The Karoubi envelope or idempotent completion of a category is the universal enlargement of the category with the property that every idempotent is a split idempotent. This is the Set-enriched version of the more general notion of Cauchy completion of an enriched category.
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.
There is an
that characterizes idempotent completions. In particular the idempotent completion always exists and is unique up to equivalence of categories. Explicit constructions include:
For more constructions and equivalent characterizations see at Cauchy complete category in the section In ordinary category theory.
For $\mathcal{C}$ a category, a functor $\mathcal{C} \to \tilde \mathcal{C}$ exhibits $\tilde \mathcal{C}$ as an idempotent completion of $\mathcal{C}$ if
$\tilde \mathcal{C}$ is an idempotent complete category;
$\mathcal{C} \to \tilde \mathcal{C}$ is a full and faithful functor;
every object in $\tilde \mathcal{C}$ is the retract of an object in $\mathcal{C}$, under this embedding.
See e.g. (Lurie, def. 5.1.4.1).
For a fully faithful embedding $i \colon \mathcal{C} \to \mathcal{D}$ to exhibit an idempotent(-splitting) completion of $\mathcal{C}$, it suffices that
$i(p)$ splits in $\mathcal{D}$ for every idempotent $p$ in $\mathcal{C}$, and
every object in $\mathcal{D}$ is the retract of an object in $\mathcal{C}$ under $i$.
We must show that these conditions imply that every idempotent $e \colon D \to D$ in $\mathcal{D}$ splits. Write $D$ as a retract of some $i(C)$, say $r: i(C) \to D$ with right inverse $s$ ($r s = 1_D$). Then $p = s e r \colon i(C) \to i(C)$ is idempotent, and we may split $p$, say as $p = \sigma \pi$ with $\pi \sigma = 1_E$ for some $E$. We claim that the pair
provides a splitting of $e$. Certainly we have
and we also have
whence
as desired.
Let $C$ be a category. We give an elementary construction of the Karoubi envelope $\bar{C}$ which formally splits idempotents in $C$.
The objects of $\bar{C}$ are pairs $(c, e: c \to c)$ where $e$ is an idempotent on an object $c$ of $C$. Morphisms $(c, e) \to (d, f)$ are morphisms $\phi: c \to d$ in $C$ such that $f \circ \phi = \phi = \phi \circ e$. NB: the identity on $(c, e)$ in $\bar{C}$ is the morphism $e: c \to c$.
There is a functor
which maps an object $c$ to $(c, 1_c)$. This functor is full and faithful: it fully embeds $C$ in $\bar{C}$. If $e: c \to c$ is an idempotent in $C$, then in $\bar{C}$ there are maps
both given by $e: c \to c$. It is clear that $p \circ j$ is the identity $e: (c, e) \to (c, e)$, and that $j \circ p$ is the idempotent $E(e): E(c) \to E(c)$. Thus the pair $(p, j)$ formally splits the idempotent $e: c \to c$. The same argument shows that every idempotent $\phi: (c, e) \to (c, e)$ in $\bar{C}$ 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 $(p_h: d \to d_h, j_h: d_h \to d)$ (using identities to split identities), and let $F: C \to D$ be a functor. Define an extension
by sending $(c, e: c \to c)$ to the underlying object $F(c)_{F(e)}$ of the splitting of $F(e): F(c) \to F(c)$ in $D$. For morphisms $\phi: (c, e) \to (c', e')$, define $\bar{F}(\phi)$ to be the composite
Then $\bar{F}$ 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 $\bar{F}$ depends on how we do this but is unique up to unique natural isomorphism. Alternatively, we can define $\bar{F}$ as an anafunctor; then no AC is needed, and we still have $\bar{F}$ unique up to unique natural isomorphism. (It is key here that a splitting of an idempotent is unique up to a coherent isomorphism.)
Essentially the same argument shows that for any $D$ in which idempotents split, the restriction functor $[E, D]: [\bar{C}, D] \to [C, D]$ is an equivalence. The details are spelled out here.
For $\mathcal{C}$ a small category, write $PSh(\mathcal{C})$ for its category of presheaves and write $\tilde \mathcal{C} \hookrightarrow PSh(\mathcal{C})$ for the full subcategory on those presheaves which are retracts of objects in $\mathcal{C}$, under the Yoneda embedding. Then the Yoneda embedding
exhibits $\tilde \mathcal{C}$ as the idempotent completion of $\mathcal{C}$.
For instance (Lurie, proof of prop. 5.1.4.2).
A functor $\mathcal{C}\to \tilde \mathcal{C}$ exhibiting an idempotent completion, def. 1, is a final functor.
For instance (Lurie, lemma 5.1.4.6).
The functor that forms idempotent completion is the monad induced from the adjunction between categories and semicategories given by the forgetful functor $Cat \to SemiCat$ and its right adjoint. More details on this are at Semicategory - Relation to categories.
Let $Man$ be the category of smooth manifolds and smooth maps, where by a βsmooth manifoldβ, we mean a finite-dimensional, second-countable, Hausdorff, $C^\infty$ manifold without boundary. Let $i: Open \hookrightarrow Man$ be the full subcategory whose objects are the open subspaces of finite-dimensional Cartesian spaces.
The subcategory $i: Open \hookrightarrow Man$ exhibits $Man$ as an idempotent-splitting completion of $Open$.
By lemma 1, it suffices to prove that
Every smooth manifold is a smooth retract of an open set in Euclidean space;
If $p : U \to U$ is a smooth idempotent on an open set $U \subseteq \mathbb{R}^n$, then the subset $Fix(p) \hookrightarrow U$ is an embedded submanifold.
For the first statement, we use the fact that any manifold $M$ can be realized as a closed submanifold of some $\mathbb{R}^n$, and every closed submanifold has a tubular neighborhood $U \subseteq \mathbb{R}^n$. In this case $U$ carries a structure of vector bundle over $M$ in such a way that the inclusion $M \hookrightarrow U$ is identified with the zero section, so that the bundle projection $U \to M$ provides a retraction, with right inverse given by the zero section.
For the second statement, assume that the origin $0$ is a fixed point of $p$, and let $T_0(U) \cong \mathbb{R}^n$ be its tangent space (observe the presence of a canonical isomorphism to $\mathbb{R}^n$). Thus we have idempotent linear maps $d p(0), Id-d p(0): T_0(U) \to T_0(U)$ where the latter factors through the inclusion $\ker \; d p(0) \hookrightarrow T_0(U)$ via a projection map $\pi: T_0(U) \to \ker \; d p(0)$. We have a map $f: U \to \mathbb{R}^n$ that takes $x \in U$ to $x - p(x)$; let $g$ denote the composite
Now we make some easy observations:
$Fix(p) \subseteq g^{-1}(0)$.
The map $p: U \to U$ restricts to a map $p: g^{-1}(0) \to g^{-1}(0)$, by idempotence of $p$.
The derivative $d g(0): T_0(U) \to T_0(\ker \; d p(0)) \cong \ker \; d p(0)$ is $\pi$ again since $Id - d p(0)$ is idempotent. Thus $d g(0)$ has full rank ($m$ say), and so the restriction of $g$ to some neighborhood $V$ has $0$ as a regular value, and $g^{-1}(0) \cap V$ is a manifold of dimension $m$ by the implicit function theorem. The tangent space $T_0(g^{-1}(0) \cap V)$ is canonically identified with $im(d p(0))$.
There are smaller neighborhoods $V'' \subseteq V' \subseteq V$ so that $p$ restricts to maps $p_1, p_2$ as in the following diagram ($i, i', i''$ are inclusion maps, all taking a domain element $x$ to itself):
and such that $p_1, p_2$ are diffeomorphisms by the inverse function theorem (noting here that $d p_i(0): im(d p(0)) \to im(d p(0))$ is the identity map, by idempotence of $p$).
Letting $q: g^{-1}(0) \cap V' \to g^{-1}(0) \cap V''$ denote the smooth inverse to $p_2$, we calculate $i' = p \circ i'' \circ q$, and
so that $p_1(x) = x$ for every $x \in g^{-1}(0) \cap V'$. Hence $g^{-1}(0) \cap V' \subseteq Fix(p)$.
From all this it follows that $Fix(p) \cap V' = g^{-1}(0) \cap V'$, meaning $Fix(p)$ is locally diffeomorphic to $\mathbb{R}^m$, and so $Fix(p)$ is an embedded submanifold of $\mathbb{R}^n$.
Lawvere comments on this fact as follows: βFor example, if $\mathbf{C}$ is the category of all smooth maps between all open subsets of all Euclidean spaces, then $\widebar{\mathbf{C}}$ $[$the Karoubi envelope$]$ is the category of all smooth manifolds. This powerful theorem justifies bypassing the complicated considerations of charts, coordinate transformations, and atlases commonly offered as a βbasicβ definition of the concept of manifold. For example the 2-sphere, a manifold but not an open set of any Euclidean space, may be fully specified with its smooth structure by considering any open set $A$ in 3-space $E$ which contains it but not its center (taken to be $0$) and the smooth idempotent endomap of $A$ given by $e(x) = x/{|x|}$. All general constructions (i.e., functors into categories which are Cauchy complete) on manifolds now follow easily (without any need to check whether they are compatible with coverings, etc.) provided they are known on the opens of Euclidean spaces: for example, the tangent bundle on the sphere is obtained by splitting the idempotent $e'$ on the tangent bundle $A \times V$ of $A$ ($V$ being the vector space of translations of $E$) which is obtained by differentiating $e$. The same for cohomology groups, etc.β
The category of projective modules over any ring is the Karoubi envelope of its full subcategory of free modules.
The category of (locally trivial, finite dimensional) vector bundles over any fixed paracompact space is the Karoubi envelope of its full subcategory of trivial bundles.
Both examples are related by the Serre-Swan theorem. In fact both these facts together with the observation that the global sections functor is an equivalence from the category trivial bundles over $X$ to the category of free modules over $C(X)$ prove the Serre-Swan theorem itself.
A classical account is for instance in
For more classical references see the references at Cauchy complete category.
Karoubi envelopes for (β,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 Neemanβs about Karoubi closure and compact generators.
The Karoubi envelope for the additive case (see also additive envelope) is covered at
Discussion for triangulated categories is in
The proof that idempotents split in the category of manifolds was adapted from this MO answer:
Which provides a solution to exercise 3.21 in
The accompanying above remark of Lawvere appears on page 267 of
A comparison of the Karoubi envelope to other completions can be found here:
Formalization in homotopy type theory: