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    • I have deleted some sentences that are not correct. See the last discussion on cellular model category.
    • I have deleted some statements that are not correct. It is quite obvious that what I have deleted is wrong, in fact the restatement given a few lines later is the verification of the condition that that map is onto (which is definitely not automatic).
    • started a minimum at analytification, mainly interested for the moment in collecting the references now given there which discuss analytification of algebraic (etc.) stacks

    • I have expanded just a little at KR-theory by giving it an actual Idea-paragraph and adding some more references.

    • I gather (via this nice MO comment) that

      The functor that takes linear algebraic groups G to their -points G() constitutes an equivalence of categories between compact Lie groups and -aniosotropic reductive algebraic groups over all whose connected components have -points.

      For G as in this equivalence, then then complex Lie group G() is the complexification of G().

      I have a gap in my education here and would like to fill it. What’s a good source that discusses this statement a bit more? And which one of Chevalley’s articles is this result originally due?

    • I noticed tha tthe entry separable closure existed but was effectively devoid of content. I have now copy-and-pasted the relevant paragraphs from the entry Galois theory into it.

    • I wrote a bit at heap about the empty heap (and its automorphism group, the empty group, which I put in the headline for maximum shock value).

    • Over on MO (in the comments here) Stefan Wendt kindly reminds me of an old nLab entry I once started on B1-homotopy theory. Have added a reference and hope to be adding more.

    • started Hodge cycle, but my battery is dying right this moment….

    • Larusson formulates the Oka principle homotopy-theoretically as: a complex manifold X is Oka if for every Stein manifold Σ the canonical map

      Mapshol(Σ,X)Mapstop(Σ,X)

      between the mapping spaces is a weak homotopy equivalence (see here).

      It is natural to wonder what this looks like in terms of the cohesion of the -topos AnlyticGrpd over CplxMfd.

      If we write Π:AnlyticGrpdGrpd, then up to possible technicalities to be checked, it should simply mean

      Π[Σ,X][ΠΣ,ΠX]

      where [,] is the internal hom.

      (Something close to this (but not quite the same) is what Lawvere calls the “axiom of continuity” in a cohesive topos.)

      If instead we work internally and let Π:AnlyticGrpdAnlyticGrpd be the shape modality, then the above is equivalently

      Π[Σ,X][ΠΣ,ΠX].

      In either case, it is a very natural condition to ask for in general cohesive -toposes. Maybe one should call it the Oka-Larusson property or something…

    • I split off an entry applications of (higher) category theory from the entry nPOV.

      Hopefully we find the energy to further improve this entry in various ways. For the moment I just added a 1-line intro. And a quote, which I think hits the nail of this entry on the head.

    • added pointers to Fornaess-Stout on complex polydiscs here

    • Somebody emailed me highlighting that the text green here, revision 69 of Dold-Kan correspondence does not quite parse.

      I didn’t write this,though. There is a definition meant to be equivalent to that at combinatorial spectrum, but at least some indices need renamining, and it seems maybe more needs to be fixed or at least added. Not sure. Also I absolutely don’t have the leisure to look into this right now. I hope somebody finds the energy to look into it.

    • created Serre duality with a simple minimum of content

      (I have also briefly touched a bunch of related entries on Dolbeault cohomology etc. but most of them are still in a sad state and need work)

    • I created at equivariant cohomology separate subsections for, so far, Borel equivariant and Bredon equivariant cohomology.

      At Bredon cohomology I added a sentence about the coefficient objects.

    • The nLab entry Spectral Schemes has existed for a long time, now finally the article with that title exists, too. ;-) See the link there

    • only now realized that Zoran had an old entry moduli space of bundles. Have now vigorously cross-linked it with a bunch of related entries

    • I added some material on arc-connected spaces to connected space.

      I added also a reference to Willard’s General Topology, together with this online link to a Scribd document: Willard. Is this kosher (I am guessing this document is not “pirated”, but I’m not sure)?

    • Prompted by a question which I received, I went and tried to streamline the old entry Lie infinity-algebroid representation a little:

      • moved the pevious “Properties”-discussion of complexes of holomorphic bundles to the Examples-section;

      • added the example of L-algebra extensions

      • added more information to the References-section

      • cross-linked a bit more with infinity-action and with L-infinity algebra cohomology etc.

    • I started a separate page for Picard stack (which used to be just a redirect to Picard scheme), stated the general nonsense idea with a pointer to Lurie’s thesis, where this essentially appears.

      (BWT, where in the DAG series did this end up? I forget.)

      Of course the upshot is that it’s simply the internal hom/mapping stack Pic(X)=[X,B𝔾m]. I have a question here: it seems clear that the higher versions [X,Bk𝔾m] want to be called the higher intermediate Jacobians (their deformation theory at 0 are the Artin-Mazur formal groups). Why does nobody say this? (Or if they do, where?)

    • Added stub for GAGA.
    • Over in the thread on “Picard infinity-stack” we turned to discussion of Brauer stack. Just for completeness I should probably make this a separate thread here: I had created Brauer stack for the moment only with the following Idea-section


      It is traditional to speak, for a suitable scheme X, of its Picard group and of its Brauer group. Moreover, it is a classical fact that under suitable conditions the former admits itself a canonical geometric structure that makes it the Picard scheme of X. Still well known, if maybe less commonly highlighted, is that this is just the 0-truncation of the Picard stack of X, which is simply the mapping stack [X,B𝔾m] into the delooping of the multiplicative group. In this form this applies immediately also to more general context such as E-∞ geometry ("spectral geometry") and gives a concept of Picard ∞-stack ("derived Picard stack"). Given this and the relation of the Brauer group to étale cohomology it is clear that the Brauer group similarly arises as the torsion subgroup of the 0-truncation of the ∞-stack which ought to be called the Brauer stack, given as the mapping stack

      Br(X)[X,B2𝔾m]

      into the second delooping of the multiplicative group (modulating line 2-bundles). Indeed, just as the Picard stack turns under Lie integration (evaluation on infinitesimally thickened points) and 0-truncation into what is commonly called the formal Picard group, so this Brauer -stack similarly gives what is commonly called the formal Brauer group.

      However, while therefore the terminology "Brauer stack" is the evident continuation of a traditional pattern (which in the other direction continues with the group of units and the mapping scheme [X,𝔾,]), it seems that this terminology has never been introduced in the literature (at time of this writing). (?)

    • I have just deleted a large number of dollar \ , dollar from the bottom of Blakers-Massey theorem. The effect of such is to add a large ammount of blank space at the end of the page. Was this intentional extra space for something? If not, what is causing it? I should add that I have found similar blank space before and deleted that as well.

    • started a minimum at Calabi-Yau cohomology.

      This is an obvious idea that must have been studied before (for n2) but I have had no luck with finding much detail so far.

    • ;-). I found a typo ‘gorup’ and did a search on the n-Lab…. great fun! It is good to know others have disobient fingers!

    • added references to 3d supergravity, with brief comments, and added a paragraph on how maximally supersymmetric 3d supergravity does admit an E8(8)-gauge field (while fluxed compactification from 11d allows only proper subgroups of the global U-duality E8(8) to be gauged)

    • wrote an entry cubical structure in M-theory.

      This reviews two stories from the literature, and points out that these two stories may be related.

      I am not sure yet exactly how much they are related. I am asking that here on PO