Not signed in (Sign In)

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 book bundles calculus categorical 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 foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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 set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack 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).

nLab > Latest Changes: Prof

Bottom of Page

1 to 5 of 5

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2021
    • (edited Oct 21st 2021)

    added pointer to

    (via user varkor, here)

    diff, v10, current

    • CommentRowNumber2.
    • CommentAuthornonemenon
    • CommentTimeSep 25th 2023
    • (edited Sep 25th 2023)

    From what I can gather, sheaves on (some) Vect (with some Grothendieck topology I don’t recall) constitute a standard model for generalized smooth spaces. I still recall my alarm when I first learned that this produces a synthetic differential geometry equivalent to that obtained by taking sheaves on the category of cartesian spaces and all smooth maps between them, which I believe is called Cart when equipped with (some Grothendieck topology I don’t recall). Resolving my surprise concerning this identification was formative in developing my intuition for the various flavors of completion 2-functors such as Karoubi envelopes and Cauchy completion in the enriched context (and Yoneda etc).

    Forgive me if a flaw in my understanding has resulted in a silly question, but I am quite curious what would be obtained by extending this recipe along the analogy/abstraction [R : R-Bimod] :: [V : V-Prof]. In short: what are “spaces locally modelled on enriched categories” in the sense that tangent spaces are (enriched) categories, tensor fields are diagrams of profunctors indexed by the shape ∫ of the base, etc.

    Does anyone know if this has been addressed in the literature? I would love to read more! Thank you

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 25th 2023

    sheaves on (some) Vect\mathbf{Vect} (with some Grothendieck topology I don’t recall) constitute a standard model for generalized smooth spaces.

    Do you have a pointer to what you have in mind here?

    the category of cartesian spaces and all smooth maps between them, which I believe is called Cart\mathbf{Cart}

    Yes. I had turned attention to this category as a site for smooth spaces in Def. 1.2.197 of dcct (p. 144) and the idea stuck (previous authors had insisted on using categories of open subsets of Euclidean spaces, instead).

    • CommentRowNumber4.
    • CommentAuthornonemenon
    • CommentTimeSep 25th 2023

    Hmm. After struggling to track down where I got the idea about Vect serving as a nice site for smooth spaces, I wonder if I got confused reading an older version of CartSp or smooth set. Maybe my alarm was actually good sensibility all along, haha.

    Thank you for the reference. I will keep chewing on the relationship between Vect and Prof; right now I’m toying with slices Prof /1Prof_{/1} as analogs of Vect /k[1]≃AffVect_{/k[1]} \simeq Aff.

    • CommentRowNumber5.
    • CommentAuthoranuyts
    • CommentTimeFeb 16th 2024

    Prof = Kl(PSh)

    diff, v11, current

1 to 5 of 5

  1.