Definition

Let for a linearly ordered set $J$ the simplicial set ${\Delta }^J:=N(J)$ be defined to be the nerve of $J$; i.e. ${\Delta }^J$ is given in degree $n$ given by nondecreasing maps $\{0,...,n\}\to J$.

The simplicial set ${\mathrm{Idem}}^{+}$ is defined as follows: for every nonempty, finite, linearly ordered set J, $\mathrm{sSet}({\Delta }^J,{\mathrm{Idem}}^{+})$ is the set of pairs $(J_0,\sim )$, where $J_0\subseteq J$ and $\sim$ is an equivalence relation on $J_0$ which satisfies the following condition:

• Let $i\le j\le k$ be elements of $J$ such that $i,k\in J_0$ , and $i\sim k$. Then $j\in J_0$ , and $i\sim j\sim k$.

Let $\mathrm{Idem}$ denote the simplicial subset of ${\mathrm{Idem}}^{+}$, corresponding to those pairs $(J_0,\sim )$ such that $J=J_0$ . Let $\mathrm{Ret}\subseteq {\mathrm{Idem}}^{+}$ denote the simplicial subset corresponding to those pairs $(J_0,\sim )$ such that the quotient $J_0/\sim$ has at most one element.

Definition

Let C be an $\mathrm{\infty }$-category, incarnated as a quasi-category.

1. An idempotent morphism in C is a map of simplicial sets $\mathrm{Idem}\to C$. We will refer to $\mathrm{Fun}(\mathrm{Idem},C)$ as the $(\mathrm{\infty },1)$-category of idempotents in $C$.

2. A weak retraction diagram in $C$ is a homomorphism of simplicial sets $\mathrm{Ret}\to C$. We refer to $\mathrm{Fun}(\mathrm{Ret},C)$ as the $(\mathrm{\infty },1)$-category of weak retraction diagrams in $C$.

3. A strong retraction diagram in $C$ is a map of simplicial sets ${\mathrm{Idem}}^{+}\to C$. We will refer to $\mathrm{Fun}(\mathrm{Idem}+,C)$ as the $(\mathrm{\infty },1)$-category of strong retraction diagrams in $C$.

Definition

An idempotent $F:\mathrm{Idem}\to C$ is effective if it extends to a map ${\mathrm{Idem}}^{+}\to C$.

Definition

$C$ is called an idempotent complete $(\mathrm{\infty },1)$ if every idempotent is effective.