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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?
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.
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.
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.
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.
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