# nLab idempotent complete (infinity,1)-category

### Context

#### $\left(\infty ,1\right)$-Category theory

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

## Definition

###### Deﬁnition

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

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

• Let $i\le j\le k$ be elements of $J$ such that $i,k\in {J}_{0}$ , and $i\sim k$. Then $j\in {J}_{0}$ , and $i\sim j\sim k$.

Let $\mathrm{Idem}$ denote the simplicial subset of ${\mathrm{Idem}}^{+}$, corresponding to those pairs $\left({J}_{0},\sim \right)$ such that $J={J}_{0}$ . Let $\mathrm{Ret}\subseteq {\mathrm{Idem}}^{+}$ denote the simplicial subset corresponding to those pairs $\left({J}_{0},\sim \right)$ such that the quotient ${J}_{0}/\sim$ has at most one element.

###### Deﬁnition

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

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

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

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

###### Deﬁnition

An idempotent $F:\mathrm{Idem}\to C$ is effective if it extends to a map ${\mathrm{Idem}}^{+}\to C$.

###### Proposition

An idempotent diagram $F:\mathrm{Idem}\to C$ 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 $\left(\infty ,1\right)$ 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 $C\to C\prime ↪{\mathrm{Ind}}_{\kappa }\left(C\right)$ with $C\prime$ the full subcategory on $\kappa$-compact objects exhibits $C\prime$ as the idempotent completion of $C$.

This is HTT, lemma 5.4.2.4.

## References

Section 4.4.5 of

Revised on November 16, 2012 00:42:53 by Guillaume Brunerie (192.16.204.210)