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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 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 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

    • CommentRowNumber12.
    • CommentAuthorTim Campion
    • CommentTimeFeb 25th 2020
    • (edited Feb 25th 2020)

    Somehow, I can’t find in Higher Algebra, nor can I find any discussion of this coherence issue anywhere else in HA. Presumably it was removed in some update to HA (or else I’m just colossally failing). Does anybody know where I can find an exposition of this (i.e. the fact that an incoherent idempotent plus a finite number of coherences automatically lifts to a coherent idempotent) which isn’t phrased in type theory like Mike’s paper or blog post? It’s reminiscent of HTT, which basically shows that this coherence statement is solvable for idempotent endofunctors.

    • CommentRowNumber13.
    • CommentAuthorTim Campion
    • CommentTimeFeb 25th 2020
    • (edited Feb 25th 2020)

    I figured it out – when Lurie rewrote the idempotents section in HTT, he seems to have moved this there – it’s HTT now.

    The fun thing about this coherence question is that it tells us the following: the walking idempotent IdemIdem is a compact object in the \infty-category of \infty-categories – even though it has infinitely many nondegenerate simplices as a simplicial set! In particular, writing IdemIdem as the sequential colimit of its skeleta, we see that the identity map factors through some stage of the colimit, \infty-categorically. More precisely, I think what we can say is that IdemIdem is a retract of a Joyal-fibrant replacement of its 3-skeleton.

    Now, I believe it’s shown that if XX is a finite simplicial set, or just Joyal-equivalent to a finite simplicial set, then any \infty-category with an initial object and pushouts has XX-indexed colimits. We know that not every \infty-category with an initial object and pushouts admits splitting of idempotents (e.g. finite spaces, by the Wall finiteness obstruction). Therefore, IdemIdem is not Joyal-equivalent to a finite simplicial set, Therefore IdemIdem is an example of a compact object in Cat Cat_\infty, the \infty-category of \infty-categories, which cannot be obtained as a finite colimit of Δ[n]\Delta[n]’s, even though every compact object is a retract of a finite colimit of Δ[n]\Delta[n]’s. The point is that this is analogous to the phenomenon in the \infty-category Gpd Gpd_\infty of spaces, where not every compact object is a finite colimit of contractible spaces, even though every compact object is a retract of such – as seen via the Wall finiteness obstruction.

    That is, if you’re looking for examples of the Wall phenomenon in Gpd Gpd_\infty, it’s a bit exotic. But if you look for examples of it in Cat Cat_\infty, you have this great, familiar example in the form of IdemIdem.

    Relatedly, the fact that the skeleta of IdemIdem can’t be equivalent to IdemIdem implies that there must exist examples of inequivalent coher-ifications of homotopy idempotents of all orders, as Mike asked about here.

    Do these arguments show that the inclusion of the 3-skeleton into IdemIdem is \infty-categorically cofinal and co-cofinal?

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeMar 12th 2020

    Nice observation. I don’t know about finality.

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 12th 2020

    Corrected a link.

    diff, v19, current

  2. This is possibly a silly question, but what do we know about the weak homotopy type of Idem\mathsf{Idem}?

    Lurie mentions in HTT that Idem\mathsf{Idem} is precisely the classifying space of the “Boolean monoid” 𝔹=({0,1},OR,1)\mathbb{B}=(\{0,1\},\text{OR},1), and thus clearly we have

    π 0(Idem) π 0(B𝔹) *, \begin{aligned} \pi_0(\mathsf{Idem}) &\cong \pi_0(\mathbf{B}\mathbb{B})\\ &\cong *, \end{aligned}

    but also

    π 1(Idem) π 1(B𝔹) K 0(𝔹) 0. \begin{aligned} \pi_1(\mathsf{Idem}) &\cong \pi_1(\mathbf{B}\mathbb{B})\\ &\cong \mathrm{K}_0(\mathbb{B})\\ &\cong 0. \end{aligned}

    What about π n(Idem)\pi_n(\mathsf{Idem}) for n2n\geq2?

    • CommentRowNumber17.
    • CommentAuthorHurkyl
    • CommentTimeSep 19th 2021

    HTT says that Idem is weakly contractible; its weak homotopy type is the point.

  3. Ah, perfect! Thanks, Hurkyl! :)

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