In the published version of HTT (Def 4.4.5.2) and [on the nlab](https://ncatlab.org/nlab/show/idempotent+complete+%28infinity%2C1%29-category) one finds one definition of a split homotopy coherent idempotent corepresented by a quasicategory $Idem^+\_\mathrm{old}$. A few months ago, Gal Dor [found](https://mathoverflow.net/questions/266384/is-the-category-idem-filtered) 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?




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**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.




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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.



Did Lurie's model of a homotopy coherent idempotent change?
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 $\{1,2\}$.


