nLab
idempotent complete (infinity,1)-category

Idea

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 $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-category can have all finite limits without being idempotent-complete.

Definition

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Properties

The following properties generalize those of idempotent-complete 1-categories.

Theorem

A small (∞,1)-category is idempotent-complete if and only if it is accessible.

Proof

This is HTT, 5.5.3.6.

Theorem

For $C$ a small (∞,1)-category and $\kappa$ a regular cardinal, the (∞,1)-Yoneda embedding $C\to C\prime \hookrightarrow {\mathrm{Ind}}_{\kappa }(C)$ with $C\prime$ the full subcategory on $\kappa$-compact objects exhibits $C\prime$ as the idempotent completion of $C$.

Proof

This is HTT, lemma 5.4.2.4.

References

Section 4.4.5 of

• Jacob Lurie, Higher Topos Theory