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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.
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...
Lurie defines a simplicial set $Idem$ such that an idempotent in an $(\infty,1)$-category $C$ is the same as a functor $Idem\to C$. Is this $Idem$ the same as the nerve of the free 1-category containing an idempotent?
Suppose I have an $(\infty,1)$-category $C$ and an idempotent in its homotopy 1-category. Can it be lifted to a “coherent” idempotent in $C$ itself? Is there an obstruction theory?
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.)
Awesome, together those answer the question completely, thanks! I’ve recorded these facts on the page.
I have added pointer to Mike’s HoTT wrapup here
Updated idempotent complete (infinity,1)-category with the new, simpler definitions (and updated page references).
Somehow, I can’t find 7.3.5.14 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 5.2.7.4, which basically shows that this coherence statement is solvable for idempotent endofunctors.
I figured it out – when Lurie rewrote the idempotents section in HTT, he seems to have moved this there – it’s HTT 4.4.5.20 now.
The fun thing about this coherence question is that it tells us the following: the walking idempotent $Idem$ 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 $Idem$ 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 $Idem$ is a retract of a Joyal-fibrant replacement of its 3-skeleton.
Now, I believe it’s shown that if $X$ 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 $X$-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, $Idem$ is not Joyal-equivalent to a finite simplicial set, Therefore $Idem$ is an example of a compact object in $Cat_\infty$, the $\infty$-category of $\infty$-categories, which cannot be obtained as a finite colimit of $\Delta[n]$’s, even though every compact object is a retract of a finite colimit of $\Delta[n]$’s. The point is that this is analogous to the phenomenon in the $\infty$-category $Gpd_\infty$ 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 $Gpd_\infty$, it’s a bit exotic. But if you look for examples of it in $Cat_\infty$, you have this great, familiar example in the form of $Idem$.
Relatedly, the fact that the skeleta of $Idem$ can’t be equivalent to $Idem$ 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 $Idem$ is $\infty$-categorically cofinal and co-cofinal?
Nice observation. I don’t know about finality.
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