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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2009
    • (edited May 13th 2010)

    I removed my recent material at simplex in a lined topos and instead inserted this now, expanded, at

    interval object

    where it belongs. There is now a section there that discusses how interval objects gives rise to cubical and simplicial path oo-categories.

    The proposition I state there I have carefully checked. Should be correct. But haven't typed the proof, it doesn't lend itself to being typed (straightforward but tedious, as one says).

    But if it is indeed correct, this must be standard well-known stuff. Does anyone have a reference?!

    I also restructured and edited the rest of the entry a bit, trying to make it a bit nicer. THis entry deserves more attention, it is a pivotal entry.

    Tomorrw when I am more awake I'll remove simplex in a lined topos and redirect links to it suitably to interval oject.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeOct 8th 2009

    Last link: interval object

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeOct 8th 2009
    Added remark to interval object that the affine line A^1 is an interval object in the A^1 homotopy theory of schemes, but is obviously not like the interval in the topological sense.
    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeOct 8th 2009
    Whoops, apologies, that last comment was me.

    -David Roberts

    (tardy in signing up to the forum, I know)
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2009

    Thanks David.

    I had known about this from Zoran, but never looked into it in any detail.

    But I have now at least picked up this thread and created stubs for

    homotopy localization


    A1 homotopy theory

    and linked to them from the paragraph that you added to interval object.

    I also restructzred interval object a but further to amplify this point.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2009

    I also worked further on interval object:

    expanded the introduction further, renames some headlines, added formal definition/theorem/proof environments, expanded the remarks on the meaning of the construction of that cosimplicial object.