equivalences in/of $(\infty,1)$-categories
An ordinary category is idempotent complete, aka Karoubi complete or Cauchy complete , if every idempotent splits. Since the splitting of an idempotent is a limit or colimit of that idempotent, any category with all finite limits or all finite colimits is idempotent complete.
In an (∞,1)-category the idea is the same, except that the notion of idempotent is more complicated. Instead of just requiring that $e\circ e = e$, we need an equivalence $e\circ e \simeq e$, together with higher coherence data saying that, for instance, the two derived equivalences $e\circ e\circ e \simeq e$ are equivalent, and so on up. In particular, being idempotent is no longer a property of a morphism, but structure on it.
It is still true that a splitting of an idempotent in an $(\infty,1)$-category is a limit or colimit of that idempotent (now regarded as a diagram with all its higher coherence data), but this limit is no longer a finite limit; thus an $(\infty,1)$-category can have all finite limits without being idempotent-complete.
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 $Idem^+$ is deﬁned as follows: for every nonempty, ﬁnite, linearly ordered set J, $sSet(\Delta^J,Idem^+)$ 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 $Idem$ denote the simplicial subset of $Idem^+$, corresponding to those pairs $(J_0 , \sim)$ such that $J = J_0$ . Let $Ret \subseteq Idem^+$ denote the simplicial subset corresponding to those pairs $(J_0 , \sim)$ such that the quotient $J_0 / \sim$ has at most one element.
Let C be an $\infty$-category, incarnated as a quasi-category.
An idempotent morphism in C is a map of simplicial sets $Idem \to C$. We will refer to $Fun(Idem, C)$ as the $(\infty,1)$-category of idempotents in $C$.
A weak retraction diagram in $C$ is a homomorphism of simplicial sets $Ret \to C$. We refer to $Fun(Ret, C)$ as the $(\infty,1)$-category of weak retraction diagrams in $C$.
A strong retraction diagram in $C$ is a map of simplicial sets $Idem^+ \to C$. We will refer to $Fun(Idem+, C)$ as the $(\infty,1)$-category of strong retraction diagrams in $C$.
An idempotent $F \colon Idem \to C$ is effective if it extends to a map $Idem^+ \to C$.
(Lurie, above corollary 4.4.5.14)
An idempotent diagram $F \colon Idem \to C$ is effective precisely if it admits an (∞,1)-limit, equivalently if it admits an (∞,1)-colimit.
By (Lurie, lemma 4.3.2.13).
$C$ is called an idempotent complete $(\infty,1)$ if every idempotent is effective.
(Lurie, above corollary 4.4.5.14)
The following properties generalize those of idempotent-complete 1-categories.
A small (∞,1)-category is idempotent-complete if and only if it is accessible.
This is HTT, 5.4.3.6.
For $C$ a small (∞,1)-category and $\kappa$ a regular cardinal, the (∞,1)-Yoneda embedding $C \to C' \hookrightarrow Ind_\kappa(C)$ with $C'$ the full subcategory on $\kappa$-compact objects exhibits $C'$ as the idempotent completion of $C$.
This is HTT, lemma 5.4.2.4.
Section 4.4.5 of