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    • Now I am working on the next chapter of “geometry of physics”: geometry of physics – supersymmetry.

      A fair bit of material is in place now, but much is missing still. This here is mainly in case you are watching the logs and are wondering. At this point, if anyone has any edits to suggest (typo fixing or more substantial) maybe best to not touch the file yet but to tell me about it. Thanks!

    • I have renamed the entry on the ∞\infty-topos on CartSp topCartSp_{top} into Euclidean-topological infinity-groupoid.

      Then in the section Geometric homotopy I have written out statement and proof that

      1. the intrinsic fundamental ∞\infty-groupoid functor in ETop∞GrpdETop \infty Grpd sends paracompact topological spaces to their traditional fundamental ∞\infty-groupoid

        Π ETop∞Grpd(X)≃Π Top(X)≃SingX \Pi_{ETop \infty Grpd}(X) \simeq \Pi_{Top}(X) \simeq Sing X;

      2. more generally, for X •X_\bullet a simplicial topological space we have

        |Π ETop∞Grpd(X •)|≃|X •| |\Pi_{ETop \infty Grpd}(X_\bullet)| \simeq |X_\bullet| ,

        where on the left we hve geometric realization of simplicial sets, and on the right of (good) simplicial topological spaces.

    • am giving this, finally, its own little page

      v1, current

    • have added the original articles on geometric quantization and on diffeological spaces to the list of “Selected writings”

      diff, v5, current

    • just to make edit signatures be hyperlinked

      v1, current

    • Just a definition (hope I got it right) and a couple properties. I wasn’t sure how to set up the redirects; currently “modest set” redirects here while “PER” redirects to partial equivalence relation, but other suggestions are welcome.

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • the old entry representation contained an old query box with some discussion.

      I am hereby moving this old discussion from there to here:

      +– {: .query} I don't agree with this D≔Aut(V)D \coloneqq Aut(V) business. A kk-linear representation of a group GG is a functor from BG\mathbf{B}G to kVectk Vect, period. Because BG\mathbf{B}G has one object (or is pointed), we can pick out an object VV of kVectk Vect, and it was remiss of me not to mention this (and the language ‘on VV’ vs ‘in DD’. But we usually don't want DD to actually be Aut(V)Aut(V) instead of kVectk Vect; when doing representation theory, we fix GG and fix kk (or fix DD in some other way), but we don't fix VV. —Toby

      If you look at the textbooks of representation of groups, then they start with representation of groups as homomorphisms of groups, that is just functors. Then they say, that usually the target groups are groups of automorphisms of some other objects. And at the end they say that one usually restricts just to linear automorphisms of linear objects when linearizing the general problematics to the linear one. Now the fact that in some special case there is a category which expresses the same fact does not extend to other symmetry objects, like for representations of vertex operator algebras, pseudotensor categories etc. I mean End(something) or Aut(something) is just inner end in some setup like in closed monoidal category, but there are symmetries in mathematics which have a notion of End of Aut for a single object but do not have good notion of category one level up which has inner homs leading to the same End or Aut. Conceptually actions are about endosymmetries or symmetries (automorphisms) being reducable to categorical ones but not necessarily, I think. In a way you say that you are sure that any symmetry of another object can be expressed internally in some sort of a higher category of such objects, what is to large extent true, but I am sure not for absolutely all examples.

      • I can’t recall ever seeing group homomorphisms ρ:G→H\rho\colon G \to H described in general as ’representations’, but I have limited experience; I should look at some more textbooks. The one that I learnt the subject from, Serre's Linear Representations of Finite Groups, looked only at representations on vector spaces from the beginning, but its title suggests a bias that might explain that. (^_^)

      (for “on” terminology:) Ross Street uses monads in a 2-category and monads on a 1-category and I know of no objects in category theory.

      • Yes, this is analogous to representation in a category vs on an object in such a category. (But what do you mean by ’I know of no objects in category theory’?)

      Another important thing is that the endomorphisms are by definitions often equipped with some additional (e.g. topological) structure which is not necessarily coming from some enrichement of the category of objects. –Zoran

      • Good point.

      (Zoran on word “classical representation” being just for groups: so the representations of associative algebras, Lie algebras, Leibniz algebras, topological groups, quivers, are not classical ??).

      • I thought that they came later, but maybe not. I added ’of groups’ to fix/clarify. —Toby =–

    • I am starting to bring (infinity,n)-category into shape. So far I have

      • rewritten the Idea-section

      • added a bare minimum of the axiomatic characterization

      • added references

      • also polished n-category a bit.

      My plan ist to add now technical details to the entry. Let’s see how far I get.

    • I noticed that augmented simplicial set did not point anywhere, so i created the entry. But have no energy to put anything of substance there right now.

    • Added a description of the Sweedler hom and Sweedler product.

      diff, v29, current

    • added list of “Selected writings” and cross-links from all entries citing these

      diff, v2, current

    • I saw that the entry strict omega-groupoid was not in good shape, so I have edited a little. Expanded a bit, restructured a bit. More could be done.

    • I am working on the entry topological manifold.

      I gave it a subsection locally Euclidean spaces, which maybe eventually wants to be split off as an entry in its own right.

      Now I have added statement and proof that locally Euclidean spaces are T 1T_1, sober and locally compact (in the compact neighbourhood base sense): here.

    • Created ZFA, about ZF with atoms. I’ve added it to the foundations side bar under material set theories and a stack of links.

    • added this historical pointer (dug out by David C. in another thread) where the term “representation group” is used to refer to a GG-set:

      • William Burnside, On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups, Proceedings of the American Mathematical Society 1901 (doi:10.1112/plms/s1-34.1.159)

      diff, v16, current

    • Page created, but author did not leave any comments.

      v1, current

    • Added a concrete example: equifiers of natural transformations.

      diff, v4, current

    • Page created, but author did not leave any comments.

      v1, current

    • Characterized full faithfulness in terms of a pullback

      diff, v15, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • The entry

      http://ncatlab.org/nlab/show/topological+modular+form

      has an outdated link to the reading list:
      http://www.math.uiuc.edu/~ganter/talbot/index.html

      It should be replaced by:
      http://www.ms.unimelb.edu.au/~nganter/talbot/index.html

      I have tried several times to edit the page accordingly, but upon
      submission (as "Marc Olschok") alway got the message that
      "Anonymous Coward" is already editing the page.
      So now I give up and let the nlab-elves do the edit.

    • added an Idea-section and a further reference to geometric stack.

      I hope we can eventually fully harmonize the definitions. It seems to me that the definitions in the literature vary slightly on the strength of their conditions.

    • am starting something. Not done yet, but need to save

      v1, current

    • created the book-entry Supergravity and Superstrings - A Geometric Perspective as the usual recipient, eventually, for a link list of entries on supergravity.

      This book is a jewel, but out of print. I happen to have an electronic djvu copy of it. I am feeling tempted to upload that to the lab, for the sake of mankind in general and of future students in particular. Probably legally a bad idea. Or is it?

    • I have started working on Diff: added a subsection with discussion of its properties as a site.

      Added statement and proof of the fact that

      • Sh(Diff)Sh(Diff) is a cohesive topos (direct consequence of the comparison lemma)

      • Sh (∞,1)(Diff)Sh_{(\infty,1)}(Diff) is a cohesive (infinity,1)-topos (easy with some lemmas from the literature, but not immediate (I think))

    • brief category:people-entry for hyperlinking references

      v1, current

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