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    • when creating a stub entry local Langlands conjectures I noticed that it has already become hard to know which entries on the Langlands program we already have. I always take this as a sign that a summary table of contents is called for. So I started

      Langlands correspondence – contents

      and added it as a “floating table of contents” to the relevant entries.

      (Even though all of these entries are still more or less stubs.)

    • I noticed that there was some wild formatting at building. I have tried to tame it a bit.

    • I have started a table of contents measure theory - contents and started adding it as a floating toc to the relevant entries

    • currently the bulk of the entry analytic geometry is occupied by a long section on “Holomorphic functions of several complex variables”. Should that not better be moved to some dedicated entry of its own? Any opinions?

    • I created branched manifold -linked from orbifold- with a definition from ”expanding attractors” by Robert F. Williams (1974) quoted in wikipedia. This description is -as it stands- not precisely compatible to that given in Dusa McDuffs ”Groupoids, Branched Manifolds and Multisections” which I am rather interested in. So I plan to comment on this as a side note in the -yet to be written-article orbifold groupoid.

    • New article: cofinality, with the basic case being the cofinality of a quasiorder as a collection of cardinal numbers, a variation as a collection of ordinal numbers (or equivalently an ordinal number), and an apparently separate case of the cofinality of a collection of cardinal numbers, all of these tied together and interpreted as a single cardinal number if one assumes the axiom of choice.

    • added to diffeomorphism group statements and references for the case of 3-manifolds (Smale conjecture etc.)

    • I was surprised to discover that we had no page finite (infinity,1)-limit yet, especially given that they are slightly subtle in relation to the 1-categorical version. So I made one.

    • From supercompact cardinal:


      Theorem: The existence of arbitrarily large supercompact cardinals implies the statement:

      Every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class.

      In other words, if LL is a reflective localization functor on a balanced accessible category such that the unit morphism XLXX \to L X is an epimorphism for all XX and the class of LL-local objects is defined by an absolute formula, then the existence of a suficciently large supercompact cardinal implies that LL is a localization with respect to some set of morphisms.

      This is in BagariaCasacubertaMathias

      Urs Schreiber: I am being told in prvivate communication that the assumption of epis can actually be dropped. A refined result is due out soon.


      does anyone know about this refined result?

    • this is a message to Zoran:

      I have tried to refine the section-outline at localizing subcategory a bit. Can you live with the result? Let me know.

    • discovered the following remnant discussion at full functor, which hereby I move from there to here


      Mathieu says: I agree that, for functors, there is no reason to say “fully faithful” rather than “full and faithful”. But for arrows in a 2-category (like in the new version of the entry on subcategories), there are reasons. I quote myself (from my thesis): «Remark: we say fully faithful and not full and faithful, because the condition that, for all X:CX:\C, C(X,f)C(X,f) be full is not equivalent in Grpd\Grpd to ff being full. Moreover, in Grpd\Grpd, this condition implies faithfulness. We will define (Definition 197) a notion of full arrow in a Grpd\Grpd-category which, in Grpd\Grpd and Symm2Grp\Symm2\Grp (symmetric 2-groups), gives back the ordinary full functors.» Note that this works only for some good groupoid enriched categories, not for Cat\Cat, for example.

      Mike says: Do you have a reason to care about full functors which are not also faithful? I’ve never seen a very compelling one. (Maybe I should just read your thesis…) I agree that “full morphism” (in the representable sense) is not really a useful/correct concept in a general 2-category, and that therefore “full and faithful” is not entirely appropriate, so I usually use “ff” in that context. I’ve changed the entry above a bit to reflect your comment; is it satisfactory now? Maybe all this should actually go at full and faithful functor (and/or fully faithful functor)?

    • created an entry mapping cocone, following a suggestion by Zoran, that this is the right technical term for what is discussed in more detail at generalized universal bundle.

      (the examples section needs more attention, though...)

    • I have created final lift, and added to adjoint triple a proof that in a fully faithful adjoint triple between cocomplete categories, the middle functor admits final lifts of small structured sinks (and dually). This means that it is kind of like a topological concrete category, except that the forgetful functor need not be faithful.

      I find this interesting because it means that in the situation of axiomatic cohesion, where the forgetful functor from “spaces” is not necessarily faithful, we can still construct such “spaces” in “initial” and “final” ways, as long as we restrict to small sources and sinks.

    • If you're not following the categories mailing list, then you're missing out on a great discussion of evil. Peter Selinger has come from the list to the Lab to discuss it here too!

    • Thought I’d write up some old notes at symmetric product of circles (linked from unitary group, explanation to come on symmetric product of circles). Not finished yet, but have to leave it for now.

      (I was incensed to discover that to look at the source article for the material for this to check that I’m remembering it right - I last looked at it about 10 years ago - I have to pay 30 UKP. The article is 3 pages long. That’s 10UKP per page! So I’m going from vague memories and “working it out afresh”.)

    • Added Thom-Federer and Gottlieb thorems to Eilenberg-MacLane space; added the remark “ΩC(X,Y)C(X,ΩY)\Omega\mathbf{C}(X,Y)\simeq \mathbf{C}(X,\Omega Y) in any (oo,1)-category with homotopy pullbacks” in loop space object.

    • Partially spurred on by an MO question, I have started an entry on simple homotopy theory. I am also intrigued as to whether there is a constructive simple homotopy theory that may apply in homotopy type theory, but know so little (as yet) about that subject that this may be far fetched.

    • Steve (Lack) has put a comment box on AT category. I have not been following that entry so am not able to reply to his point.

    • I filled in content at n-truncated object of an (infinity,1)-category.

      to go with my discussion with David Roberts. I had planned to go further and also write the entry on Postnikov twoers, but got distracted all day.

      Apart from that I just added this link to Higher Topos Theory and did some editing there, added a table of contents, expanded the floating toc.

    • I started editing the page on reflexive Banach spaces - in particular I corrected the definition and stuck in a mention of "James space". A link or reference is needed but I am currently a bit too frazzled/stressed to do further editing today.

    • I have expanded at DHR category the Idea-section and added more hyperlinks.

    • Joel Hamkins and Andy Putman made some comments about the nLab on MathOverflow, beneath an answer by Andrew.

      It’s interesting to know what people’s perceptions are, even if they’re wrong. (And I would think that Andy P’s perception is wrong.) I don’t know what Andrew S has in mind when he says that Joel’s point is extremely easy to answer.

    • Following a discussion on the algebraic topology list, I’ve written a proof of the contractibility of the space of embeddings of a smooth manifold in a reasonably arbitrary locally convex topological vector space. The details are on embedding of smooth manifolds and it also led to me creating shift space (I checked on MO to see if there was an existing name for this, and Bill Johnson said he hadn’t heard of it).

    • added the recent Barwick/Schommer-Pries preprint to (infinity,n)Cat, together with a few more brief remarks.

    • created Lie bialgebra, but so far just a comment on their quantization.

    • while polishing up type theory - contents I felt the need for entries called syntax and semantics. I have created these just so that the links to them are not grey, but I put in only something minimalistic . I could add some general blah-blah, but I’d rather hope some actual expert feels inspired to start with some decent paragraphs.

    • Added to pasting diagram a reference to the bicategorical pasting theorem given by Verity in his thesis.

    • In the Definition-section at reflective factorization system I found the “Ψ\Psi” and “Φ\Phi” used in the text oppositely to how they appear in the displayed diagram. I think I have fixed this.

    • After contributing to the article on parallelogram identity, I added to isometry and created Mazur-Ulam theorem. The easy proof added at isometry, that shows an isometry EFE \to F between normed vector spaces is affine if FF is strictly convex, might lead one to suspect that the proof under parallelogram identity was overkill, but I think that’s an illusion. Ultimately, I believe the parallelogram identity is secretly an expression of the perfect ’roundness’ of spheres, connected with the fact observed by Tom Leinster recently at the Café that the group of isometries for the l 2l_2 norm is a continuum, whereas for other pp in the range 1<p<1 \lt p \lt \infty, you get just a finite reflection group (this is for the finite-dimensional case, but there’s an analogue in the infinite-dimensional case as well).

      The Mazur-Ulam theorem removes the strict convexity hypothesis, but adds the hypothesis that the isometry is surjective. The conclusion is generally false if this hypothesis is omitted.

    • I have split off universal quantifier and existential quantifier from quantifier in order to expose the idea in a more pronounced way in dedicated entries.

      Mainly I wanted to further amplify the idea of how these notions are modeled by adjunctions, and how, when formulated suitably, the whole concept immediately and seamlessly generalizes to (infinity,1)-logic.

      But I am not a logic expert. Please check if I got all the terminology right, etc. Also, there is clearly much more room for expanding the discussion.