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    • I have added the following reference to Berkovich space. Judging from the abstract this sounds like I nice unifying perspective. But I haven’t studied it yet

      • Oren Ben-Bassat, Kobi Kremnizer, Non-Archimedean analytic geometry as relative algebraic geometry (arXiv:1312.0338)

      We show that Berkovich analytic geometry can be viewed as algebraic geometry in the sense of Toën-Vaquié-Vezzosi over various categories. The objects in these categories are vector spaces over complete valued fields which are equipped with additional structure. The categories themselves will be quasi-abelian and this is needed to define certain topologies on the categories of affine schemes. We give new definitions of categories of Berkovich analytic spaces and in this way we also define (higher) analytic stacks. We characterize in a categorical way the G-topology or the topology of admissible subsets used in analytic geometry. We demonstrate that the category of Berkovich analytic spaces embeds fully faithfully into the categories which we introduce. We also include a treatment of quasi-coherent sheaf theory in analytic geometry proving Tate’s acyclicity theorem for quasi-coherent sheaves. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

    • I wanted to collect some of the stuff recently added to a bunch of chromatic entries in a way that forms an at least semi-coherent story, so I made an entry

      This is built mostly from copy-and-pasting stuff that I had added to dedicated entries, equipped with a bit of glue to make it stick together and form a story.

      (Special thanks to Marc Hoyois for general discussion and in particular for working on the text on the Lurie spectral sequence.)

      I want to further fine-tune this. But not tonight.

    • as you may have seen in the logs, I am working on an entry Higher toposes of laws of motion, something like extended talk notes.

      I am running a bit out of time, and so the entry is unpolished and turns into just a list of keywords towards the end, for the moment. But in case anyone is wondering about the logs, here is the announcement.

      Don’t look at this yet if you feel like just reading. Of course if you feel like joining in with the editing a bit, that’s welcome, as usual.

    • I split off an entry dg-geometry from the entry on Hochschild geometry, since it really deserves a stand-alone discussion.

      Eventually somebody should add the references by Kapranov et al on dg-schemes etc. And much more.

    • stub for infinitesimal extension (and cross-linked a bit)

      Zoran: sorry, I know I should cite that article of yours/your colleagues, could you please add it? Thanks.

    • as last week, I have created an entry that collects some of the recent edits scattered over the nLab supposedly in one coherent story, it’s

      basics of étale cohomology

      Should be expanded a bit more. But not tonight.

    • At the entry classifying morphism, there is a query (not displayed as one):

      Where can I find a construction of the classifying morphism to a classifying space for a G-bundle with connection - using the connections as a 1-form?

      Given the subject matter and the name ‘jim’ plus the location of 96.245.205.76 (Pennsylvania …)????

    • at Artin-Schreier sequence I spelled out the existence proof. This derserves/demands to be further polished and streamlined, but i’ll leave it at that for the moment.

    • in my search for a good way of introducing basics of étale cohomology I switched from Milne to Tamme, and started some hyperlinked index for the latter’s Introduction to Étale Cohomology. As before, in the course of this I created some brief entries for keywords there, if they didn’t exist yet.

      Hope to expand this now…

    • Fixed a couple incorrect statements at hypercomplete (infinity,1)-topos:

      • Remark 1 claimed that having enough points in the 1-topos sense implies having enough points in the ∞-sense. I replaced it with a counterexample from HTT.
      • In Proposition 1 I replaced “finite homotopy dimension” by “locally of homotopy dimension n”.
    • I have started a hyperlinked index for Milne’s Lectures on Étale Cohomology

      (Up to and including section 27, where the proof of the Weil conjectures starts. )

      In the course I have created a bunch of brief entries, if the corresponding keyword didn’t yet have one. Also cross-linked vigorously.

    • added to the list of equivalent conditions in the definition at étale morphism of schemes the pair “smooth+unramified”. Added a remark after the definition on how to read these pairs of conditions.

    • we had an entry spectrum (geometry) which wasn’t linked to from almost anywhere, in particular not from spectrum - disambiguation.

      I have now added a bunch of cross-links, between these two entries and between the entries that they link to. It’s better now than it was before, but could still do with further improvement.

    • started entries

      Question: We have the implications

      étale morphism weakly étale morphism formally étale morphism

      but can one say more specifically what kind of generalized finite presentability condition makes a formally étale morphism a weakly étale morphism?

    • I have fixed some dead links relating to Loday which were occurring n several pages.

    • made a note of a simple observation:

      for X an atom in a cohesive -topos H over Grpd, then also the slice H/X is cohesive … except possibly for the property that shape preserves binary products (but it does preserve the terminal object):

    • I noted an entry on generalized Eilenberg-MacLane spaces, but note that there is another use of this term in the literature, namely the representing fibrations for cohomology with local coefficients. These are the fibrations used by Gitler and then by Alan Robinson, Hans Baues and others more recently. What would be the preferred name for these latter things. (I personally find the idea of giving a name to products of Eilenberg- Mac Lane spaces other that ‘products of Eilenberg - Mac Lane spaces’ a bit strange, but I know that there is some strange terminology around!)

    • I noticed by accident that we have an entry coinvariant. Then I noticed that we also have an entry homotopy coinvariant functor.

      I have now added cross-links between these entries and with invariant and orbit, so that they no longer remain hidden.

      I also edited the first case of group representation coinvariants at coinvariant a little.

    • I noticed that some old entries were requesting a keyword link for brave new algebra, so I created it and filled in a default-paragraph. Please feel invited to expand.

      In that context I have a question: the dual generlized Steenrod algebras have been called “brave new Hopf algebroids” in articles including

      • Andrew Baker, Brave new Hopf algebroids (pdf)

      • Andrew Baker and Alain Jeanneret, Brave new Hopf algebroids and extensions of MU-algebras, Homology Homotopy Appl. Volume 4, Number 1 (2002), 163-173. (Euclid)

      • Mark Hovey, Homotopy theory of comodules over a Hopf algebroid (arXiv:math/0301229)

      But the Hopf algebroids considered in these articles are ordinary Hopf algebroids, they are given not by Hopf -algebras (E,EE) but by their homotopy groups (E,E(E)), unless I am missing something.

      So at least without further discussion, calling (E,E(E)) “brave new” is a bit of a stretch. The brave new thing would be (E,EE) (if indeed it is a “Hopf -algebroid”).

      Can anyone say more about this? I can’t seem to find any source talking about this. The canonical guess of googling for “derived Hopf algebroid” doesn’t show relevant results.

    • some basics at Lubin-Tate theory

      (wanted to do more, but the nLab is giving me a really hard (down-)time )

    • created a table-for-inclusion image of J – table listing pertinent information in low degree, and included it in some relevant entries

    • created Einstein’s equation, only to record a writeup by Gonzalo Reyes which I just came across by chance, who gives a discussion in terms of synthetic differential geometry.

    • finally created the category:reference-entry for Lurie’s chromatic lecture. See Chromatic Homotopy Theory

      (And as a special service to the community… with lecture titles. ;-)


      • Lecture 1 Introduction (pdf)

      • Lecture 2 Lazard’s theorem (pdf)

      • Lecture 3 Lazard’s theorem (continued) (pdf)

      • Lecture 4 Complex-oriented cohomology theories (pdf)

      • Lecture 5 Complex bordism (pdf)

      • Lecture 6 MU and complex orientations (pdf)

      • Lecture 7 The homology of MU (pdf)

      • Lecture 8 The Adams spectral sequence (pdf)

      • Lecture 9 The Adams spectral sequence for MU (pdf)

      • Lecture 10 The proof of Quillen’s theorem (pdf)

      • Lecture 11 Formal groups (pdf)

      • Lecture 12 Heights and formal groups (pdf)

      • Lecture 13 The stratification of FG (pdf)

      • Lecture 14 Classification of formal groups (pdf)

      • Lecture 15 Flat modules over FG (pdf)

      • Lecture 16 The Landweber exact functor theorem (pdf)

      • Lecture 17 Phanton maps (pdf)

      • Lecture 18 Even periodic cohomology theories (pdf)

      • Lecture 19 Morava stabilizer groups (pdf)

      • Lecture 20 Bousfield localization (pdf)

      • Lecture 21 Lubin-Tate theory (pdf)

      • Lecture 22 Morava E-theory and Morava K-theory (pdf)

      • Lecture 23 The Bousfield Classes of E(n) and K(n) (pdf)

      • Lecture 24 Uniqueness of Morava K-theory (pdf)

      • Lecture 25 The Nilpotence lemma (pdf)

      • Lecture 26 Thick subcategories (pdf)

      • Lecture 27 The periodicity theorem (pdf)

      • Lecture 28 Telescopic localization (pdf)

      • Lecture 29 Telescopic vs En-localization (pdf)

      • Lecture 30 Localizations and the Adams-Novikov spectral sequence (pdf)

      • Lecture 31 The smash product theorem (pdf)

      • Lecture 32 The chromatic convergence theorem (pdf)

      • Lecture 33 Complex bordism and E(n)-localization (pdf)

      • Lecture 34 Monochromatic layers (pdf)

      • Lecture 35 The image of J (pdf)

    • I recently created entry Bol loop. Now I made some corrections and treated the notion of a core of a right Bol loop (the term coming allegedly from Russian term сердцевина).

    • started stubs E-∞ geometry, E-∞ scheme.

      To be filled with more content, for the moment I just need to be able to use the links.