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    • New entry defining ideal of topologizing subcategory (of an abelian category), wanted at conormal bundle. It is in fact a subfunctor of the identity functor and if we evaluate it on projective generator in the case of a module category then we get the usual ideal in the corresponding ring.

    • Affinity in the context of D-modules, as defined by Alexander Beilinson is the subject of a new stub D-affinity. There is a categorical generalization in the MPI1996-53 preprint (pdf) of Lunts and Rosenberg in terms of differential monads. Many generalizations of Beilinson-Bernstein localization theorem have their intuitive explanation in a two-step reasoning. First the noncommutative algebra in question is understood as a noncommutative (or maybe categorical) resolution of singularities of a commutative object. Then the latter satisfies D-affinity and one can localize.

    • New entry domain globalization of functors (zoranskoda) under development. The codomain globalization is more trivial. This are questions of extending the constructions related to Beck’s comonadicity from categories to functors. Our interest with Gabi Bohm are mainly for covers by localizations with some equivariance/compatibility with respect to additional (co)monad, which are a matter of ongoing work. This compatibility is like, or some dual of the one in the definition of morphisms of Q-categories and also the compatibility of differential monads and localization, studied by Lunts and Rosenberg. The latter is related to the classical fact that the assignment of ring of regular differential operators to a commutative ring RDiff(R)R\mapsto Diff(R) is compatible with exact localizations, in the sense that S 1RS 1Diff(R)S^{-1}R \mapsto S^{-1}Diff(R); and also to Beilinson’s notion of D-affinity.

    • I have created stubs for the missing entries to complete this table:

      The main actual content I added are, (at 2-type theory and 2-logic): pointers to Dan Licata’s thesis and to Mike’s personal wiki pages.

      I’d hope that one outcome of the present nnCafé discussion is that eventually some of these entries get equipped with some useful content.

      (P.S. I would have linked to material by Mike Stay, too, but I don’t know what to link to.)

    • I am about to create an entry called locally algebra-ed topos in the spirit of the section for local algebras at classifying topos.

      I tend to think this terminology is better than the undescriptive “structured topos”, but please let me know what you think.

      I would like to amplify the following fact:

      if we agree to say (which is reasonable) that

      • an algebra is a model of some essentially algebraic theory, hence a lex functor out of a finite-limite category;

      • a local algebra with respect to a coverage on the category is such a lex functor that preserves covers.

      then the statement is:

      • geometric theories are equivalently theories of local algebras.
    • I started an important entry differential monad. According to Lunts-Rosenberg MPI 1996-53 pdf differential calculus on schemes and noncommutative schemes can be derived from the yoga of coreflective topologizing subcategories in the abelian category of quasicoherent sheaves on the scheme, like the 𝕋\mathbb{T}-filtration, and 𝕋\mathbb{T}-part, in the case when the topologizing subcategory is the diagonal in the sense of the smallest subcategory of the category of additive endofunctors having right adjoint which contains the identity functor – in that case we say differential filtration and differential part. The regular differential operators are the elements of the differential part of the bimodule of endomorphisms. Similarly, one can define the conormal bundle etc.

    • A point of information. These constructions are due to Charles Wells in this particular setting and to Jonathan Leech, (H-coextensions of monoids, vol. 1, Mem. Amer. Math. Soc, no. 157, American Mathematical Society, 1975) in the single object case, and McLane introduces the category of factorisations I think. Charlie Wells even pushes things a bit further than Baues. Hans does not seem to have known of that work. (Charles Wells, Extension theories for categories (preliminary report), (available from http://www.cwru.edu/artsci/math/wells/pub/pdf/catext.pdf), 1979. ) I have been meaning to have a go at this entry as I have written up a modern version of Wells especially in the non-Abelian case. There is a very nice interpretation of Natural System as a lax functor. (I will do this some time…. but I can make the notes available to anyone interested.)

    • Urs created Frechet manifold, so I created Frechet space. (We violated the naming conventions too, but I guess it's OK since we have the redirects in.)

    • I am trying to begin to coherently add some of the topics of part D of the Elephant into the Lab.

      Currently I am creating lots of stub entries, splitting them off from existing entries if necessary, cross-link them appropriately, and then eventually add content to them.

      so far I have for instance created new (mostly stub) entries for things like

      I have created

      and made it a disambiguation page.

      I have edited the linked table of contents at Elephant, etc.

      (or rather I will have in a few minutes. All my save-windows are currently stalled. Will have to restart the server.)

    • created standard site (maybe not a great term, but since I am nnLabifying the Elephant). Added the theorem that every sheaf topos has a standard site of definition to site

    • I have created degeneration conjecture required at Dmitri Kaledin. In my memory, I never heard ofthis degeneration conjecture by precisely that name and I do not like it (there are so many degeneration conjectures in other fields, some of which I heard under that name). It is usually said the degeneration of Hodge to de Rham spectral sequence (conjecture). It has a classical analogue. I put redirect degeneration of Hodge to de Rham spectral sequence.

    • In differential cohomology in an (∞,1)-topos – survey, I can’t guess what ’nothing’ should be here:

      The curvature characteristic forms / Chern characters in the traditional formulation of differential cohomology take values in abelian \infty-Lie algebras and are therefore effectively nothing differential forms with values in a complex of vector spaces

    • This is an excerpt I wrote at logical functor:

      As far as cartesian morphism there are two different universal properties in the literature, which are equivalent for Grothendieck fibered categories but not in general. In what Urs calls the “traditional definition” (but is in fact a later one) one has for every xx', for every hh, for every gg such that … there exist a unique da da da. This way it is spelled in Vistoli’s lectures. This is in fact the strongly cartesian property, stronger than one in Gabriel-Grothendieck SGA I.6. The usual, Grothendieck, or weak property takes for gg the identity, and the unique lift is of the identity at p(x 1)p(x_1). Then a Grothendieck fibered category is the one which has cartesian lifts for all morphisms below and all targets, and cartesian morphisms are closed under composition. With the strong cartesian property one does not need to require the closedness under composition. Now a theorem says that in a Grothendieck fibered category, a morphism is strongly cartesian iff it is cartesian.


      Now I have made some changes to cartesian morphism, so that the entry is aware of the two variants of the universal property, which are not equivalent in general but are equivalent for Grothendieck fibered categories.

      There was also a statement there

      In words: for all commuting triangles in Y and all lifts through p of its 2-horn to X, there is a unique refinement to a lift of the entire commuting triangle.

      which is too vague and I am not happy with, as it does not involve the essential parameter: the morphism for which we test cartesianess. I made a hack to it, and still it is not something I happy with (I like the idea of horn mentioned, however not the lack of appropriate quantifiers/conditions etc.). It is cumbersome to talk horn. (Maybe we could skip the whole statement in this imprecise form, and just mention please note the filling of the horn in XX with prescribed projection in YY or alike). Here is the temporary hack:

      In imprecise words: for all commuting triangles in YY (involving p(f)p(f) as above) and all lifts through pp of its 2-horn to XX (involving ff as above), there is a unique refinement to a lift of the entire commuting triangle.

    • I have started a puny disambiguation page projection

    • After Zoran had emphasized it for years without me ever really looking into it, now I have finally read the beginning of Kontsevich-Rosenberg’s article on “Q-categories” in more details… and was struck:

      their notion of “generalized sheaves” is essentially nothing but the kind of condition that Lawvere considered in cohesive toposes (u !u *u *u !):TS(u_! \dashv u^* \dashv u_* \dashv u^!) : T \to S. More precisely, Lawvere considered the objects xx for which the canonical morphism u *xu !xu_* x \to u_! x is an isomorphism. What Kontsevich-Rosenberg call generalized sheaves are those objects for which the other canonical morphism is an isomorphism: u *xu !xu^* x \to u^! x.

      There are mainly two kinds of applications in Kontsevich-Rosenberg:

      1. the original one was to find the right notion of sheaves over formal duals of non-commutative algebras. Apparently Rosenberg is fond of the insight that for a suitable cohesive presheaf topos (my words of course) the right condition is that u *xu !xu^* x \to u^! x is an iso.

      2. Apparently (if I remember correctly what Zoran told me) Kontsevich added the observation that formal smoothness and hence infinitesimal thickening is naturally described in this context. Now that I looked through it, I realize that what they talk about in this context is really pretty much exactly what I axiomatized as infinitesimal cohesion.

      So I am happy: at once now the entire 79 page article by Kontsevich-Rosenberg turns out to be a great resource of examples and applications of cohesive topos technology! Notably they shed more light on the role of those infamous extra axioms that involve the two canoical natural transformations that come with any cohesive topos.

      For that reason I have now begun expanding the nnLab entry Q-category that Zoran once started

    • This was the query in topologizing subcategory which I summarized shortly:

      Mike: Where does the word ’topologizing’ come from?

      Zoran Skoda: I am not completely sure anymore, but I think it is from ring theory, where people looked at the localizations at topologizing categories. There exist some topologies on various sets of ideals like Jacobson topology, so it is something of that sort in the language of subcategories instead of the language of filters of ideals. I’ll consult old references like Popescu, maybe I recall better. In any case it is pretty standard and has long history in usage: both classical and modern. No, it is not in Popescu…old related term is in fact talking about topologizing filters of ideals in a ring, so that must be the source…for example, the classical algebra by Faith, vol I, page 520 defines when the set of right ideals is topologizing. I am not good with that notion, but I can make an entry with quotation to be improved later.

    • New microstubs S-category, separable coring and finally some substantial material at separable functor at last. The monograph by Caenapeel, Militaru and Zhu listed at separable functor studies Frobenius functors and separable functors in parallel; there are relations in a number of interesting situations. Frobenius functors are those where left and right adjoint are the same (hence in particular we have adjoint n-tuple for every nn). Separable is a notion which is about certain spliting condition. This spliting is of the kind as spliting in Galois theory, I mean the Grothendieck’s version of classical Galois theory involves separable algebras at one side of Galois equivalence.

      S-category due Tomasz Brzeziński is a formalism something similar to Q-categories of Alexander Rosenberg. Tomasz studies formal smoothness and separability in the setup of abelian categories, motivated by corings, Hopf algebras and similar applications. I would guess that understanding those could be useful into better understanding the Galois theory in cohesive topos, but I do not know.

      I also created Maschke’s theorem which is one of the motivations for separable functors.

    • I have expanded the entry formally smooth morphism:

      I have first of all added the general-abstract formalization by Kontsevich-Rosenberg, taking the liberty of polishing it a bit from Q-category language to genuine (cohesive) topos-theoretic language and making contact with the notion of infinitesimal cohesion .

      Then I added their theorems about how the general abstract topos-theoretic definitions do reproduce the traditional explicit notions.

      Except for one clause : in prop. 5.8.1 of Noncommutative spaces they show that the correct notion of formal smoothness for morphisms is reproduced in the non-commutative case (via the relative Cuntz-Quillen condition). But for the commutative case I see the corresponding statement only for objects (in section 4.1) not for morphisms.

      Zoran, do you know if they also discuss the relative version in the commutative case? Maybe it’s trivial, I haven’t thought it through yet.

    • Expanded Vassiliev invariant, started Kontsevich integral, did a bit of reorganisation on knot theory (in particular, linking to more pages).

      In case anyone’s wondering, there was a book put on the arXiv a couple of days ago touting itself as an introduction to Vassiliev invariants. I’m reading through it and taking notes as I go. I left in a bit of a rush today so the formatting of the Kontsevich integral went a bit haywire, and I made a statement on the Vassiliev invariant page that I know I didn’t say quite right.

      In the arXiv book, Vassiliev invariants are introduced first using the Vassiliev skein relations, not their “proper” way (which I haven’t gotten to yet so I don’t know it). The formula looked very like a boundary map on a complex, but I think it has to be a cubical complex rather than a simplicial one. Only it isn’t the full boundary map, rather a partial boundary map (going to opposite faces), but I didn’t get it straight in my head until later. But now I think I’m going to wait until I read the bit about the true definition - which I guess will be something like this - before correcting it (unless anyone gets there before me, of course).

      Drew a few more SVGs relevant for knots as well. The code for producing the trefoil knot is very nice now, though I say so myself!

    • Bas Spitters has kindly pointed out to me that the proof by Banaschewski and Mulvey of Gelfand duality is not actually constructive, as it invokes Barr’s theorem, and that he has a genuine constructive and also simpler proof with Coquand. I have added that to the refrences at constructive Gelfand duality theorem