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.
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