nLab idempotent complete (infinity,1)-category




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 ee=ee\circ e = e, we need an equivalence eeee\circ e \simeq e, together with higher coherence data saying that, for instance, the two derived equivalences eeeee\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 (,1)(\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 (,1)(\infty,1)-category can have all finite limits without being idempotent-complete.



Let Idem +Idem^+ be the nerve of the free 1-category containing a retraction, with e:XXe:X\to X the idempotent, r:XYr:X\to Y the retraction, and i:YXi:Y\to X the section (and e=ire = i r and ri=1 Yr i = 1_Y). Let IdemIdem be the similar nerve of the free 1-category containing an idempotent, which is the full sub-simplicial set of Idem +Idem^+ spanned by the object XX. Let RetRet be the image in Idem +Idem^+ of the 2-simplex Δ 2Idem +\Delta^2 \to Idem^+ exhibiting the composite ri=1 Yr i = 1_Y; thus RetRet is also the quotient of Δ 2\Delta^2 that collapses the 1-face to a point.

(Lurie, p.304)


Let C be an \infty-category, incarnated as a quasi-category.

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

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

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

(Lurie, p.304)


An idempotent F:IdemCF \colon Idem \to C is effective if it extends to a map Idem +CIdem^+ \to C.

(Lurie, above corollary


An idempotent diagram F:IdemCF \colon Idem \to C is effective precisely if it admits an (∞,1)-limit, equivalently if it admits an (∞,1)-colimit.

By (Lurie, lemma


CC is called an idempotent complete (,1)(\infty,1) if every idempotent is effective.

(Lurie, above corollary


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,


For CC a small (∞,1)-category and κ\kappa a regular cardinal, the (∞,1)-Yoneda embedding CCInd κ(C)C \to C' \hookrightarrow Ind_\kappa(C) with CC' the full subcategory on κ\kappa-compact objects exhibits CC' as the idempotent completion of CC.

This is HTT, lemma

Coherent vs incoherent idempotents

We may also ask how idempotent-completeness of CC is related to that of its homotopy category hCh C. An idempotent in hCh C is an “incoherent idempotent” in CC, i.e. a map e:XXe:X\to X such that ee 2e \sim e^2, but without any higher coherence conditions. In this case we have:


(HA Lemma If CC is stable, then CC is idempotent-complete (i.e. every coherent idempotent is effective) if and only if hCh C is (as a 1-category).

However, if CC is not stable, this is false. The following counterexample in ∞Gpd is constructed in Warning of HA. Let λ:GG\lambda : G \to G be an injective but non-bijective group homomorphism such that λ\lambda and λ 2\lambda^2 are conjugate. (One such is obtained by letting GG be the group of endpoint-fixing homeomorphisms of [0,1][0,1], with λ(g)\lambda(g) acting as a scaled version of gg on [0,12][0,\frac 1 2] and the identity on [12,1][\frac 1 2,1]. Then λ(g)h=hλ 2(g)\lambda(g) \circ h = h \circ \lambda^2(g) for any hh such that h(t)=2th(t) = 2t for t[0,14]t \in [0,\frac 1 4].)

Then Bλ:BGBGB\lambda : B G \to B G is homotopic to Bλ 2B\lambda^2, hence idempotent in the homotopy category. If it could be lifted to a coherent idempotent, then the colimit of the diagram

BGBλBGBλBGB G \xrightarrow{B \lambda} B G \xrightarrow{B \lambda} BG \to \cdots

would be its splitting, and hence the map BGcolim(BGBλBG)B G \to \colim (B G\xrightarrow{B \lambda} B G \to\cdots) would have a section. Passing to fundamental groups, Gcolim(GλG)G \to \colim (G\xrightarrow{\lambda} G \to\cdots) would also have a section; but this is impossible as λ\lambda is injective but not surjective.

However, we do have the following:


(HA Lemma in older versions, HTT Proposition in newer versions) A morphism ee in an (,1)(\infty,1)-category CC is idempotent (i.e. e:Δ 1Ce:\Delta^1 \to C extends to IdemIdem) if and only if there is a homotopy h:ee 2h : e \sim e^2 such that h11hh\circ 1 \sim 1\circ h.

In other words, an incoherent idempotent can be fully coherentified as soon as it admits one additional coherence datum.

However, there may be multiple inequivialent ways of doing so.


Section 4.4.5 of

Formalization in homotopy type theory:

Last revised on June 22, 2023 at 21:49:41. See the history of this page for a list of all contributions to it.