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    • 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 ]

    • Added to reflected limit the example that fully faithful functors reflect limits and colimits (here).

    • I think I've found a mistake in the entry on exponential ideals (https://ncatlab.org/nlab/show/exponential+ideal).

      It states that any exponential ideal is itself cartesian closed, but I don't see why it would necessarily
      be closed under products.

      I've removed the statement for now, let me know if I missed something obvious.

    • I have recently created an entry definable set. It is usually defined as an equivalence class of formulas satisfied by the same elements in each structure of the first order language (or model for a theory). But some works recently, like the lectures of David Kazhdan (pdf), take for granted the observation that in fact the relation of equivalence implies that the evaluation of definable sets in a model extends to a functor on the category of models and elementary monomorphismsm, say ℳ el\mathcal{M}_{el}. There are now many other notions in model theory which have ’definable” as a prefix, and they do not fill uniformly in the same pattern. For example, the category of “definable spaces” and “definable continuous maps” is not the same as the functor from ℳ el\mathcal{M}_{el} to topological spaces and continuous maps, though possibly some examples would probably fit in the latter description. Also some “definable” notions are for a fixed model and not functors on ℳ\mathcal{M}.

      Now, Hrushovski in his 2006 work looked at definable groupoids, about which I just plan to create an entry. Now I was cautious not to say that it is simply an internal groupoid in the category of definable sets, as it is at first defined differently. Besides dealing with internal objects in a large functor category, one should possibly make care about this as well, regarding that we are dealing with a setup where one should be maybe careful about tools like large cardinals (what would be the elegant way to do?).

      But in any case, the first problem (that the definition was not given as a functor) can easily be dealt.

      So the definition roughly says that one has definable set of objects and of morphisms (just as we wanted), but then the structure maps like target, composition etc. should be definable functions, hence, as relations they are definable subsets of cartesian products, so when presented from ℳ el\mathcal{M}_{el} as functors into the categories of sets and relations, which are functional in every model. So, now I just checked a very simple fact, specific to the category of sets and functions, that

      Proposition. Given a small category ℳ\mathcal{M} and two functors F,G:ℳ→SetF,G:\mathcal{M}\to Set the natural transformations α:F→G\alpha : F\to G are in 1-1 correspondence with functors β:ℳ→Set\beta : \mathcal{M}\to Set such that β(A)⊂F(A)×G(A)\beta(A) \subset F(A)\times G(A) and β(A)\beta(A) is a functional relation.

      In other words, functoriality of β\beta is the same as α\alpha being natural. That is new to me.

      This immediately implies that the definable groupoid as in Hrushovski arxiv/math.LO/0603413 is (up to the delicacies dealing with functor categories) indeed an internal groupoid in the category of functors ℳ el→Set\mathcal{M}_{el}\to Set and natural transformations.

    • I’ve expanded Zoran’s entry on type (in model theory), with an ideas section in the language of categorical logic. I’ve also moved definitions to a definition section and briefly mentioned saturation, monster models, and how Barr-exactness of the syntactic category (i.e. elimination of imaginaries) leads to a model-theoretic account of Galois theory—there’s a grey link there which I will flesh out soon.

    • Hi everyone!

      I’ve created relative adjoint functor, and linked to it from the local definition of adjoint functor (a partially defined adjoint yields an adjoint relative to the inclusion of a full subcategory).

      L J⊣RL {\,\,}_J\!\dashv R and L⊣ JRL \dashv_J R is as far as I know nonstandard notation, but I think it’s ok, even if the left subscript feels a bit kludgy. I will add more stuff in the next few days.

      PS: Thanks a lot to all the nLab contributors; in the past few years I’ve learn a lot through here :) I now have the time and a little bit of confidence to contribute, so any pointers, tips, formatting, style suggestions, whatever will be greatly appreciated

    • There is a new personal page created with the name Debapriyay, it doesn’t look great. I haven’t got time at the moment to address it.

    • started a minimum at de dicto and de re, mainly such as to have a place to point to section 4 of

      • Michael Makkai, Gonzalo Reyes, Completeness results for intuitionistic and modal logic in a categorical setting, Annals of Pure and Applied Logic 72 (1995) 25-101

    • at universal construction there used to be a little chat between me and Toby along the lines of "would be nice if somebody added something here".

      Since I think by now we have plenty of pointers to this entry, I thought it should present itself in a slighly more decent fashion. So I removed our chat and left a stubby but honest entry.

    • I created a short entry Banach-Alaoglu theorem to fulfil a grey link. While the general theorem is equivalent to the Tychonoff theorem for Hausdorff spaces, the case of separable Banach spaces is constructive. I do wonder how constructive, but apparently this gets used to construct solutions to PDEs, so I guess it’s quite concrete.

    • created a stub for dihedral homology, for the moment just so as to record a recent reference

    • Am starting an entry pro-manifold. Have added statement and proof that pro-Cartesian spaces are fully faithful in smooth loci (here).

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