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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJan 27th 2010

    At first Zoran's reply to my query at structured (infinity,1)-topos sounded as though he were saying "being idempotent-complete" were a structure on an (oo,1)-category rather than just a property of it. That had me worried for a while. It looks, though, like what he meant is that "being idempotent" is structure rather than a property, and that makes perfect sense. So I created idempotent complete (infinity,1)-category.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJan 27th 2010

    Sorry for the typo, I was originally planning to say the thing in two steps, but then decided it is better to just quote the reference and the phrase was left...

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 11th 2010

    Lurie defines a simplicial set IdemIdem such that an idempotent in an (,1)(\infty,1)-category CC is the same as a functor IdemCIdem\to C. Is this IdemIdem the same as the nerve of the free 1-category containing an idempotent?

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeDec 4th 2014

    Suppose I have an (,1)(\infty,1)-category CC and an idempotent in its homotopy 1-category. Can it be lifted to a “coherent” idempotent in CC itself? Is there an obstruction theory?

  1. @Mike: the closest result I know of is Lemma 7.3.5.14 of Higher Algebra. More or less, it says that if e: X -> X is an morphism, then e can be enhanced to a coherent idempotent if and only there is an equivalence e -> e^2 making the obvious maps e^2 -> e^3 determined by the two choices of parentheses commute up to homotopy.
    • CommentRowNumber6.
    • CommentAuthorDylan Wilson
    • CommentTimeDec 5th 2014

    cf. also Warning 1.2.4.8 for a counterexample if we try to weaken this criterion. (though there are no counterexamples if C is stable.)

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeDec 5th 2014

    Awesome, together those answer the question completely, thanks! I’ve recorded these facts on the page.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 8th 2014

    I have added pointer to Mike’s HoTT wrapup here

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeAug 16th 2017

    Apparently the answer to the question asked (over 7 years ago) in #3 above is “if it isn’t, it should have been”. Lurie has now changed the definition of IdemIdem to be the nerve of the free category containing an idempotent; see here and here. We should update the definition on the lab.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeAug 16th 2017

    Updated idempotent complete (infinity,1)-category with the new, simpler definitions (and updated page references).

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMay 3rd 2018

    updated link to point to my paper in addition to blog post

    diff, v17, current

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