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    • Suppose somebody formally minded is looking for a good problem in the topic of contextual categories (C-systems). How about this:

      fix a given C-system, with your preferred set of extra type constructor data on it, and then ask the question: for any given small site, is the category of sheaves on that site with values in that C-system again canonically a C-system with the same collection of extra type constructors?

      I gather aspects of this play a role in most discussions of type theory model building, but is there any systematic discussion?

      I suppose the difficulty and interest in this question considerably varies with what the set of “extra type constructors” includes. A while back I had asked a similar question where “extra type constructors” was “modalities”. Maybe that was overambitious for the person who I am asking for, so I am trying to see if something along these lines but more tractable would be a good thing to aim at.

    • The entry closed subspace was a bit weird. I have touched it to try to improve a little. But if anyone has ten minutes to spare, it might still be good to bring this into more decent shape.

    • To make sense of this in my mind as a general concept, I have written semidefinite element. This gives a general context in which to define ‘positive definite’, ‘negative semidefinite’, ‘indefinite’ etc.

      (This seems the safest page title, as the least likely to have any conflicts. I've also put in a lot of redirects, but possibly some of these will have to go elsewhere; definite seems the most dangerous.)

    • Somehow I was under the impression that I had written out on the nnLab at several places how the traditional physics way to talk about instantons connects to the correct maths discussion. But now that I wanted to point a physicist to this, I realize that in each entry that touches on this, I just gave a quick remark pointing to Cech cohomology, clutching construction, one-point compactification and Chern-Simons 2-gerbes, but not actually giving an exposition.

      So I went ahead and wrote such an exposition finally:

      SU2-instantons from the correct maths to the traditional physics story

      Beware two things:

      1) this entry is meant to be included as a subsection into other entries (such as Yang-Mills instanton, BPTS instanton) therefore it is intentionally lacking toc, headlines and other introductory stuff

      2) I just wrote this in one go (trying to get back to somebody waiting for me), and now I am out of steam. This hasn’t been proof-read even once yet. So unless you feel energetic about joining in the editing, better wait until a little later when this has stabilized.

      Ideally this kind of account would eventually be beautified with some pictures and the like.

    • the entry category of sheaves had been a bit shy about giving away full information where it came to the recognition of epi/mono/isos. I have now expanded the proposition here.

      Maybe somebody feels inspired to add pointer to proof, or, better yet, add proof.

    • contrapositive now exists. It doesn't say much (unless you were unsure which versions of this rule are intuitionistically valid, in which case, you can take a look). But there was a link to it, so I made it.

    • Hi all,

      I've recently written my first all-on-my-own article, graded fusion category. I'll gladly welcome anyone looking over it and improving it (adding links, contents, improving formatting)
      I went on to work on a larger article, $G$-crossed braided fusion category, which is not quite finished yet. Here, help is appreciated as well. I'll go on adding more content over the next weeks, whenever I find time.

      I'm underwhelmed with the quality of the commutative diagram I could produce. I followed the example in monoidal category, which I thought must be state-of-the-art since it's a page about a really important concept. But I'm appalled with my result for the G-crossed braiding axiom in $G$-crossed braided fusion category. How can I improve it?
      I'll have a look at https://ncatlab.org/nlab/show/HowTo again later and see whether there is anything else that I can do.

      Manuel

    • started Fierz identity to collect some references. Am still searching for the good reference for the general case…

    • Thanks to Manuel Bärenz for writing graded fusion category. It’s worth announcing such things here.

      One strange thing with itex is that you need to separate letters for them to italicize, e.g., 𝒞 gh\mathcal{C}_{gh} needs to be 𝒞 gh\mathcal{C}_{g h}.

    • I have added at combinatorial spectrum the missing bibliographical information for Kan’s original article.

      While doing so I noticed old forgotten discussion sitting there, which hereby I move from there to here:

      — begin forwarded discussion —

      A previous version of this entry triggered the following discussion:

      +–{: .query} Mike: Are you sure about that last condition? I remember a condition more like “for each x∈E nx\in E_n there is some finite m<nm \lt n such that all faces of xx in E mE_m are the basepoint.

      Urs: on the bottom of page 437 in the reference by Brown it says: “each simplex of EE has only finitely many faces different from **”.

      I see that my original phrasing reflected this only very imprecisely. I have tried to improve that now. But it also seems that this condition m<nm \lt n which you mention is not implied by Brown(?) In particular, it seems this condition does not harmoize with the fact that nn may be negative.

      But this looks like the condition which does appear in the definition of the nn-simplex spectra (next page of Brown). I have added that in the list of examples now.

      Another question: what’s the established term for these things here? I made up both “combinatorial spectrum” and “simplicial spectrum” after reading Brown’s article, which just calls this “spectrum” without qualification. I am tending to think that “simplicial spectrum” would be a good term.

      Related to that: what’s a more recent good reference on these combinatorial version of spectra?

      Mike: I was remembering a condition like that from Kan’s original article “Semisimplicial spectra,” which I unfortunately don’t have access to a copy of right now. I think the idea is that a spectrum of this sort is built out of a naive prespectrum of simplicial sets (that is, a sequence of based simplicial sets X nX_n with maps ΣX n→X n+1\Sigma X_n \to X_{n+1}) by making the kk-simplices of X nX_n into (k−n)(k-n)-simplices in the spectrum. I thought the condition on m<nm\lt n is sort of saying that each simplex comes from X nX_n for some n<∞n\lt \infty. But possibly my memory is just wrong.

      Since Kan’s original term was “semisimplicial spectrum” back when “semisimplicial set” meant what we now call a “simplicial set,” it’s hard to argue with “simplicial spectrum.” As far as I know, however, no algebraic topologist has really thought seriously about these things for quite some time, probably due largely to the appearance of symmetric monoidal categories of spectra (EKMM SS-modules, orthogonal spectra, symmetric spectra, etc.) of which there is no known analogue for this sort of spectra. It’s kind of a shame, I think, since these spectra give a really good intuition of “an object with kk-cells for all k∈ℤk\in\mathbb{Z}.” I spent a little while once trying to come up with a version of these that would have a symmetric monoidal smash product, maybe starting with simplicial symmetric spectra instead of naive prespectra, but I failed.

      Urs: thanks, very useful. That’s a piece of information that I was looking for.

      Yes, this combinatorial spectrum is nicely suggestive of a ℤ\mathbb{Z}-category. It seems surprising that there shouldn’t be a symmetric monoidal product on that. What goes wrong?

      Concerning terminology: now that I thought about it I feel that “simplicial spectrum” may tend to be misleading, as it collides with the use of “simplicial xyz” as a simplicial object internal to the category of xyzxyzs. Surely some people out there will already be looking at functors Δ op→Spectra\Delta^{op} \to Spectra and call them “simplicial spectra” (?)

      Mike: Yes, you’re quite right that “simplicial spectrum” should probably be reserved for a simplicial object in spectra; I wasn’t thinking. What we really need is a name for the shape category that arises here, analogous to “simplex category,” “cube category,” and so on. Like “spectrix category.” Then combinatorial spectra would be “spectricial sets.” (I’m only half joking.)

      The thing that goes wrong with the symmetric monoidal product is, as far as I can tell, sort of the same thing that goes wrong for naive prespectra: there are automorphisms that don’t get taken into account. But it’s possible that no one has just been clever enough.

      =–

      — end forwarded discussion —

    • There are now two different pages: old D-branes and new one created by Urs today D-brane. I found D-branes as I recalled we had something and I looked under nLab search.

    • When I made inhabitant redirect to term a few minutes ago, I noticed a bunch of orphaned related entries.

      For instance inhabited type since long ago redirect to inhabited set. I haven’t changed that yet, but I added some cross links and comments to make clear that and how the three

      are related. The state of these entries deserves to be improved on, but I won’t do anything further right now.

    • Noticed there was no page for the left adjoint to joins. Added subtraction.

    • As a postscript to some discussion on virtual knot theory we had here a little under a year ago, I met Victoria Lebed yesterday and it turns out that she studied categorification questions (among other things) in her thesis! I’ve only skimmed it so far, but it seems very nice, and her approach to virtual braids is very much in the spirit of John Baez’s n-Café comment. In particular, the thesis shows that the virtual braid group VB nVB_n is isomorphic to the group of endomorphisms End 𝒞 2br(V ⊗n)End_{\mathcal{C}_{2br}}(V^{\otimes n}), where 𝒞 2br\mathcal{C}_{2br} is the free symmetric monoidal category generated by a single braided object VV. (The thesis also talks a lot about positive braids, which are interpreted in terms of “pre-braided” objects, i.e., an object VV equipped with a not necessarily invertible morphism σ:V⊗V→V⊗V\sigma : V \otimes V \to V \otimes V satisfying the Yang-Baxter equation. This definition even allows for the possibility of “idempotent” braidings, which she mentioned in her talk yesterday.)

    • (hm, wait)

    • I have added something to causal structure, for the moment mainly so as to record references to definitions of causal manifolds and to proposals for axioms of local nets over these.

    • At uniform space, we have 6 axioms defining a uniform structure (or 4 to define a covering uniformity); I had added an axiom 0 [ETA: which I probably got from Douglas Bridges] which is classically trivial but useful in constructive mathematics. I now think that it's wrong to insist on this axiom; rather, those uniform spaces that satisfy it should be singled out as particularly nice. A good word for this, which may be used in similar contexts related to generalizations of metric spaces, is ‘located’; and since there are no Google hits for "located uniform space" (except for one which says ‘neatly located’ and so does not technically conflict), I am going to use that.

      The result is that there are no more references to axiom (0) at uniform space; instead, we have a definition of a located uniform space.

      I thought that were references to this axiom (0) on at least on other page pointing to uniform space, but the search function doesn't find any; if you do, though, then it would be nice to fix them.

    • Made the chemistry page slightly less stubby.

      Slightly off topic but… I’ve been working on the category of recipes as side project as a way to bring the layman to category theory, the general idea also appears in Bob Coecke’s Quantum Pictoralism. In some exchanges I pointed out to Bob that recipes are just the edible full subcategory of chemistry. Btw, here’s the first ever (buggy) visual recipe made using monoidal category theory and string diagrams.

      Would anyone be offended if I added in that recipes are a full subcategory of chemistry as a sidenote?

      May I also start some pages on recipes… in some way? I think it really is a great way to teach people category theory. After all, everyone eats, and if you can follow a recipe, you can do monoidal category theory.

    • I added the definition of uniformly/almost located to located subspace, as well as attempted proofs that covert and compact subspaces of uniformly regular uniform spaces are uniformly located. Unfortunately covert sets and compact spaces — as opposed to compact locales — seem to be hard to come by constructively. It would be nice to have some more concrete examples. I also don’t know what to write for the “idea” section, nor what Toby had in mind for “topologically located”.

    • Had need to note down Witten’s quick argument for how brane/anti-brane annihilation suggests that D-brane charge is in K-theory: at anti-D-brane.

    • I gave kappa-symmetry a decent Idea-section (I hope), highlighting the super-geometric interpretation due to Sorokin et. al. Also added more references.

    • Urs added a picture of the Fano plane there, and I gave a description and some common notation. For some reason, whenever I wrote ℙ\mathbb{P} ($\mathbb{P}$) it turns out slanted, which is annoying. I don’t know why.

    • Started SVect as it was requested by various links. But I see ’sVect’ being written too. Any preference?

    • Created covert space, a concept that is to closed maps the way overts are to open maps and compacts are to proper maps. Dubuc and Penon call covert discrete topologies “compact objects” but that seems possibly misleading in general, and “covert” seems a natural analogue of “overt” when we replace “open” by “closed”. But terminological objections are welcome…

    • I have put the definition of this at proper map.

    • I started typing into length of an object when I felt that we had an entry on this already somewhere. Where?

    • I added to allegory a section on division allegories and power allegories.

    • Hello, I was curious about how to describe involutions as algebras of a monad, so I worked it out and added some simple stuff to the article involution. As always, corrections and/or generalizations are welcome.

    • I started a page Eff. I need to link from ’effects’ and ’handlers’, but have a query on terminology. Is it ’algebraic effect’ rather than ’algebraic side effect’ that’s more commonly used?

    • Not exactly in my comfort zone, but let’s hope someone will expand this stub for effect handler.

      Given that I started out poking around these constructions because of monad (in linguistics), it’s interesting to find people using algebraic effects and handlers there.

      • Jirka Maršík, Maxime Amblard, Algebraic Effects and Handlers in Natural Language Interpretation, pdf
      • Jirka Maršík, Maxime Amblard, Introducing a Calculus of Effects and Handlers for Natural Language Semantics, arXiv:1606.06125

    • I am polishing the entry category algebra. In the course of this I noticed that there was old and long-forgotten discussion sitting there, which now first of all I hereby move from there to here:

      [ begin old discussion ]

      I use k[S]k[S] to stand for the free vector space on the set SS. This is compatible with the notation k[G]k[G] for group algebra of GG. Urs’ notation k[C]k[C] for the category algebra is also compatible, but in a different way.

      Why is my notation better? First, because I don’t like the clunky notation span k(C)span_k(C) for the free vector space on the set SS. Second, because the equation k[BG]=k[G]k[B G] = k[G] is inconsistent unless Urs is finally willing to admit that BG=GB G = G.

      So what would I call the category algebra of CC? I guess k[C 1]k[C_1] or k[Mor(C)]k[Mor(C)]. You might complain that this notation is clunky, and I’d see your point. However, it’s a fact that whenever the category algebra is important, its representation on k[C 0]=k[Ob(C)]k[C_0] = k[Ob(C)] also tends to be important — so I think the benefits of a notation that handles both structures outweigh the disadvantages of a slight clunkiness. – John

      Urs says: It is good that you said this, because we need to talk about this: I am puzzled by your attitude towards BG\mathbf{B}G vs GG. It is not the least a remark in your lecture notes with Mike that it is important to distinguish between a kk-tuply monoidal structure and the corresponding kk-tuply degenerate category, even though there is a map identifying them. The issue appears here for instance when discussing the universal GG-bundle in its groupoid-incarnation. It is

      G→EG→BG G \to \mathbf{E}G \to \mathbf{B}G

      (where EG=G//G\mathbf{E}G = G//G is the action groupoid of GG acting on itself). On the left we crucially have GG as a monoidal 0-category, on the right as a once-degenerate 1-category. In your notation you cannot even write down the universal GG-bundle! ;-)

      Or take the important difference between group representations and group 2-algebras, the former being functors BG→Vect\mathbf{B}G \to Vect, the latter functors G→VectG \to Vect. This is important all over the place, as you know better than me.

      Or take an abelian group AA and a codomain like 2Vect2Vect. Then there are 3 different things we can sensibly consider, namely 2-functors

      A→2Vect A \to 2Vect BA→2Vect \mathbf{B}A \to 2Vect B 2A→2Vect. \mathbf{B}^2A \to 2Vect \,.

      All of this is different. All of this is needed. The first one is the group 3-algebra of AA. The second is pseudo-representations of the group AA. The third is representations of the 2-group BA\mathbf{B}A. We have notation to distinguish this, and we should use it.

      Finally, writing BG\mathbf{B}G for the 1-object nn-groupoid version of an nn-monoid GG makes notation behave nicely with respect to nerves, because then realization bars |⋅||\cdot| simply commute with the BBs in the game: |BG|=B|G||\mathbf{B}G| = B|G|. I think this makes for instance your theorem with Danny appear in a prettier way.

      This behaviour under nerves shows also that, generally, writing BG\mathbf{B}G gives the right intuition for what an expression means. For instance, what’s the “geometric” reason that a group representation is an arrow ρ:BG→Vect\rho : \mathbf{B}G \to Vect? It’s because this is, literally, equivalently thought of as the corresponding classifying map of the vector bundle on BG\mathbf{B}G which is ρ\rho-associated to the universal GG-bundle:

      the ρ\rho-associated vector bundle to the universal GG-bundle is, in its groupoid incarnations,

      V ↓ V//G ↓ BG, \array{ V \\ \downarrow \\ V//G \\ \downarrow \\ \mathbf{B}G } \,,

      where VV is the vector space that ρ\rho is representing on, and this is classified by the representation ρ:BG→Vect\rho : \mathbf{B}G \to Vect in that this is the pullback of the universal VectVect-bundle

      V//G → Vect * ↓ ↓ BG →ρ Vect, \array{ V//G &\to& Vect_* \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Vect } \,,

      In summary, I think it is important to make people understand that groups can be identified with one-object groupoids. But next it is important to make clear that not everything that can be identified is actually equal.

      For instance concerning the crucial difference between the category in which GG lives and the 2-category in which BG\mathbf{B}G lives.

      [ continued in next message ]

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