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idempotent complete (infinity,1)-category

Context

Idempotents

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

Contents

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 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.

Definition

Definition

Let for a linearly ordered set JJ the simplicial set Δ J:=N(J)\Delta^J:=N(J) be defined to be the nerve of JJ; i.e. Δ J\Delta^J is given in degree nn given by nondecreasing maps {0,...,n}J\{0,...,n\}\to J.

The simplicial set Idem +Idem^+ is defined as follows: for every nonempty, finite, linearly ordered set J, sSet(Δ J,Idem +)sSet(\Delta^J,Idem^+) is the set of pairs (J 0,)(J_0,\sim), where J 0JJ_0\subseteq J and \sim is an equivalence relation on J 0J_0 which satisfies the following condition:

  • Let ijki \le j \le k be elements of JJ such that i,kJ 0i, k \in J_0 , and iki \sim k. Then jJ 0j \in J_0 , and ijki \sim j \sim k.

Let IdemIdem denote the simplicial subset of Idem +Idem^+, corresponding to those pairs (J 0,)(J_0 , \sim) such that J=J 0J = J_0 . Let RetIdem +Ret \subseteq Idem^+ denote the simplicial subset corresponding to those pairs (J 0,)(J_0 , \sim) such that the quotient J 0/J_0 / \sim has at most one element.

(Lurie, 4.4.5.2 p.249)

Definition

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, 4.4.5.4 p.250)

Definition

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

(Lurie, above corollary 4.4.5.14)

Proposition

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 4.3.2.13).

Definition

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

(Lurie, above corollary 4.4.5.14)

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.

This is HTT, 5.4.3.6.

Theorem

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 5.4.2.4.

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:

Theorem

(HA Lemma 1.2.4.6) 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 1.2.4.8 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:

Theorem

(HA Lemma 7.3.5.14) 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.

References

Section 4.4.5 of

Formalization in homotopy type theory:

Revised on December 8, 2014 18:38:55 by Urs Schreiber (87.183.144.209)