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 , we need an equivalence , together with higher coherence data saying that, for instance, the two derived equivalences 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 -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 -category can have all finite limits without being idempotent-complete.
Let denote the simplicial subset of , corresponding to those pairs such that . Let denote the simplicial subset corresponding to those pairs such that the quotient has at most one element.
Let C be an -category, incarnated as a quasi-category.
An idempotent morphism in C is a map of simplicial sets . We will refer to as the -category of idempotents in .
A strong retraction diagram in is a map of simplicial sets . We will refer to as the -category of strong retraction diagrams in .
An idempotent is effective if it extends to a map .
By (Lurie, lemma 18.104.22.168).
is called an idempotent complete if every idempotent is effective.
The following properties generalize those of idempotent-complete 1-categories.
This is HTT, 22.214.171.124.
This is HTT, lemma 126.96.36.199.
Section 4.4.5 of