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:**

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

Is there at least a map between them?

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.