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In the published version of HTT (Def 4.4.5.2) and on the nlab one finds one definition of a split homotopy coherent idempotent corepresented by a quasicategory $Idem^+_\mathrm{old}$. A few months ago, Gal Dor found an issue in Lurie's treatment of idempotents, and Lurie rewrote the section. The definition of an idempotent in a quasicategory has changed: in the updated version of HTT, it's a map from $Idem^+_\mathrm{new}$, which is simply the nerve of the free split idempotent in $\mathsf{Cat}$.

In his question, Dor suggested that these two definitions are the same, but I'm not so sure. I count 8 nondegenerate 3-simplices in $Idem^+_\mathrm{new}$, but 6 nondegenerate 3-simplices in $Idem^+_\mathrm{old}$, so I don't think they're isomorphic. I don't see an obvious bijection between the simplices of the two models.

Questions:

  1. Are $Idem^+_\mathrm{old}$ and $Idem^+_\mathrm{new}$ equivalent quasicategories?

  2. Is there at least a map between them?

  3. It seems that all the foundational results about the old definition have been re-proved for the new definition (including a cohrence result -- the new Prop 4.4.5.20 which originally appeared under the old definition in Higher Algebra). Are there any results in the literature which rely on the details of the old construction?


EDIT

Just for concreteness, here are the non-degenerate simplices in dimension 3:

For $Idem^+_\mathrm{old}$:

A 3-simplex is an (unlabeled) set of disjoint nonempty subintervals of $[3]$, which is nondegenerate iff each subinterval has exactly one element and each "gap:" between subintervals (including the gaps at the beginning and end) have at most one element. So a nondegenerate 3-simplex can be specified by a set of numbers between 0 and 3 (each representing a singleton interval). They are:

$\begin{align*} \{0,1,2\}, \{0,1,3\}, \{0,2,3\}, \{1,2,3\},\{0,2\}, \{1,3\}, \{1,2\}, \{0,1,2,3\} \end{align*}$

for a total of 6 8.

For $Idem^+_\mathrm{new}$:

A 3-simplex is a sequence of 3 composable morphisms in the free split idempotent $A^{\overset{i}{\to}}_{\underset{r}{\leftarrow}} X \overset{e}{\to} X$, and it's nondegenerate if none of them are the identity. They are:

$\begin{align*} (i,r,i), (i,e,r), (i,e,e), (r,i,r), (r,i,e), (e,r,i), (e,e,r), (e,e,e) \end{align*}$

for a total of 8.


Side note: I personally find the new definition both much cleaner and much more transparent. I'm still puzzled about the motivation of the old definition.

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    $\begingroup$ I asked essentially this question 7 years ago on the nForum, but got no answers: nforum.ncatlab.org/discussion/683/…. $\endgroup$ – Mike Shulman Aug 16 '17 at 6:24
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    $\begingroup$ My understanding is that Lurie initially did not notice that $Idem$ is really a 1-category and so went with the more complicated definition, but he recently updated HTT with the much more streamlined description we have now. I'll try to cook up a proof of the equivalence of the two definitions today. $\endgroup$ – Denis Nardin Aug 16 '17 at 7:56
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    $\begingroup$ I can (sort of) confirm @DenisNardin's version of events. A few years ago reading that section of Higher Topos Theory I convinced myself that Idem and Idem^+ were nerves of ordinary categories. I asked Jacob about it and he said something like "I think that's right. I should rewrite that whole section some day." Then a few years later Gal Dor's question prompted him to do it. $\endgroup$ – Omar Antolín-Camarena Aug 16 '17 at 17:31
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A bijection of simplices is not too hard. Just note that a chain of composable non-identity morphisms in $A^{\overset{i}{\to}}_{\underset{r}{\leftarrow}} X \overset{e}{\to} X$ can be recovered from the list of morphisms, or from the list of objects which can be specified as indices of $X$'s. So the 3-simplex example $A \overset{i}{\to} X \overset{e}{\to} X \overset{r}{\to} A$ corresponds to $(i,e,r)$ and $\{1,2\}$.

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    $\begingroup$ Oh, I see. And more generally for a degenerate simplex, you label the points inside partitions with $X$, outside the partitions with $A$. To fill in the morphisms, you just need to decide where $e$ should go versus $\mathrm{id}_X$. The answer will be that if two adjacent $X$'s are in the same partition, put $\mathrm{id}_X$ between them; if they're in distinct neighboring partitions, but $e$ between them. This is an isomorphism of simplicial sets. $\endgroup$ – Tim Campion Aug 16 '17 at 15:46

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