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    • 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.

    • I am trying to write up an elementary exposition for how the Hochschild chain complex for a commutative associate algebra is the normalized chains/Moore complex of the simplicial algebra that one gets by tensoring the algebra AA with the simplicial set Δ[1]/∂Δ[1]\Delta[1]/\partial \Delta[1]:

      C •(A,A)=N •((Δ[1]/∂Δ[1])⋅A). C_\bullet(A,A) = N_\bullet( (\Delta[1]/\partial \Delta[1]) \cdot A ) \,.

      I would like to get feedback on whether or not my exposition is in fact understandable in an elementary way.

      The section that contains this material is the section

      The simplicial circle algebra

      at the entry Hochschild cohomology. Just this one section. It’s not long.

      It describes first the simplicial set Δ[1]/∂Δ[1]\Delta[1]/\partial \Delta[1], then discusses how the coproduct in CAlg kCAlg_k is given by the tensor product over kk, and deduces from that what the simplicial algebra (Δ[1]/∂Δ[1])(\Delta[1]/\partial \Delta[1]) is like.

      After taking the normalized chains of that, the result is Pirashvili’s construction of a chain complex from a simplicial set and a commutative algebra. I just think it is important to amplify that this construction of Pirashvili’s is a categorical tensoring=copower operation. Because that connects the construction to general abstract constructions. That’s what the beginning of the above entry is about. But for the moment I would just like to make the elementary exposition of the tensoring operation itself pretty and understandable.

    • tried to bring the old neglected entry sSet-category roughly into some kind of stubby shape. Added Porter-Cordier and LurieA.3 as references. The former was my motivation for doing this. Eventually it would be good to have here a detailed discussion of sSetsSet-category models for (∞,1)(\infty,1)-category theory. See the discussion with Tim over in the other thread on the (∞,1)(\infty,1)-Yoneda lemma.

      (I don’t have time for this now. I am saying all this in the hope that somebody else has.)

    • I’ve cleaned up diffeological space a little. In particular:

      1. I’ve removed all references to Chen spaces. There is a relationship, but not what was implied on that page.
      2. I’ve tried to clean up the distinction between the definition in the literature (which uses all open subsets of Euclidean spaces) and the preferred nLab definition (which uses CartSp).
      3. Other minor cleaning.

    • I have been advising Herman Stel on his master thesis, which is due out in a few days. I thought it would be nice to have an nLab entry on the topic of the thesis, and so I started one: function algebras on infinity-stacks.

      For TT any abelian Lawvere theory, we establish a simplicial Quillen adjunction between model category structures on cosimplicial TT-algebras and on simplicial presheaves over duals of TT-algebras. We find mild general conditions under which this descends to the local model structure that models ∞\infty-stacks over duals of TT-algebras. In these cases the Quillen adjunction models small objects relative to a choice of a small full subcategory C⊂TAlg opC \subset T Alg^{op} of the localization

      L↪←LH=Sh (∞,1)(C) \mathbf{L} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} \mathbf{H} = Sh_{(\infty,1)}(C )

      of the (∞,1)(\infty,1)-topos of (∞,1)(\infty,1)-sheaves over duals of TT-algebras at those morphisms that induce isomorphisms in cohomology with coefficients the canonical TT-line object. In as far as objects of H\mathbf{H} have the interpretation of ∞-Lie groupoids the objects of L\mathbf{L} have the interpretatin of ∞-Lie algebroids.

      For the special case where TT is the theory of ordinary commutative algebras this reproduces the situation of (Toën) and many statements are straightforward generalizations from that situation. For the case that TT is the theory of smooth algebras (C ∞C^\infty-rings) we obtain a refinement of this to the context of synthetic differential geometry.

      As an application, we show how Anders Kock’s simplicial model for synthetic combinatorial differential forms finds a natural interpretation as the differentiable ∞\infty-stack of infinitesimal paths of a manifold. This construction is an ∞\infty-categorical and synthetic differential resolution of the de Rham space functor introduced by Grothendieck for the cohomological description of flat connections. We observe that also the construction of the ∞\infty-stack of modules lifts to the synthetic differential setup and thus obtain a notion of synthetic ∞\infty-vector bundles with flat connection.

      The entry is of course as yet incomplete, as you will see.

    • I created split coequalizer and absolute coequalizer, the latter including a characterization of all absolute coequalizers via an “nn-ary splitting.” While I was doing this, I noticed that monadic adjunction included a statement of the monadicity theorem without a link to the corresponding page, so I added one. (The discussion at the bottom of monadic adjunction should probably be merged into the page somehow.) Then I noticed that while we had a page preserved limit, we didn’t have reflected limit or created limit, so I created them. They could use some examples, however.

      I would also like to include an example of how to actually use the monadicity theorem to prove that a functor is monadic. Something simpler than the classic example in CWM about compact Hausdorff spaces; maybe monadicity of categories over quivers? Probably not something that you would need the monadicity theorem for in practice, so that it can be simple and easy to understand.

    • Under the Definitions at topos, I gave definitions of Grothendieck toposes and W-toposes, since these are two very important kinds of toposes that some authors (at least) often call simply ‘toposes’. (Also it gave me a place to redirect W-topos and its synonym topos with NNO.)

    • On Tim’s suggestion, I have been in contact with Ronnie Brown on some of the history behind “convenient categories”, and received a wealth of information from him. I have made an initial attempt to summarize what I have learned in the Historical Remarks section of convenient category of topological spaces, but it might be somewhat garbled still. Hopefully Ronnie and/or Tim will have a look. I will be adding more references by and by.

      I also added in the follow-up discussion with Callot under Counterexamples.

    • Francois Métayer already has a lab-entry. (NB the acute accent is missing on the new one! That was the cause of the error.)

    • I added some more details to the section on ultrafilter monad at ultrafilter. Incidentally, it seems to me that the bit on Barr’s observation (“topological spaces = relational β\beta-modules”) is too terse. There is a lot of generalized topology via abstract nonsense that deserves more explanation.

      I also added some elementary material to cartesian closed category, mainly to indicate to the novice how exponentials deserve to be thought of as function spaces, how internal composition works, and so forth. I left the job somewhat unfinished.

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