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 bundle bundles calculus categories category category-theory chern-weil-theory cohesion 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 galois-theory 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 homology homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory itex k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory 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 planar 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 set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft 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).
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 25th 2020

    Trivial edit to create a discussion page.

    diff, v8, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 25th 2020

    I find this article extremely confusing.

    After Definition 2.1, it immediately moves to discuss the more general case of differential concretifications.

    There is no definition of a differential concretification given.

    There are only ad hoc constructions in Defintiion 3.6 and Definition 3.8, but these do not make sense for arbitrary ∞-sheaves since the construction involves B^{p+1} U(1) and B(B^p U(1))_conn, and no indication is given how to construct one from the other.

    Continuing the confusion, I also do not see how the parameter p is supposed to be chosen for a given abstract ∞-sheaf.

    So what exactly is this article trying to do? What is the definition of a differential concretification?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2020

    Thanks for looking into it. One day this deserves to be straightened out. It’s highly interesting and important, but also complicated.

    My first version of differential concretification in dcct had a technical mistake, and I (ab)used this entry as a scratch-pad for working out a correct version.

    So I wrote the stuff there in an effort to work it out for myself, not as an exposition. If it is too unpolished for public domain, let’s just clear it. One day I’ll pick it up from the entry history and do justice to it. And be it in a the next life.

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 26th 2020
    • (edited Jun 26th 2020)

    Okay, thanks for clarifying this.

    But do you actually have a precise definition of differential concretification as of right now?

    My worry is that having a fixed h-level appears to be crucial for any construction I can think of.

    Consider the following 3 examples, where in each example we have an ∞-sheaf F we look at the map Hom(M,F)→#Hom(M,F), and we apply the n-connected / n-truncated factorization system (e.g., for n=0 we factor it as the essential image followed by an inclusion of connected components) and characterize the resulting intermediate object G: Hom(M,F)→G→#Hom(M,F).

    Example 1: F(S) = the set of Riemannian metrics on S.

    1a) n=0: G(U) = smooth U-indexed families of Riemannian metrics on M (correct answer).

    1b) n=1: G(U) = Riemannian metrics on M⨯U (wrong answer).

    Example 2: F(S) = the groupoid of principal G-bundles with connection on S.

    2a) n=0: G(U) = the groupoid whose objects are smooth U-indexed families of principal G-bundles with connection on M, whereas morphisms are discontinuous U-indexed families of connection-preserving isomorphisms of principal G-bundles on M (wrong answer).

    2b) n=1: G(U) = the groupoid whose objects are principal G-bundles with connection on M⨯U (wrong answer!), whereas morphisms are smooth U-indexed families of connection-preserving isomorphisms of principal G-bundles on M (correct answer only for 1-morphism, not objects).

    So both 2a) and 2b) are incorrect and a more involved procedure is necessary, I guess.

    But even with this procedure in mind, consider now the following example:

    Example 3: Take the product of 1) and 2).

    3a) n=0: we have the product of 1a) and 2a), and 2a) is wrong.

    3b) n=1: we have the product of 1b) and 2b), and 1b) is wrong.

    I also need this in my own work, and it seems to me that concretification should be considered an additional data (i.e., multiple concretifications are possible for the same ∞-sheaf) and we probably cannot expect an abstract concretification functor.

    But I am very curious about anything you have to say about this.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2020

    That’s really great that you are looking into this now in detail. Alas, at the moment I have no time for it. (BTW, are you still on that project re shape of diffeological spaces?) I’ll try to come back to it, but it may not be too soon.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 27th 2020

    I am writing up the model structure on diffeological spaces, just finished writing down transfinite compositions and now moving on to the case of cobase changes of generating acyclic cofibrations.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)