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    • This alone would not be an improvement over point particle versions of quantum gravity except that it is possible to define a perturbative expansion of string theory around a fixed, classical background space-time that is finite and anomaly free at 1-loop

    • Added section on condensed ∞\infty-groupoids as an example of a local (∞,1)(\infty, 1)-topos

      Maus

      diff, v11, current

    • hopefully the given definition makes sense and is equivalent to the definition found in Scholze’s “Lectures on analytic geometry”, somebody more knowledgeable at (∞,1)(\infty,1)-category theory could double check.

      Anonymous

      v1, current

    • 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.

    • Started the article dependent choice, and did some editing at COSHEP to make clearer to myself the argument that COSHEP + (1 is projective) implies dependent choice. It’s not clear to me that the projectivity of 1 is removable in that argument; maybe it is.

    • Only a stub at the moment, but I thought we needed to start a page on this. Looks like it’s going to become important.

      v1, current

    • moving the following old discussion from out of the entry to here, just for the record (it concerns a bygone version of the entry):

      +– {: .query} Tim: As I read the entry on nice topological spaces, it really refers to ’nice categories’ rather than ’nice spaces’! I have always thought of spaces such as CW-complexes and polyhedra as being ’locally nice’, but the corresponding categories are certainly not ’nice’ in the sense of nice topological space. Perhaps we need to adjust that other entry in some way.

      Toby: You're right, I think I've been linking that page wrongly. (I just now did it again on homotopy type!) Perhaps we should write locally nice space or locally nice topological space (you pick), and I'll fix all of the links tomorrow.

      Tim:I suggest locally nice space. (For some time I worked in Shape Theory where local singularities were allowed so the spaces were not locally nice!) There would need to be an entry on locally nice. I suggets various meanings are discussed briefly, e.g. locally contractible, locally Euclidean, … and so on, but each with a minimum on it as the real stuff is in CW-complex etc and these are the ’ideas’.

      Mike: Why not change the page nice topological space to be about CW-complexes and so on, and move the existing material there to something like convenient category of spaces, which is also a historically valid term? I am probably to blame for the current misleading content of nice topological space and I’d be happy to have this changed.

      Toby: I thought that nice topological space was supposed to be about special kinds of spaces, such as locally compact Hausdorff spaces, whose full subcategories of Sp\Sp are also nice. (Sort of a counterpoint to the dichotomy between nice objects and nice categories, whose theme is better fit by the example of locally Euclidean spaces). CW-complexes also apply —if you're interested in the homotopy categories.

      Mike: Well, that’s not what I thought. (-: I don’t really know any type of space that is nice and whose corresponding subcategory of Top is also nice. The category of locally compact Hausdorff spaces, for instance, is not really all that nice. In fact, I can’t think of anything particularly good about it. I don’t even see any reason for it to be complete or cocomplete!

      I think it would be better, and less confusing, to have separate pages for “nice spaces” and “nice categories of spaces,” or whatever we call them. And, as I said, I don’t see any need to invent a new term like “locally nice.”

      When algebraic topologists (and, by extension, people talking about ∞\infty-groupoids) say “nice space” they usually mean either (1) an object of some convenient category of spaces, or (2) a CW-complex-like space, between which weak homotopy equivalences are homotopy equivalences. Actually, there is a precise term for the latter sort: an m-cofibrant space, aka a space of the (non-weak) homotopy type of a CW complex.

      Toby: I thought the full subcategory of locally compact Hausdorff spaces was cartesian closed? Maybe not, and it's not mentioned above.

      But you can see that most of the examples above list nice properties of their full subcategories. And the page begins by talking about what a lousy category Top\Top is. So it seems clearly wrong that you can't make Top\Top a nicer category by taking a full subcategory of nice spaces. (Not all of the examples are subcategories, of course.)

      Mike: It’s true that locally compact Hausdorff spaces are exponentiable in TopTop. However, I don’t think there’s any reason why the exponential should again be locally compact Hausdorff.

      I guess you are right that one could argue that compactly generated spaces themselves are “nice,” although I think the main reason they are important is that the category of compactly generated spaces is nice. I propose the following:

      1. Move the current content of this page to convenient category of spaces.
      2. Create m-cofibrant space (I’ll do that in a minute).
      3. Update most links to point to one or the other of the above, since I think that in most places one or the other of them is what is meant.
      4. At nice topological space, list many niceness properties of topological spaces. Some of them, like compact generation, will also produce a convenient category of spaces; others, like CW complexes, will be in particular m-cofibrant; and yet others, like locally contractible spacees, will do neither.

      Toby: I believe that the compact Hausdorff reflection (the Stone–Čech compactification) of Y XY^X is an exponential object.

      Anyway, your plan sounds fine, although nice category of spaces might be another title. (I guess that it's up to whoever gets around to writing it first.) Although I'm not sure that people really mean m-cofibrant spaces when they speak of nice topological spaces when doing homotopy theory; how do we know that they aren't referring to CW-complexes? (which is what I always assumed that I meant).

      Mike: I guess nice category of spaces would fit better with the existing cumbersomely-named dichotomy between nice objects and nice categories. I should have said that when people say “nice topological space” as a means of not having to worry about weak homotopy equivalences, they might as well mean (or maybe even “should” mean) m-cofibrant space. If people do mean CW-complex for some more precise reason (such as wanting to induct up the cells), then they can say “CW complex” instead.

      Re: exponentials, the Stone-Čech compactification of Y XY^X will (as long as Y XY^X isn’t already compact) have more points than Y XY^X; but by the isomorphism Hom(1,Y X)≅Hom(X,Y)Hom(1,Y^X)\cong Hom(X,Y), points of an exponential space have to be in bijection with continuous maps X→YX\to Y.

      Toby: OK, I'll have to check how exactly they use the category of locally compact Hausdorff spaces. (One way is to get compactly generated spaces, of course, but I thought that there was more to it than that.) But anyway, I'm happy with your plan and will help you carry it out.

      =–

      diff, v23, current

    • I had started an entry “exponentiation” but then thought better of it and instead expanded the existing exponential object: added an examples-section specifically for SetSet and made some remarks on exponentiation of numbers.

    • Yde is a full professor, not an associate one.

      Noam Cohen

      diff, v4, current

    • For some time now I’ve been bothered by an implicit redundancy spanned by the articles nice category of spaces and convenient category of topological spaces. I would like the latter to have a more precise meaning and the former to be something more vague and flexible. I have therefore been doing some rewriting at the former. But if anyone disagrees with the edits, please let’s discuss this here.

      I have removed a query box:

      +– {: .query} I’m not sure that we really want to use the terminology that way, but Ronnie already created that page, so I’m linking these together. —Toby =–

    • starting page on light profinite sets

      Maus

      v1, current

    • starting page on light condensed sets

      Maus

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

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

      v1, current

    • added pointer to

      • B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, corollary 15.3.4 of Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985

      diff, v6, current

    • category: people page for Freek Geerligs

      Maus

      v1, current

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

      Anonymous

      v1, current

    • starting page on synthetic Stone duality

      Maus

      v1, current

    • Since Felix has changed surname, I’ve changed the name of this page.

      diff, v7, current

    • I have updated this item:

      with the arXiv number. It looks like a bunch of recent anonymously-authored entries are essentially taken from this thesis (such as protoring and its many variants).

      diff, v13, current

    • Added the characterization of CompComp as the unique non-trivial pretopos which is well-pointed, filtral and admits all set-indexed copowers of its terminal object from

      • Vincenzo Marra, Luca Reggio, A characterisation of the category of compact Hausdorff spaces, (arXiv:1808.09738)

      diff, v25, current

    • After a suggestion from Toby, I added a note on the “analytic Markov’s principle” to Markov’s principle.

    • added to homotopy groups of spheres the table

      k=k = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ⋯\cdots
      π k(𝕊)=\pi_k(\mathbb{S}) = ℤ\mathbb{Z} ℤ 2\mathbb{Z}_2 ℤ 2\mathbb{Z}_2 ℤ 24\mathbb{Z}_{24} 00 00 ℤ 2\mathbb{Z}_2 ℤ 240\mathbb{Z}_{240} (ℤ 2) 2(\mathbb{Z}_2)^2 (ℤ 2) 3(\mathbb{Z}_2)^3 ℤ 6\mathbb{Z}_6 ℤ 504\mathbb{Z}_{504} 00 ℤ 3\mathbb{Z}_3 (ℤ 2) 2(\mathbb{Z}_2)^2 ℤ 480⊕ℤ 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2

    • Started page on generalized symmetries, with brief description of main Idea.

      v1, current

    • Added content, including the idea, GW/PT/GV correspondence and references. (The german Wikipedia article is now also available.)

      (I also plan to create an article for the Pandharipande-Thomas invariant in the future.)

      diff, v4, current

    • created black holes in string theory, since somebody asked me: a brief paragraph explaining how the entropy-counting works and some references.

    • Added more material to Boolean algebra, particularly the principle of duality and the connection to Boolean rings, and a wee bit of material on Stone duality.

      Stone duality deserves greater expansion, bringing out the dualities via ambimorphic (ahem, schizophrenic) structures on the 2-element set, and mentioning the connection to Chu spaces. Another day, another dollar.

    • Explained why the square of the finite power set functor is (or “can be made into”) the monad for Boolean algebras.

      diff, v14, current

    • Began stub for Tambara functor. Neil Strickland’s, Tambara Functors, arXiv:1205.2516 seems to be a good reference.

      Seems like it’s very much to do with pullpush through polynomial functors, if you look around p. 23.

      I would try to say what the idea is, but have to dash.

    • the page action is also a mess. I have added a pointer to the somewhat more comprehensive module and am hereby moving the following discussion box from there to here:

      [ begin forwarded discussion ]

      +–{.query} I am wondering if we will need the notion of action which works in categories with product, i.e. G×X→XG\times X\to X and so on. There is also an action of one Lie algebra on another (for instance in some definitions of crossed module of Lie algebra, where AutAut is replaced by the Lie algebra of derivations. (a similar situation would seem to exist in various other categories where action is needed in a slightly wider context. I think most would be covered by an enriched setting but I am not sure.) Thoughts please.Tim

      Yes, I think certainly all those types of action should eventually be described somewhere, possibly on this page. -Mike

      Tim: I have added some of this above. There should be mention of actions of a monoid in a monoidal category on other objects, perhaps.

      Mac Lane, VII.4, only requires a monoidal category to define actions. – Uday =–

      [ end forwarded discussion ]

    • I added a bit to category of simplices, including the fact that the category of nondegenerate simplices is final and thus colimits can be computed using only that, and that the nerve of the category of simplices itself is colimit-preserving.

    • Changes made only to the Universal property of the 2-category of spans section. The citations by Urs lead to another citation which, in turn, leads to another citation. With a little effort, I tracked down the a full copy of said universal property, I’ve replicated it here, added the citation used, although I left the previous citations there for convenience; a more experienced editor can remove those if they would like.

      I would like to note that the author whose work I have referenced, Hermida, also notes: “[this universal property] is folklore although we know no references for it.”

      Please make any corrections needed and clean up the language here; this is a fairly direct copy of what is written, but I imagine somebody with more knowledge of all the language used here can rewrite this universal property stuff in a cleaner way.

      Thanks!

      Anonymous

      diff, v57, current

    • Create a stub. I will expand the list shortly.

      v1, current

    • starting a dedicated entry for the category of vector bundles with homomorphisms allowed to cover non-trivial base maps (while previously we only had VectBund(B) for fixed base BB).

      For the moment the main point is to record the interesting cartesian- and tensor-monoidal structure (now here)

      v1, current

    • I will brush-up the entry homotopy hypothesis. But not right now, right now I have to run and do something else. But here is some leftover discussion that was sitting there, and which I have now removed from the entry and reproduce here, in order that we go and use it to make the entry better, but not clutter it up.

      (continued in next comment)

    • a category:reference-entry (“FGA” used to redirect to EGA)

      v1, current

    • starting a stub. Nothing here yet, but need to save.

      v1, current

    • At crossed module it seems we are missing what i think should be the prototypical example: the relative second homotopy group π 2(X,A)\pi_2(X,A) together with the bundary map δ:π 2(X,A)→π 1(A)\delta:\pi_2(X,A)\to \pi_1(A) and the π 1(A)\pi_1(A)-action on π 2(X,A)\pi_2(X,A). As someone confirms this example is correct I’ll add it to crossed module.

    • Changed typo’d “pyschoaccoustics” to correct “psychoacoustics”.

      Anonymous

      diff, v16, current

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