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    • 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 X∈HX \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.

    • 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?)

    • I extracted the definition of “n-category with all duals” from Scott Morrison and Kevin Walker’s “Blob homology” at blob n-category.

      This is to some extent a take on defining hyperstructures.

    • Someone left a query at bicategory of relations, and I put down a partial response (proving the separability condition). I plan to add a little more later, but I confess that I don’t see how to derive the other dual Frobenius condition from the one given. (Hard to believe the dual one wasn’t given by Carboni and Walters.)

      Carboni and Walters call a “bicategory of relations” a discrete cartesian bicategory (because the local posets of maps are discrete). They are equivalent to unitary pretabular allegories.

    • An Anonymous Coward changed a codecogs picture to an SVG at locale of real numbers, but it doesn’t look right to me, so I changed it back. (The SVG editor didn’t recognise it; otherwise, I’d have tried to fix it that way.)

    • There’s a note on how to do this at preset.

    • recorded at end in a new section Set-enriched coends as colimits the isomorphism

      ∫ d∈DW(d)⋅F(d)≃lim →((elW) op→D→FC). \int^{d \in D} W(d) \cdot F(d) \simeq \lim_{\to}( (el W)^{op} \to D \stackrel{F}{\to} C ) \,.

    • added characterizations of smooth kk-algebras to smooth scheme.

      Some expert please look at that and its relation to the rest of the entry.

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