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 k-theory lie-theory 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 nforum 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 sheaves 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: adjunction between topological spaces and diffeological spaces

Bottom of Page

1 to 5 of 5

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2020

    this is the bare statement of a Proposition, to be !include-ed into the relevant entries, notably at diffeological space and at Delta-generated topological space

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2020
    • (edited Jun 5th 2020)

    if I am not mistaken, then the full statement (including also Lemma 3.3 in Shimakawa-Yoshida-Haraguchi 10) is that we have an idempotent adjunction which exhibits Delta-generated spaces as coreflective inside all topological spaces, and as reflective in diffeological spaces:

    TopologicalSpacesAA⊥AA⟶Cdfflg↩DeltaGeneratedSpacesAA⊥AA↪⟵DtplgDiffeologicalSpaces TopologicalSpaces \underoverset { \underset{ Cdfflg }{\longrightarrow} } { \overset{ }{\hookleftarrow} } {\phantom{AA}\bot\phantom{AA}} DeltaGeneratedSpaces \underoverset { \underset{ }{\hookrightarrow} } { \overset{ Dtplg }{\longleftarrow} } {\phantom{AA}\bot\phantom{AA}} DiffeologicalSpaces

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2020

    I have spelled out the (elementary and straightforward) proof.

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 12th 2020
    • (edited Jun 12th 2020)

    added pointer to Haraguchi-Shimakawa 13, Sec. 7, which makes explicit the factorization of the adjunction as in #2

    diff, v7, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2020
    • (edited Nov 26th 2020)

    adjusted the wording in the cautionary remark regarding a gap in the proof of the Quillen equivalence, now that there is explicit claim that the gap has been filled (HS20)

    diff, v10, current

1 to 5 of 5

  1.