Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories itex k-theory lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • tried to polish a bit the matrial at Chern-Weil theory.

      (not that there is much, yet, but still)

    • 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