nLab idempotent complete (infinity,1)-category

Contents

Context

$(\infty,1)$-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

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 $e\circ e = e$, 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 $(\infty,1)$-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 $(\infty,1)$-category can have all finite limits without being idempotent-complete.

Definition

Definition

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

Definition

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

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

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

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

Definition

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

Proposition

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

By (Lurie, lemma 4.3.2.13).

Definition

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

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

This is HTT, lemma 5.4.2.4.

Coherent vs incoherent idempotents

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

Theorem

(HA Lemma 1.2.4.6) If $C$ is stable, then $C$ is idempotent-complete (i.e. every coherent idempotent is effective) if and only if $h C$ is (as a 1-category).

However, if $C$ is not stable, this is false. The following counterexample in ∞Gpd is constructed in Warning 1.2.4.8 of HA. Let $\lambda : G \to G$ be an injective but non-bijective group homomorphism such that $\lambda$ and $\lambda^2$ are conjugate. (One such is obtained by letting $G$ be the group of endpoint-fixing homeomorphisms of $[0,1]$, with $\lambda(g)$ acting as a scaled version of $g$ on $[0,\frac 1 2]$ and the identity on $[\frac 1 2,1]$. Then $\lambda(g) \circ h = h \circ \lambda^2(g)$ for any $h$ such that $h(t) = 2t$ for $t \in [0,\frac 1 4]$.)

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

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

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

However, we do have the following:

Theorem

(HA Lemma 7.3.5.14 in older versions, HTT Proposition 4.4.5.20 in newer versions) A morphism $e$ in an $(\infty,1)$-category $C$ is idempotent (i.e. $e:\Delta^1 \to C$ extends to $Idem$) if and only if there is a homotopy $h : e \sim e^2$ such that $h\circ 1 \sim 1\circ h$.

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.

References

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.