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nLab > Latest Changes: adjunction between topological spaces and diffeological spaces

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  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

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