idempotent complete (infinity,1)-category



(,1)(\infty,1)-Category theory



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 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.


Section 4.4.5 of

Formalization in homotopy type theory:

Revised on August 16, 2017 19:02:20 by Mike Shulman (