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Karoubi envelope

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Idea

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.

Definition

There is an

that characterizes idempotent completions. In particular the idempotent completion always exists 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.

Abstract definition

Definition

For π’ž\mathcal{C} a category, a functor π’žβ†’π’žΛœ\mathcal{C} \to \tilde \mathcal{C} exhibits π’žΛœ\tilde \mathcal{C} as an idempotent completion of π’ž\mathcal{C} if

See e.g. (Lurie, def. 5.1.4.1).

Lemma

For a fully faithful embedding i:π’žβ†’π’Ÿi \colon \mathcal{C} \to \mathcal{D} to exhibit an idempotent(-splitting) completion of π’ž\mathcal{C}, it suffices that

  • i(p)i(p) splits in π’Ÿ\mathcal{D} for every idempotent pp in π’ž\mathcal{C}, and

  • every object in π’Ÿ\mathcal{D} is the retract of an object in π’ž\mathcal{C} under ii.

Proof

We must show that these conditions imply that every idempotent e:Dβ†’De \colon D \to D in π’Ÿ\mathcal{D} splits. Write DD as a retract of some i(C)i(C), say r:i(C)β†’Dr: i(C) \to D with right inverse ss (rs=1 Dr s = 1_D). Then p=ser:i(C)β†’i(C)p = s e r \colon i(C) \to i(C) is idempotent, and we may split pp, say as p=σπp = \sigma \pi with πσ=1 E\pi \sigma = 1_E for some EE. We claim that the pair

πs:D→E,rσ:E→D\pi s \colon D \to E, \qquad r \sigma \colon E \to D

provides a splitting of ee. Certainly we have

(rσ)(πs)=rsers=e,(r \sigma)(\pi s) = r s e r s = e,

and we also have

Οƒ(Ο€s)(rΟƒ)=psrΟƒ=sersrΟƒ=serΟƒ=pΟƒ=σπσ=Οƒ\sigma(\pi s)(r \sigma) = p s r \sigma = s e r s r \sigma = s e r \sigma = p \sigma = \sigma \pi \sigma = \sigma

whence

(Ο€s)(rΟƒ)=πσ(Ο€s)(rΟƒ)=πσ=1 E,(\pi s)(r \sigma) = \pi \sigma (\pi s)(r \sigma) = \pi \sigma = 1_E,

as desired.

In components

Let CC be a category. We give an elementary construction of the Karoubi envelope CΒ―\bar{C} which formally splits idempotents in CC.

The objects of CΒ―\bar{C} are pairs (c,e:cβ†’c)(c, e: c \to c) where ee is an idempotent on an object cc of CC. Morphisms (c,e)β†’(d,f)(c, e) \to (d, f) are morphisms Ο•:cβ†’d\phi: c \to d in CC such that fβˆ˜Ο•=Ο•=Ο•βˆ˜ef \circ \phi = \phi = \phi \circ e. NB: the identity on (c,e)(c, e) in CΒ―\bar{C} is the morphism e:cβ†’ce: c \to c.

There is a functor

E:C→C¯E: C \to \bar{C}

which maps an object cc to (c,1 c)(c, 1_c). This functor is full and faithful: it fully embeds CC in C¯\bar{C}. If e:c→ce: c \to c is an idempotent in CC, then in C¯\bar{C} there are maps

p:(c,1 c)β†’(c,e),j:(c,e)β†’(c,1 c),p: (c, 1_c) \to (c, e), \, j: (c, e) \to (c, 1_c),

both given by e:cβ†’ce: c \to c. It is clear that p∘jp \circ j is the identity e:(c,e)β†’(c,e)e: (c, e) \to (c, e), and that j∘pj \circ p is the idempotent E(e):E(c)β†’E(c)E(e): E(c) \to E(c). Thus the pair (p,j)(p, j) formally splits the idempotent e:cβ†’ce: c \to c. The same argument shows that every idempotent Ο•:(c,e)β†’(c,e)\phi: (c, e) \to (c, e) in CΒ―\bar{C} splits. Actually this formal construction does more: it gives a choice of splitting for every idempotent.

Let DD be any category in which every idempotent h:d→dh: d \to d has a chosen splitting (p h:d→d h,j h:d h→d)(p_h: d \to d_h, j_h: d_h \to d) (using identities to split identities), and let F:C→DF: C \to D be a functor. Define an extension

FΒ―:CΒ―β†’D\bar{F}: \bar{C} \to D

by sending (c,e:c→c)(c, e: c \to c) to the underlying object F(c) F(e)F(c)_{F(e)} of the splitting of F(e):F(c)→F(c)F(e): F(c) \to F(c) in DD. For morphisms ϕ:(c,e)→(c′,e′)\phi: (c, e) \to (c', e'), define F¯(ϕ)\bar{F}(\phi) to be the composite

F(c) F(e)β†’F(j F(e))F(c)β†’F(Ο•)F(cβ€²)β†’F(p F(eβ€²))F(cβ€²) F(eβ€²)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 FΒ―\bar{F} is the unique extension of FF which preserves chosen splittings. Thus the Karoubi envelope is universal among functors from CC into categories DD in which every idempotent has a chosen splitting.

If DD 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 FΒ―\bar{F} depends on how we do this but is unique up to unique natural isomorphism. Alternatively, we can define FΒ―\bar{F} as an anafunctor; then no AC is needed, and we still have FΒ―\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.)

Under the Yoneda embedding

Proposition

For π’ž\mathcal{C} a small category, write PSh(π’ž)PSh(\mathcal{C}) for its category of presheaves and write π’žΛœβ†ͺPSh(π’ž)\tilde \mathcal{C} \hookrightarrow PSh(\mathcal{C}) for the full subcategory on those presheaves which are retracts of objects in π’ž\mathcal{C}, under the Yoneda embedding. Then the Yoneda embedding

π’žβ†’π’žΛœ \mathcal{C} \to \tilde \mathcal{C}

exhibits π’žΛœ\tilde \mathcal{C} as the idempotent completion of π’ž\mathcal{C}.

For instance (Lurie, proof of prop. 5.1.4.2).

Properties

Finality of the completion

Definition

A functor π’žβ†’π’žΛœ\mathcal{C}\to \tilde \mathcal{C} exhibiting an idempotent completion, def. 1, is a final functor.

For instance (Lurie, lemma 5.1.4.6).

Monadicity over semicategories

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

Examples

The category of smooth manifolds

Let ManMan be the category of smooth manifolds and smooth maps, where by a β€œsmooth manifold”, we mean a finite-dimensional, second-countable, Hausdorff, C ∞C^\infty manifold without boundary. Let i:Openβ†ͺMani: Open \hookrightarrow Man be the full subcategory whose objects are the open subspaces of finite-dimensional Cartesian spaces.

Theorem

The subcategory i:Openβ†ͺMani: Open \hookrightarrow Man exhibits ManMan as an idempotent-splitting completion of OpenOpen.

Proof

By lemma 1, it suffices to prove that

  • Every smooth manifold is a smooth retract of an open set in Euclidean space;

  • If p:Uβ†’Up : U \to U is a smooth idempotent on an open set UβŠ†β„ nU \subseteq \mathbb{R}^n, then the subset Fix(p)β†ͺUFix(p) \hookrightarrow U is an embedded submanifold.

For the first statement, we use the fact that any manifold MM can be realized as a closed submanifold of some ℝ n\mathbb{R}^n, and every closed submanifold has a tubular neighborhood UβŠ†β„ nU \subseteq \mathbb{R}^n. In this case UU carries a structure of vector bundle over MM in such a way that the inclusion Mβ†ͺUM \hookrightarrow U is identified with the zero section, so that the bundle projection Uβ†’MU \to M provides a retraction, with right inverse given by the zero section.

For the second statement, assume that the origin 00 is a fixed point of pp, and let T 0(U)≅ℝ nT_0(U) \cong \mathbb{R}^n be its tangent space (observe the presence of a canonical isomorphism to ℝ n\mathbb{R}^n). Thus we have idempotent linear maps dp(0),Idβˆ’dp(0):T 0(U)β†’T 0(U)d p(0), Id-d p(0): T_0(U) \to T_0(U) where the latter factors through the inclusion kerdp(0)β†ͺT 0(U)\ker \; d p(0) \hookrightarrow T_0(U) via a projection map Ο€:T 0(U)β†’kerdp(0)\pi: T_0(U) \to \ker \; d p(0). We have a map f:U→ℝ nf: U \to \mathbb{R}^n that takes x∈Ux \in U to xβˆ’p(x)x - p(x); let gg denote the composite

Uβ†’fℝ nβ‰…T 0(U)β†’Ο€kerdp(0).U \stackrel{f}{\to} \mathbb{R}^n \cong T_0(U) \stackrel{\pi}{\to} \ker\; d p(0).

Now we make some easy observations:

  1. Fix(p)βŠ†g βˆ’1(0)Fix(p) \subseteq g^{-1}(0).

  2. The map p:Uβ†’Up: U \to U restricts to a map p:g βˆ’1(0)β†’g βˆ’1(0)p: g^{-1}(0) \to g^{-1}(0), by idempotence of pp.

  3. The derivative dg(0):T 0(U)β†’T 0(kerdp(0))β‰…kerdp(0)d g(0): T_0(U) \to T_0(\ker \; d p(0)) \cong \ker \; d p(0) is Ο€\pi again since Idβˆ’dp(0)Id - d p(0) is idempotent. Thus dg(0)d g(0) has full rank (mm say), and so the restriction of gg to some neighborhood VV has 00 as a regular value, and g βˆ’1(0)∩Vg^{-1}(0) \cap V is a manifold of dimension mm by the implicit function theorem. The tangent space T 0(g βˆ’1(0)∩V)T_0(g^{-1}(0) \cap V) is canonically identified with im(dp(0))im(d p(0)).

  4. There are smaller neighborhoods Vβ€³βŠ†Vβ€²βŠ†VV'' \subseteq V' \subseteq V so that pp restricts to maps p 1,p 2p_1, p_2 as in the following diagram (i,iβ€²,iβ€³i, i', i'' are inclusion maps, all taking a domain element xx to itself):

    g βˆ’1(0)∩Vβ€³ β†ͺiβ€³ g βˆ’1(0) p 2↓ ↓ p g βˆ’1(0)∩Vβ€² β†ͺiβ€² g βˆ’1(0) p 1↓ ↓ p g βˆ’1(0)∩V β†ͺi g βˆ’1(0)\array{ g^{-1}(0) \cap V'' & \stackrel{i''}{\hookrightarrow} & g^{-1}(0) \\ _\mathllap{p_2} \downarrow & & \downarrow _\mathrlap{p} \\ g^{-1}(0) \cap V' & \stackrel{i'}{\hookrightarrow} & g^{-1}(0) \\ _\mathllap{p_1} \downarrow & & \downarrow _\mathrlap{p} \\ g^{-1}(0) \cap V & \stackrel{i}{\hookrightarrow} & g^{-1}(0) }

    and such that p 1,p 2p_1, p_2 are diffeomorphisms by the inverse function theorem (noting here that dp i(0):im(dp(0))β†’im(dp(0))d p_i(0): im(d p(0)) \to im(d p(0)) is the identity map, by idempotence of pp).

  5. Letting q:g βˆ’1(0)∩Vβ€²β†’g βˆ’1(0)∩Vβ€³q: g^{-1}(0) \cap V' \to g^{-1}(0) \cap V'' denote the smooth inverse to p 2p_2, we calculate iβ€²=p∘iβ€³βˆ˜qi' = p \circ i'' \circ q, and

    ip 1=piβ€²=ppiβ€³q=piβ€³q=iβ€²,i p_1 = p i' = p p i''q = p i'' q = i',

    so that p 1(x)=xp_1(x) = x for every x∈g βˆ’1(0)∩Vβ€²x \in g^{-1}(0) \cap V'. Hence g βˆ’1(0)∩Vβ€²βŠ†Fix(p)g^{-1}(0) \cap V' \subseteq Fix(p).

From all this it follows that Fix(p)∩Vβ€²=g βˆ’1(0)∩Vβ€²Fix(p) \cap V' = g^{-1}(0) \cap V', meaning Fix(p)Fix(p) is locally diffeomorphic to ℝ m\mathbb{R}^m, and so Fix(p)Fix(p) is an embedded submanifold of ℝ n\mathbb{R}^n.

Remark

Lawvere comments on this fact as follows: β€œFor example, if C\mathbf{C} is the category of all smooth maps between all open subsets of all Euclidean spaces, then CΒ―\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 ”basicβ€œ 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 AA in 3-space EE which contains it but not its center (taken to be 00) and the smooth idempotent endomap of AA given by e(x)=x/|x|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 eβ€²e' on the tangent bundle AΓ—VA \times V of AA (VV being the vector space of translations of EE) which is obtained by differentiating ee. The same for cohomology groups, etc.”

References

A classical account is for instance in

  • Francis Borceux and D. Dejean, Cauchy completion in category theory Cahiers Topologie GΓ©om. DiffΓ©rentielle CatΓ©goriques, 27:133–146, (1986) (numdam)

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

  • Paul Balmer, Marco Schlichting, Idempotent completion of triangulated categories (pdf)

The proof that idempotents split in the category of manifolds was adapted from this MO answer:

  • Zack (http://mathoverflow.net/users/300/zack), Idempotents split in category of smooth manifolds?, URL (version: 2014-04-06): http://mathoverflow.net/q/162556 (web)

The accompanying remark of Lawvere appears on page 267 of

  • F. William Lawvere, Qualitative distinctions between some toposes of generalized graphs, Contemporary Mathematics 92 (1989), 261-299. (pdf)

Revised on June 15, 2014 05:59:05 by Todd Trimble (67.81.95.215)