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:
See e.g. (Lurie, def. 220.127.116.11).
For a fully faithful embedding to exhibit an idempotent(-splitting) completion of , it suffices that
splits in for every idempotent in , and
every object in is the retract of an object in under .
We must show that these conditions imply that every idempotent in splits. Write as a retract of some , say with right inverse (). Then is idempotent, and we may split , say as with for some . We claim that the pair
provides a splitting of . Certainly we have
and we also have
The objects of are pairs where is an idempotent on an object of . Morphisms are morphisms in such that (or equivalently, such that ). NB: the identity on in is the morphism .
There is a functor
which maps an object to . This functor is full and faithful: it fully embeds in . If is an idempotent in , then in there are maps
both given by . It is clear that is the identity , and that is the idempotent . Thus the pair formally splits the idempotent . The same argument shows that every idempotent in splits. Actually this formal construction does more: it gives a choice of splitting for every idempotent.
Let be any category in which every idempotent has a chosen splitting (using identities to split identities), and let be a functor. Define an extension
by sending to the underlying object of the splitting of in . For morphisms , define to be the composite
Then is the unique extension of which preserves chosen splittings. Thus the Karoubi envelope is universal among functors from into categories in which every idempotent has a chosen splitting.
If 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 depends on how we do this but is unique up to unique natural isomorphism. Alternatively, we can define as an anafunctor; then no AC is needed, and we still have 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 in which idempotents split, the restriction functor is an equivalence. The details are spelled out here.
For a small category, write for its category of presheaves and write for the full subcategory on those presheaves which are retracts of objects in , under the Yoneda embedding. Then the Yoneda embedding
exhibits as the idempotent completion of .
For instance (Lurie, proof of prop. 18.104.22.168).
For instance (Lurie, lemma 22.214.171.124).
The functor that forms idempotent completion is the monad induced from the adjunction between categories and semicategories given by the forgetful functor and its right adjoint. More details on this are at Semicategory - Relation to categories.
Let be the category of smooth manifolds and smooth maps, where by a “smooth manifold”, we mean a finite-dimensional, second-countable, Hausdorff, manifold without boundary. Let be the full subcategory whose objects are the open subspaces of finite-dimensional Cartesian spaces.
The subcategory exhibits as an idempotent-splitting completion of .
By lemma 1, it suffices to prove that
Every smooth manifold is a smooth retract of an open set in Euclidean space;
If is a smooth idempotent on an open set , then the subset is an embedded submanifold.
For the first statement, we use the fact that any manifold can be realized as a closed submanifold of some , and every closed submanifold has a tubular neighborhood . In this case carries a structure of vector bundle over in such a way that the inclusion is identified with the zero section, so that the bundle projection provides a retraction, with right inverse given by the zero section.
For the second statement, assume that the origin is a fixed point of , and let be its tangent space (observe the presence of a canonical isomorphism to ). Thus we have idempotent linear maps where the latter factors through the inclusion via a projection map . We have a map that takes to ; let denote the composite
Now we make some easy observations:
The map restricts to a map , by idempotence of .
The derivative is again since is idempotent. Thus has full rank ( say), and so the restriction of to some neighborhood has as a regular value, and is a manifold of dimension by the implicit function theorem. The tangent space is canonically identified with .
There are smaller neighborhoods so that restricts to maps as in the following diagram ( are inclusion maps, all taking a domain element to itself):
and such that are diffeomorphisms by the inverse function theorem (noting here that is the identity map, by idempotence of ).
Letting denote the smooth inverse to , we calculate , and
so that for every . Hence .
From all this it follows that , meaning is locally diffeomorphic to , and so is an embedded submanifold of .
Lawvere comments on this fact as follows: “For example, if is the category of all smooth maps between all open subsets of all Euclidean spaces, then 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 ”basic“ 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 in 3-space which contains it but not its center (taken to be ) and the smooth idempotent endomap of given by . 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 on the tangent bundle of ( being the vector space of translations of ) which is obtained by differentiating . The same for cohomology groups, etc.”
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 to the category of free modules over 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 (∞,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.
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: