2.1 in HTT. (Beware, though, that there are typos in there. There is a general mix-up of ops already in the statement of the lemma and the first  \psi in the proof is a  \phi . )

    • I rewrote the Idea-section of structured (infinity,1)-topos.

      i tried to make it clearer and shorter. And I highlighter more the aspect that this is a way to equip little \infty-toposes with geometric structure.

    • created homotopy groups of a Lie groupoid

      The definition is the one following from the general abstract \infty-topos theoretic notion applied in ooLieGrpd, but I wanted a separate entry for this in order to record the references for that special case

    • I expanded at cohesive site the example “Families of sets”.

      I also started expanding the Examples-section at cohesive topos, but it remains stubby for the moment. I have to run now.

    • I wrote an introduction to logic. It’s pretty brief, but at least it defines the meanings (one a mass noun, one a count noun) of the word.

    • I have added to the Examples at structured (infinity,1)-topos a section Canonical structure sheaves on objects in a big topos.

      For the moment this only contains the observation that for H=Sh(𝒢)\mathbf{H} = Sh(\mathcal{G}) the big topos on a geometry 𝒢\mathcal{G}, for every object XHX \in \mathbf{H} its little topos H/X\mathbf{H}/X is canonically equipped with a 𝒢\mathcal{G}-structure sheaf.

      This is evident from the discussion at etale geometric morphism, but it nevertheless seems to be noteworthy.

      I have added also an inducation on how this canonical structure sheaf is indeed that of 𝒢\mathcal{G}-valued functions on XX. But more details on this would be desireable. But I have to interrupt now.

    • somebody signing as “Anonymous Coward” dropped a query box with a question at semigroup.

    • expanded affine line, added the definitions of 𝔾 m\mathbb{G}_m and 𝔾 a\mathbb{G}_a

    • tried to bring the entry orientation into a bit of shape

    • am starting to work on derived smooth manifold, so far just a little bit on the motivation (correction of limits of manifolds)

      I am a bit hesitant to add a lot of details from David Spivak’s article, since it seems evident that there is some room to streamline the constructions. I need to think about how to deal with this. One really wants to just specify the site as a geometry (for structured (infinity,1)-toposes) and then just say that a derived manifold is a derived scheme in the sense descrived at generalized scheme on this.

      In section 10.1 David Spivak discusses one reason that prevented him from setting things up this way: actually I think this points to the following general issue with the definition of geometry (for structured (infinity,1)-toposes): instead of a Grothendieck topology generated by admissible morphisms the definition ought to just refer to a coverage by admissible morphisms, and instead of the stability under pullback one ought to just consider the coverage-style stability condition.

      More later.

    • I worked a bit on quasicoherent (infinity,1)-sheaf:

      • I polished the account of the model-category theory presentation by Toen-Vezzosi a little

      • Then I added the slick general abstract definition in terms of the tangent (oo,1)-category that we once discussed, but which nobody had yet filled into this entry

    • The page internal ∞-groupoid claimed that the case of “internal ∞-groupoids in an (∞,1)-category” was discussed in detail at groupoid object in an (∞,1)-category. That doesn’t seem right to me—I think the groupoid objects on the latter page are really only internal 1-groupoids, not internal ∞-groupoids. They’re “∞” in that their composition is associative and unital only up to higher homotopies, but those are homotopies in the ambient (∞,1)-category; they themselves contain no “higher cells” as additional data. In particular, if the ambient (∞,1)-category is a 1-category, then an internal groupoid in the sense of groupoid object in an (∞,1)-category is just an ordinary internal groupoid, no ∞-ness about it. Does that seem right?

    • I added a bunch of entries to homotopy - contents – mostly all the variants of homotopy groups – and inserted the floating TOC to all pages listed there

    • I have started listing the contents / chapters on some of Hans Baues books and papers (that I have at hand and that are relevant to the Lab). So far I have done Algebraic Homotopy and Combinatorial Homotopy and 4-Dimensional Complexes, but so far have not tried to give section titles nor to link with other entries.

      (Edit: I have now added Homotopy Types, which is his article in the Handbook of Alg. Top. I copied the format from another similar entry but find it a bit heavy, suggestions please. I do intend to list sections and subsections and add more links later.)

    • Mr. or Mrs. Anonymous Coward created cell complex but didn’t have much to say. Maybe somebody feels like helping the Coward.

      (Is such activity failed spam or failed contribution?)