category theory

# Contents

## Idea

The Karoubi envelope of a category is the universal way to ensure that every idempotent is a split idempotent. It is the Set-enriched version of Cauchy completion.

A category in which all idempotents split is called Karoubi complete or Cauchy complete or idempotent-complete. Thus, the Karoubi envelope is a completion operation into such categories.

## Definition

Let $C$ be a category. We give an elementary construction of the Karoubi envelope $\overline{C}$ which formally splits idempotents in $C$.

The objects of $\overline{C}$ are pairs $\left(c,e:c\to c\right)$ where $e$ is an idempotent on an object $c$ of $C$. Morphisms $\left(c,e\right)\to \left(d,f\right)$ are morphisms $\varphi :c\to d$ in $C$ such that $f\circ \varphi =\varphi =\varphi \circ e$. NB: the identity on $\left(c,e\right)$ in $\overline{C}$ is the morphism $e:c\to c$.

There is a functor

$E:C\to \overline{C}$E: C \to \bar{C}

which maps an object $c$ to $\left(c,{1}_{c}\right)$. This functor is full and faithful: it fully embeds $C$ in $\overline{C}$. If $e:c\to c$ is an idempotent in $C$, then in $\overline{C}$ there are maps

$p:\left(c,{1}_{c}\right)\to \left(c,e\right),\phantom{\rule{thinmathspace}{0ex}}j:\left(c,e\right)\to \left(c,{1}_{c}\right),$p: (c, 1_c) \to (c, e), \, j: (c, e) \to (c, 1_c),

both given by $e:c\to c$. It is clear that $p\circ j$ is the identity $e:\left(c,e\right)\to \left(c,e\right)$, and that $j\circ p$ is the idempotent $E\left(e\right):E\left(c\right)\to E\left(c\right)$. Thus the pair $\left(p,j\right)$ formally splits the idempotent $e:c\to c$. The same argument shows that every idempotent $\varphi :\left(c,e\right)\to \left(c,e\right)$ in $\overline{C}$ splits. Actually this formal construction does more: it gives a choice of splitting for every idempotent.

Let $D$ be any category in which every idempotent $h:d\to d$ has a chosen splitting $\left({p}_{h}:d\to {d}_{h},{j}_{h}:{d}_{h}\to d\right)$ (using identities to split identities), and let $F:C\to D$ be a functor. Define an extension

$\overline{F}:\overline{C}\to D$\bar{F}: \bar{C} \to D

by sending $\left(c,e:c\to c\right)$ to the underlying object $F\left(c{\right)}_{F\left(e\right)}$ of the splitting of $F\left(e\right):F\left(c\right)\to F\left(c\right)$ in $D$. For morphisms $\varphi :\left(c,e\right)\to \left(c\prime ,e\prime \right)$, define $\overline{F}\left(\varphi \right)$ to be the composite

$F\left(c{\right)}_{F\left(e\right)}\stackrel{F\left({j}_{F\left(e\right)}\right)}{\to }F\left(c\right)\stackrel{F\left(\varphi \right)}{\to }F\left(c\prime \right)\stackrel{F\left({p}_{F\left(e\prime \right)}\right)}{\to }F\left(c\prime {\right)}_{F\left(e\prime \right)}$F(c)_{F(e)} \overset{F(j_{F(e)})}{\to} F(c) \overset{F(\phi)}{\to} F(c') \overset{F(p_{F(e')})}{\to} F(c')_{F(e')}

Then $\overline{F}$ is the unique extension of $F$ which preserves chosen splittings. Thus the Karoubi envelope is universal among functors from $C$ into categories $D$ in which every idempotent has a chosen splitting.

If $D$ is a category in which every idempotent splits, then we can choose a splitting for each idempotent using the axiom of choice (AC); the extension $\overline{F}$ depends on how we do this but is unique up to unique natural isomorphism. Alternatively, we can define $\overline{F}$ as an anafunctor; then no AC is needed, and we still have $\overline{F}$ unique up to unique natural isomorphism. (It is key here that a splitting of an idempotent is unique up to a coherent isomorphism.)

## Properties

The functors that forms idempotent completion is the monad induced from the adjunction between categories and semicategories given by the forgetful functor $\mathrm{Cat}\to \mathrm{SemiCat}$ and its right adjoint. More details on this are at Semicategory - Relation to categories.

## References

Karoubi envelopes for (∞,1)-categories are discussed in section 4.4.5 of

Some discussion of the stable version is in section 4.1.2 of

and section 2.3 of

In section 3.1.2 of latter are also given references (to Neeman and Lurie) for an important result of Neeman’s about Karoubi closure and compact generators.