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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 9th 2010
    • (edited Feb 9th 2010)

    I thought I'd amuse myself with creating a succinct list of all the useful structures that we have canonically in an (oo,1)-topos without any intervention by hand:

    • principal oo-bundles, covering oo-bundles, oo-vector-bundles, fundamental groupoid, flat cohomology, deRham cohomology, Chern character, differential cohomology.

    I started typing that at structures in a gros (oo,1)-topos on my personal web.

    I think this gives a quite remarkable story of pure abstract nonsense. None of this is created "by man" in a way. It all just exists.

    Certainly my list needs lots of improvements. But I am too tired now. I thought I'd share this anyway now. Comments are welcome.

    Main point missing in the list currently is the free loop space object, Hochschild cohomology and Domenico's proposal to define the Chern character along that route. I am still puzzled by how exactly the derived loop space should interact with  \Pi^{inf}(X) and \Pi(X).

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2010

    expanded my collection of canonical structurs in an (oo1,)-topos a bit further

    structures in a gros (oo,1)-topos

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2010

    have been further expanding and streamlining formal structures in an (oo,1)-topos.

    Find it quite pleasing what a whealth of structures comes out for free here.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2010
    • (edited Feb 19th 2010)

    one more:

    I think an object X in an (oo,1)-topos is

    • formally smooth if  X \to \mathbf{\Pi}_{inf}(X) is an effective epimorphism .

    With the notation as in the entry.

    This is supposed to be the derived and generralized version of what Simpson-teleman have on p . 7