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    • 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] to stand for the free vector space on the set S. This is compatible with the notation k[G] for group algebra of G. Urs’ notation 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 spank(C) for the free vector space on the set S. Second, because the equation k[BG]=k[G] is inconsistent unless Urs is finally willing to admit that BG=G.

      So what would I call the category algebra of C? I guess k[C1] or 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[C0]=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 vs G. It is not the least a remark in your lecture notes with Mike that it is important to distinguish between a k-tuply monoidal structure and the corresponding k-tuply degenerate category, even though there is a map identifying them. The issue appears here for instance when discussing the universal G-bundle in its groupoid-incarnation. It is

      G→EG→BG

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

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

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

      A→2Vect
      BA→2Vect
      B2A→2Vect.

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

      Finally, writing BG for the 1-object n-groupoid version of an n-monoid G makes notation behave nicely with respect to nerves, because then realization bars |⋅| simply commute with the Bs in the game: |BG|=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 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? It’s because this is, literally, equivalently thought of as the corresponding classifying map of the vector bundle on BG which is ρ-associated to the universal G-bundle:

      the ρ-associated vector bundle to the universal G-bundle is, in its groupoid incarnations,

      V↓V//G↓BG,

      where V is the vector space that ρ is representing on, and this is classified by the representation ρ:BG→Vect in that this is the pullback of the universal Vect-bundle

      V//G→Vect*↓↓BGρ→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 G lives and the 2-category in which BG 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 ℳel. 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 ℳel 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.

      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 ℳel 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 and two functors F,G:ℳ→Set the natural transformations α:F→G are in 1-1 correspondence with functors β:ℳ→Set such that β(A)⊂F(A)×G(A) and β(A) is a functional relation.

      In other words, functoriality of β is the same as α 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 ℳel→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).

      LJ⊣R and L⊣JR 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

    • As promised in this thread, I’ve written up how model theorists treat Galois theory at model-theoretic Galois theory. I briefly mention the connection with the theory of internal covers and the Tannakian formalism. The proof of the fundamental order-reversing bijection between closed subgroups and intermediate extensions is essentially Poizat’s, but I don’t mention the corresponding Galois theory for the strong Stone space (types taken in Teq), which is the language he prefers.

      I’ve also filled out necessary references at monster model, model-theoretic algebraic closure, and theory of algebraically closed fields.

    • 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).

    • There is a dead link at ionad (the one to ‘web’. This is repeated at Richard Garner. There is a second link, (to an ArXiv version), so I will delete the dead ones. Is there a published version somewhere as well?

    • I made some progress understanding the concept of Albanese variety and updated this page.

      The miracle here is that being an abelian variety is just a property of a pointed connected projective algebraic variety, not extra structure.

      As Qiaochu Yuan pointed out on MathOverflow, any basepoint-preserving continuous map between tori is homotopic to a group homomorphism. But when these tori are abelian varieties, and the map preserves that structure, it’s actually equal to a group homomorphism!

    • Expanded the examples section at end (prodded by discussion in the blog), stating Kan extension and geometric realization.

      Also added a toc.

    • There is a strange entry Metadata properties which was created bt ‘The User’ quite some time ago. Does anyone know what this was? (I happened on it when searching for something else.)

    • I’ve added a proof that rank 0 virtual vector bundles are nilpotent elements in K-theory to virtual vector bundle, and made some small edits to topological K-theory. The latter page had the definition of K-theory mixed up with the definition of reduced K-theory, so I made a small correction there, but the proof that K(X) is a ring is not there now, as the existing proof was for reduced K-theory only. I will edit this in the coming days.

      Incidentally, for ’compact support’ vector bundle K-theory of a non-compact space (so not representable K-theory), this implies that the ˜K(X)-adic topology is discrete, as opposed to the case where for non-compact spaces representable K-theory has something interesting going on (cf work of Atiyah-Segal on equivariant K-theory and completion, where they compute K(BG) in the case of representable K-theory for G compact).

    • Someone calling themselves‘Realigned arrows for clarification and fixed composition order’ has changed the composition order on hom-functor.

    • I see Thomas has revived construction in philosophy after it was emptied a while ago from the initial spam. If it’s naming a piece of writing, i.e., by Schelling, we tend to capitilize. But perhaps it’s a more general development within philosophical logic. It would help to relate this entry to constructive mathematics.

      In that we’re told its influence continues to our times through Dummett, etc., are we to think of that strand of constructivism which runs through to Martin-Lof? I recently noted him say

      …mathematical knowledge through the construction of concepts, Ger. mathematische Erkenntnis durch die Konstucion der Begriffe, a splendid formulation which no doubt had a fruitful influence on Brouwer, and to my mind it is justifiable to say that intuitionism is a development of an essentially Kantian position in the foundations of mathematics. (Martin-Löof, Analytic and synthetic judgements in type theory, p. 99).

    • After the third time typing "pretopology" into my nlab-goto box and ending up at pretopological space instead of where I wanted to be, namely Grothendieck pretopology, I changed the redirect. It seems likely to me that the latter notion will be of more interest to more of our clientele than the former. But if you disagree, speak up.

      I also added a Wikipedia-style "see also" note to the top of Grothendieck pretopology. Should we do that sort of thing in general?

    • hilbertthm90 has mentioned he wants to write some substance on abelian variety, so to encourage and facilitate this, I have created a short stub for it, mainly the bibliography and short idea.

    • New stub tangent map.

      It uses the link differential of a map which does not direct to anything at the moment as it is hard to decide. The entry differential is dedicated to differential of a chain complex, hence neglecting the term usage for the differential of a map of Banach spaces or the differential of a map of differentiable manifolds. Now the nLab mostly uses derivative for a differential and at th moment derivative points to differentiable map. Now there is an entry differentiation which is covering mostly the same as differentiable map but in the way of Lawvere-Kock synthetic differential geometry, Inside the entry differentiation there is a place whete derivative and differential are contrasted in a way which is exactly opposite to the traditional analysis: the entry calls derivative an infinitesimal difference and differential the ratio, while all classical textbooks do it the opposite to that. Moreover, in that entry, the link differential is used which points to chain complexes, hence nothing to do (Urs was always complaining that the expression derived functor is not motivated although differentials in chain complexes are used to do it), hence we should not mix differentials in homological algebra and differential of a map, which should become a good redirect, once we agree upon conventions or possible mergers of entries. Attention Urs, Todd, Toby, Mike.

    • There is a stub adjoint representation which in my opinion should be the same entry as adjoint action, hence should be merged. Words representation and action are in general equivalent; to each action G×M→M one assigns a representation G→EndM and viceversa (up to nicetess of inner hom spaces etc.). True, the specialists in Lie theory like to prefer calling representation when they have a linear representation but their own textbooks start with nonlinear case. Thus action of a Lie group on a Lie group is nonlinear hance usually the action terminology used while on a Lie algebra more often the representation is used, but it is not a rule, and the distinction does not survive in generalizations (like the Hopf algebra); any sensible entry, as the main entry “adjoint action” should relate th nonlinear case and its linearization hence should not be in separate entries. I wanted to write some references for adjoint action for quantum groups but gave up as I do not know into which of the entries and expect a decision on fiture fate of the two entries first.

    • Someone created a page called characteristic variety, but with silly content. It has been renamed empty 160. (The content was TJM-37MOONG BEAN and TJM-37 would seem to be a disease resistant variety of Mung bean!)

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