Idempotents in Monoidal Categories

IDEMPOTENTS IN MONOIDAL CATEGORIES

MITYA BOYARCHENKO AND VLADIMIR DRINFELD

Contents

Introduction 11. Basic properties of closed idempotents 32. Idempotents and idempotent monads 113. Hecke subcategories 144. Idempotents in braided monoidal categories 205. Idempotents in triangulated monoidal categories 21References 23

Introduction

This paper is devoted to an abstract study of idempotents in monoidal categories, buildinga foundation for the definition of character sheaves for unipotent groups [BD06].

Let us fix a monoidal category M with unit object 1. We define a closed (resp., open)

idempotent in M as a morphism 1π−→ e (resp., e

π−→ 1) in M which becomes an isomor-phism after tensoring with e either on the left or on the right. This definition is motivatedby the following example: if M is the category of sheaves of complex vector spaces on atopological space X with the usual tensor product, and ξ : Y ↪→ X is the inclusion of aclosed (resp., open) subspace, then the natural arrow CX −→ ξ!CY (resp., ξ!CY −→ CX),obtained by adjunction from the natural isomorphism CX

∣∣Y∼= CY , is a closed (resp., open)

idempotent in the category M. The definition of a locally closed idempotent, modelling thesheaf ξ!CY in the case where ξ is the inclusion of a locally closed subspace, can be obtainedas a mixture of the first two definitions; it is studied in some detail in Section 5.

In view of the applications we have in mind, most of our paper is devoted to the studyof closed idempotents. (This reflects the fact that coadjoint orbits for a unipotent algebraicgroup are closed.) Besides, every result about closed idempotents can be formally translatedinto a result about open idempotents, since if M is a monoidal category, then its oppositecategory Mop, equipped with the same monoidal structure ⊗, is also a monoidal category,

Both authors were supported by NSF grant DMS-0401164.Address: Department of Mathematics, University of Chicago, Chicago, IL 60637.E-mail : [email protected] (M.B.), [email protected] (V.D.).

1

2 M. BOYARCHENKO AND V. DRINFELD

and an arrow π in M is a closed idempotent if and only if the corresponding arrow in Mop

is an open idempotent.

In Section 1 we study morphisms between closed idempotents, two of the main resultsbeing that a closed idempotent 1

π−→ e is determined by its codomain e up to unique

isomorphism, and if 1π−→ e and 1

π′−→ e′ are two closed idempotents in M, then thereexists at most one morphism f : e −→ e′ in M such that f ◦π = π′. This leads to a naturaldefinition of a partial order on the set of isomorphism classes of closed idempotents in M.Section 2 begins with the important observation that the notion of a closed idempotentis equivalent to the notion of an idempotent monoid, i.e., a monoid whose multiplicationmap is an isomorphism. This allows us to relate our notion of a closed idempotent to thenotion of an idempotent functor studied by Adams [Ad73] and to the equivalent notion ofan idempotent monad studied by Deleanu-Frei-Hilton [DFH1, DFH2].

Let A be an associative unital ring. We say that two idempotents e1, e2 ∈ A are equivalentif e1 = e2e1 and e2 = e1e2. There is a bijection between the set of equivalence classes ofidempotents in A and the set of right ideals I ⊆ A that are direct summands of A as rightA-modules, given by e 7→ eA. Moreover, if e ∈ A is an idempotent, then eAe is a subring ofA which has e as the multiplicative identity, and we have a canonical isomorphism of rings

eAe'−→ Endmod−A(eA). The categorification of these statements, where A is replaced by

a monoidal category M and e is replaced by a closed idempotent 1π−→ e, is explained in

Section 3; in particular, the subring eAe is replaced by the strictly full subcategory eMe ofM consisting of all objects X of M such that e⊗X ⊗ e ∼= X.

With the notation of the previous paragraph, if A is moreover commutative, then the mapA −→ eAe given by a 7→ eae is a ring homomorphism. The categorification of this result isproved in Section 4, namely, we show that if M is braided, then the functor X 7→ e⊗X⊗ ecan be upgraded to a (strong) braided monoidal functor M −→ eMe. We also study theproperties of closed idempotents from the point of view of the braiding; it turns out thatif 1 −→ e is a closed idempotent in M, then the braiding automorphism of e ⊗ e is theidentity, and, moreover, the square of the braiding is the identity on X⊗e for each X ∈M.

In Section 5 we give two definitions of a locally closed idempotent in a monoidal category,and prove that they are equivalent in a triangulated tensor category.

In conclusion, we would like to mention that even though this article contains manynew results, it is written more in the style of an expository paper rather than that ofa research paper. In particular, it is essentially self-contained, and includes recollectionsof some standard material, as well as precise definitions and statements of results takenfrom works by other authors. The rest of the article can be read independently from thisintroduction.

Warning. This is a first and very preliminary draft of our paper. While the main resultsare formulated in detail, many of the proofs are missing. On the other hand, Section 1, onwhich the definition of character sheaves given in [BD06] is based, is complete.

IDEMPOTENTS IN MONOIDAL CATEGORIES 3

1. Basic properties of closed idempotents

1.1. Let M be a monoidal category with unit object 1, and let 1π−→ e be a morphism in

M. We define two full subcategories of M as follows:

πM ={m ∈M

∣∣ π ⊗ idm : 1⊗m −→ e⊗m is an isomorphism}

and

Mπ ={m ∈M

∣∣ idm⊗π : m⊗ 1 −→ m⊗ e is an isomorphism}.

Definition 1.1. We say that π is a closed idempotent in M if e ∈π M∩Mπ. We say that anobject e of M is a closed idempotent if there exists a morphism π : 1→ e which is a closedidempotent.

The second part of the definition may seem like a bad idea; however, later on we will

see that if 1π−→ e and 1

π′−→ e are two closed idempotents with the same codomain e,then there exists a unique morphism σ : e → e such that π′ = σ ◦ π (of course, σ is thennecessarily an isomorphism, since the roles of π and π′ can be interchanged).

1.2. The definition above is motivated in part by the following example. Let X be atopological space, let M be the category of sheaves of C-vector spaces on X with the usual

tensor product, and let Yi

↪→ X denote the inclusion of a closed subspace. If CX and CY

denote the constant sheaves with stalks C on X and Y , respectively, then CX is a unit

object of M, and the natural isomorphism i∗CX'−→ CY gives, by adjunction, a morphism

CX −→ i∗CY , which is easily seen to be a closed idempotent in the category M. Thisexample also explains the terminology.

In Section 5 we will introduce two other types of idempotents (“open” and “locally

closed”); the definition will be such that an arrow 1π−→ e can only denote a closed idem-

potent. In any case, from now on until the rest of this section, the word “idempotent” willmean “closed idempotent”. Also, for the time being, we will only work with idempotents asarrows 1→ e satisfying a certain property, as opposed to idempotents as objects of M.

1.3. Before we proceed with a basic study of idempotents, we need to make an importanttechnical remark. Note that the definition of an idempotent 1

π−→ e, as well as of thesubcategories πM and Mπ, makes no use of the associativity constraint on M. Likewise,most of our results (at least the ones in this section) do not involve the associativity con-straint in their statements. On the other hand, their proofs, of course, make constant useof associativity, in one form or another. Fortunately, it is known that every monoidal cate-gory is equivalent to a strictly associative one (see Theorem 3.6 below). Thus, for the mostpart of these notes, we will only deal with strictly associative monoidal categories, with theunderstanding that all our results that do not involve associativity in their statements carryover automatically to the general case without any modifications.


On the other hand, we do not assume that the unit 1 is strict, i.e., we prefer to keeptrack of the left and right unit isomorphisms. There are two reasons for doing this. First,since the unit object plays such an important role in our definitions, keeping track of it asexplicitly as possible will help us avoid needless confusion. Second, even if we start with astrictly associative and strictly unital monoidal category M, certain “Hecke subcategories”of M (to be defined in Section 3) associated to idempotents of M will not be strictly unitalin general. Thus our approach makes the situation look “more symmetric.”

By contrast, keeping track of all the associativity isomorphisms would only make theproofs of our results much more complicated than the ones given below and obscure ourunderstanding of the subject.

1.4. For future reference, we will now define precisely the type of categories we will beworking with. This avoids all possible confusion as to “how strict” our monoidal structuresare, and also makes this text essentially self-contained. Other basic definitions (monoidalfunctors, braidings, etc.) will be recalled when they are needed. As a general reference wehave used Chapters VII and XI of [McL98].

Definition 1.2. A strictly associative monoidal category is a 5-tuple M = (M,⊗,1, λ, ρ)consisting of a category M, a bifunctor ⊗ : M×M −→ M which is strictly associativein the obvious sense (both on objects and on morphisms), a distinguished object 1 ∈ M(called the unit object), and two collections of isomorphisms,

λm : 1⊗m'−→ m and ρm : m⊗ 1

'−→ m,

for all objects m ∈ M, both of which are functorial with respect to m, these data beingsubject to the following conditions:

(i) the two isomorphisms

ida⊗λc : a⊗ 1⊗ c'−→ a⊗ c and ρa ⊗ idc : a⊗ 1⊗ c

'−→ a⊗ c

agree for all a, c ∈M, and(ii) we have

λ1 = ρ1 : 1⊗ 1'−→ 1.

These axioms imply also:

(iii) λb⊗c = λb ⊗ idc : 1⊗ b⊗ c'−→ b⊗ c for all b, c ∈M and

(iv) ρa⊗b = ida⊗ρb : a⊗ b⊗ 1'−→ a⊗ b for all a, b ∈M.

We refer the reader to op. cit., §VII.1 for more details.

1.5. Until the end of this section we fix, once and for all, a strictly associative monoidalcategory M. We begin our analysis of idempotents with two simple lemmas.

Lemma 1.3. Let 1π−→ e be an idempotent in M. The following conditions are equivalent

for an object m of M:

(i) m ∈π M (resp., m ∈Mπ);


(ii) m ∼= e⊗m (resp., m ∼= m⊗ e);(iii) m ∼= e⊗ x for some x ∈M (resp., m ∼= x⊗ e for some x ∈M).

The proof is too trivial to write down, and the lemma will be used implicitly from nowon without further mention. It shows that if π is an idempotent, then the subcategories

πM and Mπ depend only on the object e and not on the arrow π, and hence the followingnotation, which we will use below, is unambiguous:

πM =: eM, Mπ =: Me, πM∩Mπ =: eMe.

Lemma 1.4. Let 1π−→ e be an idempotent in M, and let x

α−→ yβ−→ x be morphisms in

M such that β ◦ α = idx. If y ∈ eM (resp., y ∈Me), then x ∈ eM (resp., x ∈Me).

This lemma implies, for example, that if M is an additive monoidal category which isKaroubi complete, then the categories eM, Me and eMe obtained from it are also Karoubicomplete.

Proof. It is enough to consider the case where y ∈ eM. We have a commutative diagram

1⊗ xid1⊗α //

π⊗idx

��

1⊗ yid1⊗β //

π⊗idy

��

1⊗ x

π⊗idx

��e⊗ x

ide ⊗α // e⊗ yide ⊗β // e⊗ x

This implies that the arrow π ⊗ idx is a retract of π ⊗ idy (in the obvious sense). It is easyto check that in an arbitrary category, the retract of an isomorphism is an isomorphism,which proves the lemma. �

1.6. We now begin proving a sequence of results on the uniqueness of certain morphismsbetween idempotents.

Lemma 1.5. Let 1π−→ e be an idempotent in M, let m ∈M be arbitrary, and let x ∈Me.

If f, g : m⊗ e → x are two morphisms such that

f ◦ (idm⊗π) = g ◦ (idm⊗π) : m⊗ 1→ x,

then f = g.

Thus the arrow idm⊗π : m⊗ 1 −→ m⊗ e enjoys a certain weakening of the property ofbeing a categorical epimorphism. We leave it to the reader to formulate and prove the dualversion of this lemma, where x ∈ eM and idm⊗π is replaced by π ⊗ idm.

Proof. The assumption implies that

(f ⊗ ide) ◦ (idm⊗π ⊗ ide) = (g ⊗ ide) ◦ (idm⊗π ⊗ ide) : m⊗ 1⊗ e −→ x⊗ e.


Since π ⊗ ide, and hence also idm⊗π ⊗ ide, is an isomorphism, it follows that

f ⊗ ide = g ⊗ ide : m⊗ e⊗ e −→ x⊗ e.

On the other hand, since x ∈Me, the commutative diagram

m⊗ e⊗ 1f⊗id1 //

idm ⊗ ide ⊗π

��

x⊗ 1

idx ⊗π

��m⊗ e⊗ e

f⊗ide // x⊗ e

shows that f ⊗ id1, and hence f , is determined by f ⊗ ide:

f ⊗ id1 = (idx⊗π)−1 ◦ (f ⊗ ide) ◦ (idm⊗e⊗π).

Since the roles of f and g can be interchanged, it follows that f = g. �

Corollary 1.6. If 1π−→ e is an idempotent in M and f : e → e is a morphism such that

f ◦ π = π, then f = ide. (In particular, an idempotent has no nontrivial automorphisms.)

Proof. Apply the lemma to the special case m = id1 (identifying 1 ⊗ e with e) and x = e,with f being the given morphism and g = ide. �

1.7. The following construction will be needed for proving several important results below.

Consider two arrows, 1π−→ e and 1

π′−→ e′, in M, which are not necessarily idempotents

(just arbitrary morphisms). We define a new arrow, denoted 1ππ′−→ e⊗ e′, as

ππ′ := (π ⊗ π′) ◦ λ−11

= (π ⊗ π′) ◦ ρ−11

: 1 −→ 1⊗ 1 −→ e⊗ e′.

The commutative diagram

1⊗ 1

λ1

��

π⊗π′ //

id1⊗π′ $$IIIIIIIII e⊗ e′

1⊗ e′π⊗ide′

::uuuuuuuuu

λe′

��

1

π′

%%JJJJJJJJJJJ

e′


leads to a commutative diagram

1ππ′ //

π′ ��>>>

>>>>

e⊗ e′

e′(π⊗ide′ )◦λ

−1e′

<<yyyyyyyyy

(1.1)

By symmetry, we also have a commutative diagram

1ππ′ //

π��>

>>>>

>>> e⊗ e′

e(ide ⊗π′)◦ρ−1

e

<<yyyyyyyyy

(1.2)

We now use this construction to prove a few other uniqueness results.

Lemma 1.7. If 1π−→ e is an idempotent in M, then so is 1

π2

−→ e⊗ e.

Proof. We must show that

1⊗ 1⊗ e⊗ eπ⊗π⊗ide⊗e−−−−−−→ e⊗ e⊗ e⊗ e

and

e⊗ e⊗ 1⊗ 1ide⊗e ⊗π⊗π−−−−−−−→ e⊗ e⊗ e⊗ e

are isomorphisms. But

π ⊗ π ⊗ ide⊗e =[(π ⊗ ide)⊗ ide⊗ ide

]◦

[id1⊗(π ⊗ ide)⊗ ide

]and

ide⊗e⊗π ⊗ π =[ide⊗(ide⊗π)⊗ id1

]◦

[ide⊗ ide⊗(ide⊗π)

],

which proves the lemma. �

Proposition 1.8. If 1π−→ e is an idempotent in M, then the following diagram commutes:

1⊗ eπ⊗ide

$$IIIIIIIII

e

λ−1e

<<zzzzzzzzz

ρ−1e ""DD

DDDD

DDD e⊗ e

e⊗ 1

ide ⊗π

::uuuuuuuuu

(1.3)

Proof. We apply the construction above to e′ = e, π′ = π. By assumption, both π⊗ ide andide⊗π are isomorphisms; thus (1.1) and (1.2) together imply that[

(ide⊗π) ◦ ρ−1e

]−1 ◦[(π ⊗ ide) ◦ λ−1

e

]◦ π = π.

In view of Corollary 1.6, this proves the proposition. �


1.8. We are now ready to prove that an idempotent 1π−→ e is determined by its codomain

up to unique isomorphism. We begin with

Proposition 1.9. If 1π−→ e is an idempotent in M, then the map

π∗ : Hom(e, e) −→ Hom(1, e), f 7−→ f ◦ π,

is bijective.

Proof. Injectivity of π∗ is immediate from Lemma 1.5. To prove that π∗ is surjective,consider an arbitrary morphism π′ : 1→ e. We apply the construction of §1.7 to e′ = e andthe two morphisms π and π′. Even though the map ide⊗π′ may not be an isomorphism,the map π ⊗ ide still is, so we obtain

π′ =[(π ⊗ ide) ◦ λ−1

e

]−1 ◦[(ide⊗π′) ◦ ρ−1

e

]◦ π,

proving the surjectivity of π∗. �

Corollary 1.10. If 1π−→ e and 1

π′−→ e are two idempotents in M with the same codomaine, then there exists a unique morphism σ : e → e such that π′ = σ ◦ π, and σ is anisomorphism.

Proof. The first statement is a special case of Proposition 1.9. The second statement followssince the roles of π and π′ can be interchanged. �

1.9. We conclude the section with a study of morphisms between idempotents with differentcodomains, culminating in a definition of a partial order on the set of isomorphism classesof idempotents. Let us consider a commutative diagram in M,

ef // e′

1

π

^^======== π′

??��

(1.4)

Lemma 1.11. If π and π′ are idempotents, then e′ ∈ eMe.

Proof. Tensoring the diagram (1.4) with e′ on the left, we obtain

e′ ⊗ eide′ ⊗f

// e′ ⊗ e′

e′ ⊗ 1

ide′ ⊗π

ddIIIIIIIII ide′ ⊗π′

::ttttttttt

Since ide′ ⊗π′ is an isomorphism, we can define morphisms

α = (ide′ ⊗π) ◦ ρ−1e′ : e′ −→ e′ ⊗ 1 −→ e′ ⊗ e

andβ = ρe′ ◦ (ide′ ⊗π′)−1 ◦ (id′e⊗f) : e′ ⊗ e −→ e′ ⊗ e′ −→ e′ ⊗ 1 −→ e′


with the property that β ◦ α = ide′ . Since e′ ⊗ e ∈ Me, Lemma 1.4 implies that e′ ∈ Me.By symmetry, we also obtain e′ ∈ eM, which proves the desired result. �

Proposition 1.12. In the same situation, if π and π′ are idempotents and (1.4) commutes,then f is the only morphism e → e′ making that diagram commute. Furthermore, the map

f ∗ : Hom(e′, e′) −→ Hom(e, e′)

is bijective.

Proof. We have a commutative diagram

Hom(e′, e′)

π′∗ ''OOOOOOOOOOO

f∗ // Hom(e, e′)

π∗wwooooooooooo

Hom(1, e′)

In this diagram, the map π′∗ is bijective by Proposition 1.9, and the map π∗ is injective byLemmas 1.11 and 1.5. This implies that both f ∗ and π∗ are bijective. The first statementof the proposition also follows from the injectivity of π∗. �

One of the main results of this section is the following

Theorem 1.13. Let e and e′ be closed idempotents in M. The following conditions are allequivalent to one another:

(a) e′ ∈ eMe;

(a1) e′ ∈ eM;

(a2) e′ ∈Me;

(b) there exists a morphism f : e → e′ such that ide′ ⊗f : e′ ⊗ e −→ e′ ⊗ e′ andf ⊗ ide′ : e⊗ e′ −→ e′ ⊗ e′ are isomorphisms;

(b1) there exists a morphism f : e → e′ such that ide′ ⊗f : e′ ⊗ e −→ e′ ⊗ e′ is anisomorphism;

(b2) there exists a morphism f : e → e′ such that f ⊗ ide′ : e ⊗ e′ −→ e′ ⊗ e′ is anisomorphism;

(c) for every choice of closed idempotents 1π−→ e and 1

π′−→ e′, there exists a morphismf : e → e′ with π′ = f ◦ π;

(d) there exist closed idempotents 1π−→ e, 1

π′−→ e′ and a morphism f : e → e′ suchthat π′ = f ◦ π.

Proof. It is clear that (a) implies (a1) and (a2). We claim that either (a1) or (a2) implies(b). If either (a1) or (a2) holds, then either e⊗ e′ ∼= e′ or e′ ⊗ e ∼= e′, whence e′ ⊗ e⊗ e′ ∼= e′


(because e′ is also an idempotent). Choose an isomorphism γ : e′ ⊗ e⊗ e′'−→ e′. It is then

clear that the composition

f : e'−→ 1⊗ e⊗ 1

π′⊗ide ⊗π′−−−−−−→ e′ ⊗ e⊗ e′γ−→ e′

satisfies the condition of (b). Again, (b) trivially implies both (b1) and (b2).

Suppose that (b1) holds, and let f : e → e′ be a morphism such that ide′ ⊗f : e′ ⊗ e −→e′ ⊗ e′ is an isomorphism. Consider the morphism f ◦ π : 1→ e′. By Proposition 1.9, thereexists a (unique) morphism α : e′ → e′ with f◦π = α◦π′. Moreover, since e′⊗e ∼= e′⊗e′ ∼= e′,we see that e′ ∈Me; thus ide′ ⊗π : e′ ⊗ 1 −→ e′ ⊗ e is an isomorphism. But now

(ide′ ⊗f) ◦ (ide′ ⊗π) = (ide′ ⊗α) ◦ (ide′ ⊗π′)

implies that ide′ ⊗α : e′⊗ e′ −→ e′⊗ e′ is an isomorphism. The usual argument now impliesthat α is an isomorphism, since

(π′ ⊗ ide′) ◦ (id1⊗α) = (ide′ ⊗α) ◦ (π′ ⊗ ide′).

So we see that π′ = α−1 ◦ f ◦ π. Thus we have proved that (b1) implies (c). By symmetry,(b2) also implies (c). Finally, the fact that (c) implies (d) is a tautology, and the fact that(d) implies (a) is the content of Lemma 1.11. �

1.10. Partial order. If 1π−→ e and 1

π′−→ e′ are idempotents in a monoidal category M,we will write e′ ≤ e, or π′ ≤ π, or (e′, π′) ≤ (e, π), whenever the equivalent conditions ofTheorem 1.13 hold. Note that by the theorem, the condition e′ ≤ e depends only on e ande′ and not on the choice of π and π′; thus we obtain a well defined relation on the set ofisomorphism classes of closed idempotents in M.

Proposition 1.14. This relation is a partial order, i.e., if e, e′ and e′′ are idempotents inM, we have: e ≤ e; if e′ ≤ e and e ≤ e′, then e ∼= e′; and if e′′ ≤ e′ and e′ ≤ e, then e′′ ≤ e.

This proposition follows trivially from our previous results.

1.11. Another rigidity result. The following result can be thought of as a converse toCorollary 1.10.

Proposition 1.15. Let 1π−→ e be a closed idempotent in M. If there exist morphisms

π′ : 1 −→ e and f : e → e in M satisfying π = f ◦π′, then f is an isomorphism (and henceπ′ is also a closed idempotent in M).

Proof. By Proposition 1.9, there exists a (unique) morphism g : e −→ e such that π′ = g◦π.In particular, f◦g◦π = π, whence f◦g = ide by Corollary 1.6. It remains to show that g◦f =ide. Now by Proposition 3.1 below, f and g are morphisms in the monoidal category eMe,and e is a unit object of eMe, so our result follows from the observation that the monoidof endomorphisms of the unit object in any monoidal category is commutative. Indeed, ifN is a monoidal category with unit object 1N , the functor X 7→ 1N ⊗X is isomorphic tothe identity functor IdN , whence End(1N ) embeds as a submonoid of End(IdN ), the latterbeing the Bernstein center of N , which is well known to be commutative. �


2. Idempotents and idempotent monads

2.1. In this section, unlike the previous one, the associativity constraints will play a roleeven in the statements of our results; thus we prefer to work with general monoidal cate-gories. The precise definition is given in [McL98], §VII.1, but we remark that Definition 1.2has to be modified as follows. First, we do not require ⊗ to be strictly associative; instead,we need to specify the extra data of a collection α of trifunctorial isomorphisms

αX,Y,Z : (X ⊗ Y )⊗ Z'−→ X ⊗ (Y ⊗ Z)

for all objects X,Y,Z of M, called the associativity constraint, satisfying the “pentagonaxiom,” stating that for any four objects X, Y, Z, W of M, the diagram formed by the fivepossible objects of M obtained by grammatically correct insertions of parentheses in theexpression X⊗Y ⊗Z⊗W , and by the five isomorphisms between these objects arising fromthe associativity constraint, is a commutative pentagon. The only other change we need tomake is that in condition (i) and in properties (iii) and (iv) of Definition 1.2, the equality oftwo possible isomorphisms is replaced by the commutativity of the triangle formed by theseisomorphisms and the appropriate associativity constraint.

As remarked before, the definition of a closed idempotent, as well as all the results ofSection 1, remain unchanged in this setup.

2.2. Let us recall that a monoid in a monoidal category M is a triple (G, µ, η) consistingof an object G of M, a morphism µ : G ⊗ G → G, and a morphism η : 1 → G such thatthe following diagrams commute:

(G⊗G)⊗GαG,G,G //

µ⊗idG

��

G⊗ (G⊗G)

idG ⊗µ

��G⊗G

µ

&&MMMMMMMMMMM G⊗Gµ

xxqqqqqqqqqqq

G

(2.1)

(this states that µ is associative, i.e., makes G into a “semigroup”), and

G⊗G

µ

��

1⊗G

η⊗idG

99sssssssss

λG %%KKKKKKKKKK G⊗ 1

idG ⊗ηeeKKKKKKKKK

ρGyyssssssssss

G

(2.2)

(this states that η is both a left and a right unit for the multiplication µ).


Definition 2.1. An idempotent monoid in a monoidal category M is a monoid (G, µ, η) inM such that µ is an isomorphism.

If (G, µ, η) is an idempotent monoid inM, it is clear that 1η−→ G is a closed idempotent in

M, since the commutativity of (2.2) implies that both η⊗ idG and idG⊗η are isomorphisms.The converse is slightly less obvious, but is also true; more precisely:

Lemma 2.2. If 1π−→ e is a closed idempotent in M, then there exists a unique morphism

e⊗ eµ−→ e such that (e, µ, π) is a monoid; moreover, µ is an isomorphism.

Proof. The commutativity of (1.3) and the fact that ide⊗π and π ⊗ ide are isomorphisms

implies that there is a unique arrow e⊗eµ−→ e such that the diagram (2.2), with G replaced

by e, commutes; and, moreover, this µ is an isomorphism. Thus we only need to check thatµ is associative. This follows from the commutativity of the diagram

(e⊗ e)⊗ eαe,e,e //

(ide ⊗π)−1⊗ide

��

e⊗ (e⊗ e)

ide ⊗(π⊗ide)−1

��(e⊗ 1)⊗ e

αe,1,e //

ρe⊗ide

��

e⊗ (1⊗ e)

ide ⊗λe

��e⊗ e

µ%%LLLLLLLLLLL e⊗ e

µyyrrrrrrrrrrr

e

Indeed, the commutativity of the top square follows from the naturality of α, the commu-tativity of the middle square is one of the axioms of a monoidal category, and the commu-tativity of the lower triangle is obvious. Since µ = ρe ◦ (ide⊗π)−1 = λe ◦ (π ⊗ ide)

−1 byconstruction, we see that µ is associative. �

2.3. We can now relate the theory of idempotents developed above to the theory of idempo-tent functors and idempotent monads, studied in the works of Adams [Ad73] and Deleanu-Frei-Hilton [DFH1, DFH2].

Let us first recall that setup considered by Adams. Let C be a category. Adams definesan idempotent functor on C as a pair (E, η) consisting of a functor E : C −→ C and a naturaltransformation η : IdC −→ E satisfying two axioms:

Axiom 1. For every object X of C, we have

ηE(X) = E(ηX) : E(X) −→ E2(X).


Axiom 2. For every X ∈ C, the map ηE(X) : E(X) −→ E2(X) is an isomorphism.

We observe that the category Funct(C, C) of all functors C → C is in fact a strictly associa-tive and strictly unital monoidal category, the monoidal functor being given by compositionof endofunctors. Conversely, every monoidal category acts on itself via tensoring on the left,which realizes M as a monoidal subcategory of Funct(M,M). [In fact, M is equivalentto the category of endofunctors of M which “commute with the right action of M” in theappropriate sense; the details are explained in §3.3.] Thus the language of endofunctors ismore or less equivalent to the language we have been using in the previous section.

In particular, an idempotent functor on a category C in the sense of Adams is the same asa closed idempotent in the monoidal category Funct(C, C) in the sense of Definition 1.1; thisfollows from Proposition 1.8. Moreover, in view of Lemma 2.2, we see that an idempotentfunctor on C is more or less the same thing as an idempotent monad on C (which, bydefinition, is just an idempotent monoid in the category Funct(C, C))—a fact which wasasserted without proof by Adams [Ad73].

Remark 2.3. From the theory developed in the previous section (particularly Proposition1.8) it follows that in the definition of an idempotent functor (E, η) it suffices to require thea priori weaker condition that both ηE(X) : E(X) −→ E2(X) and E(ηX) : E(X) −→ E2(X)are isomorphisms for every object X of C; the equality ηE(X) = E(ηX) is then automatic.For the reader’s convenience we now give a different (and independent) proof of the samefact, namely, one which is written purely within the framework of functors, objects andmorphisms and does not involve the language of monoidal categories.

Suppose that ηE(X) and E(ηX) are isomorphisms for every X ∈ C, and fix X ∈ C. Thediagram

XηX //

ηX

��

E(X)

ηE(X)

��E(X)

E(ηX)// E2(X)

commutes because η is a morphism of functors. Applying E to this diagram and remem-bering that E(ηX) is an isomorphism, we obtain that

E2(ηX) = E(ηE(X)) : E2(X) −→ E3(X).


The diagram

E(X)ηE(X) //

ηE(X)

��

E2(X)

ηE2(X)

��E2(X)

E(ηE(X))// E3(X)

commutes for the same reason as the one above; since ηE(X) is an isomorphism, we obtainthat

E(ηE(X)) = ηE2(X) : E2(X) −→ E3(X).

Finally, the diagram

E(X)E(ηX)

//

ηE(X)

��

E2(X)

ηE2(X)

��E2(X)

E2(ηX)

// E3(X)

also commutes for the same reason. But ηE2(X) = E2(ηX) by the previous two observations.Thus E(ηX) = ηE(X), as desired.

Remark 2.4. On the other hand, it is not enough to require only one of the morphisms ηE(X)

and E(ηX) to be an isomorphism. As an example, let C be a category with one object. Thenthe data of C is equivalent to the data of a monoid G (in the category of sets). Moreover,a functor E : C → C is the same as a homomorphism e : G → G of monoids, and a naturaltransformation η : IdC → E is the same as an element η ∈ G with the property

e(g) · η = η · g ∀ g ∈ G.

However, it is trivial to find an example where these conditions are satisfied and where e(η)is invertible, but η is not. For instance, we can take e to be the constant map, defined bye(g) = 1 for all g ∈ G. Then the condition above reduces to η · g = η. Now we can takeG to be the monoid of all maps of sets {0, 1} → {0, 1} (with respect to the operation ofcomposition), and η to be the constant map 0 7→ 0, 1 7→ 0.

3. Hecke subcategories

3.1. Let M = (M,⊗,1, α, λ, ρ) be a monoidal category, and let 1π−→ e be a closed idem-

potent in M. In Section 1 we have introduced the full subcategory eMe ⊆ M consistingof all objects X of M such that X ∼= e⊗X ⊗ e. Recall that a strictly full subcategory of acategory C is a full subcategory D ⊆ C such that if X ∈ C is isomorphic to an object of D,then X ∈ D. It is clear that eMe is a strictly full subcategory of M which is closed under


the tensor product. By abuse of notation, we will denote by ⊗ the induced bifunctor oneMe, and by α the induced associativity constraint on eMe. Moreover, for every object Xof eMe we define isomorphisms

λπX = λX ◦ (π ⊗ idX)−1 : e⊗X

'−→ 1⊗X'−→ X

and

ρπX = ρX ◦ (idX ⊗π)−1 : X ⊗ e

'−→ X ⊗ 1'−→ X.

Proposition 3.1. The category eMe =(eMe,⊗, e, α, λπ, ρπ

)is monoidal.

Proof. The fact that the induced associativity constraint α on eMe satisfies the pentagonaxiom is automatic. Thus we need to verify the following two conditions:

(i) for all a, c ∈ eMe, the diagram

(a⊗ e)⊗ cαa,e,c //

ρπa⊗idc &&MMMMMMMMMM

a⊗ (e⊗ c)

ida ⊗λπcxxqqqqqqqqqq

a⊗ c

(3.1)

commutes, and

(ii) we have λπe = ρπ

e : e⊗ e −→ e.

Now property (ii) is the content of Proposition 1.8. To prove (i), we expand the diagram(3.1) into

(a⊗ e)⊗ cαa,e,c //

(ida ⊗π)−1⊗idc

��

a⊗ (e⊗ c)

ida ⊗(π⊗idc)−1

��(a⊗ 1)⊗ c

αa,1,c //

ρa⊗idc &&MMMMMMMMMMa⊗ (1⊗ c)

ida ⊗λcxxqqqqqqqqqq

a⊗ c

The commutativity of the top square follows from the naturality of α, and the commutativityof the bottom triangle is one of the axioms for the monoidal category M. �

Remark 3.2. In view of Corollary 1.10, the category eMe depends only on the idempotente and not on the arrow π, up to equivalence.

We call eMe the Hecke subcategory of M corresponding to the idempotent e, by analogywith the term Hecke subalgebra, used for the subring eAe in a ring A associated to anidempotent e.


3.2. An example. In this subsection we show that a certain naive version of Proposition3.1 is false. Namely, there exists a monoidal category M and an object e ∈ M, which is aweak idempotent in the sense that e ⊗ e ∼= e, such that the subcategory eMe ⊂ M is notmonoidal with respect to the induced convolution functor.

In other words, even though we have e⊗X ∼= X ∼= X ⊗ e for all X ∈ eMe, the functorX 7→ e⊗X is not an autoequivalence of the category eMe.

Example 3.3. Let A1 denote the affine line over an algebraically closed field k, and let M bethe bounded derived category Db

c(A1, Q`) of constructible `-adic complexes on A1, equippedwith the usual tensor product. Here ` is a prime invertible in k and Q` is a fixed algebraicclosure of the field Q` of `-adic numbers.

Put e = j!Q` ⊕ i!Q`, where j : A1 \ {0} ↪→ A1 is the open embedding and i : {0} ↪→ A1 isthe corresponding closed embedding. Then e⊗ e ∼= e, but the functor X 7→ e⊗X is not anautoequivalence of the category eMe = eM, which follows from the lemma below

Lemma 3.4. LetM be a triangulated monoidal category, let e1, e2 ∈M be weak idempotentssuch that e1⊗e2 = 0 = e2⊗e1, and suppose that there exists a nonzero morphism e1 → e2[k]for some k ∈ Z. If e = e1 ⊕ e2, then e is a weak idempotent with the property that thefunctor X 7→ e⊗X on the category eMe is not faithful.

Proof. Since e1 ⊗ e2 = 0 = e2 ⊗ e1, and since e1 and e2 are weak idempotents, it followsthat e is also a weak idempotent, and the objects e1 and e2[k] lie in eMe. Any nonzeromorphism e1 → e2[k] is annihilated by each of the functors X 7→ e1 ⊗X and X 7→ e2 ⊗X,whence it is also annihilated by the functor X 7→ e⊗X. �

Note that in the example above, the objects e1 = i!Q` and e2 = j!Q` are weak idempotentsin M, and we have (writing D for the Verdier duality functor)

Hom(e1, e2) = Hom(i!Q`, j!Q`) ∼= Hom(Di∗DQ`, Dj∗DQ`)∼= Hom(j∗DQ`, i∗DQ`) ∼= Hom(j∗Q`[2](1), i∗Q`)∼= Hom(i∗j∗Q`[2](1), Q`) ∼= Q`[−2](−1).

In our situation, Tate twists can be (noncanonically) trivialized, so we see that there existsa nonzero morphism e1 → e2[2] in M, whence the lemma can be applied.

3.3. We now review the construction of a strictly associative and strictly unital monoidalcategory equivalent to a given monoidal category, due to J. Bernstein; it was explained tous by D. Kazhdan. For future purposes, we slightly generalize our definition as follows.Consider a monoidal category M = (M,⊗,1, α, λ, ρ) and a strictly full subcategory M′ ⊆M with the property that M′M ⊆ M′, i.e., if X ∈ M′ and Y ∈ M, then X ⊗ Y ∈ M′.We will call such an M′ a right tensor ideal in M.

We define a category EndM(M′) as follows. Its objects are pairs (F, R) consisting of afunctor F : M′ −→M′ and a bifunctorial collection of isomorphisms

RX,Y : F (X)⊗ Y'−→ F (X ⊗ Y ) ∀X ∈M′, Y ∈M


satisfying an analogue of the pentagon axiom, namely, the diagram

(F (X)⊗ Y )⊗ Zα //

RX,Y ⊗idZ

��

F (X)⊗ (Y ⊗ Z)

RX,Y⊗Z

��F (X ⊗ Y )⊗ Z

RX⊗Y,Z ))SSSSSSSSSSSSSSF (X ⊗ (Y ⊗ Z))

F ((X ⊗ Y )⊗ Z)F (α)

55kkkkkkkkkkkkkk

commutes. Morphisms (F, R) → (F ′, R′) are defined as natural transformations F → F ′

that are compatible with R and R′ in the obvious sense. Moreover, EndM(M′) has a strictlyassociative and strictly unital monoidal structure given by composition of functors; moreprecisely, (F, R)⊗ (F ′, R′) = (F ◦ F ′, RR′), where, by definition,

(RR′)X,Y = F (R′X,Y ) ◦RF ′(X),Y : FF ′(X)⊗ Y

'−→ F (F ′(X)⊗ Y )'−→ FF ′(X ⊗ Y )

Remark 3.5. One can generalize the setup above by defining the category of functors betweentwo right module categories over M. However, we do not need the formalism of modulecategories in these notes, and thus we will not explain this generalization.

Bernstein’s observation can be stated as follows:

Theorem 3.6. The category EndM(M) is equivalent to M as a monoidal category.

This result is a special case of Theorem 3.7 proved below.

3.4. In order to proceed we need the notion of a monoidal functor between two monoidalcategories. We review the definition following [McL98], §XI.2. If M = (M,⊗,1, α, λ, ρ)and M′ = (M′,⊗′,1′, α′, λ′, ρ′) are monoidal categories, a (strong) monoidal functor F :M−→M′ consists of the following data:

(i) a functor F : M−→M′ between the underlying categories;(ii) a collection of bifunctorial isomorphisms

F2(a, b) : F (a)⊗′ F (b)'−→ F (a⊗ b)

for all objects a, b of M;(iii) an isomorphism

F0 : 1′'−→ F (1).

These data are required to satisfy the obvious conditions of compatibility with the associa-tivity constraints and the left and right unit isomorphisms (loc. cit.). The adjective “strong”refers to the fact that in the definition of a plain monoidal functor one only requires F0 andall the F2(a, b) to be morphisms in M′ (as opposed to isomorphisms). However, in thispaper we only use the notion of a strong monoidal functor, and thus the adjective “strong”will be omitted from now on.


The notions of a morphism of monoidal functors and the composition of monoidal functorsare defined in the obvious way. A useful fact, which can be checked trivially using thedefinition above, is that if F : M−→M′ is a monoidal functor which is an equivalence ofthe underlying categories, then it is automatically an equivalence of monoidal categories, inthe sense that if G : M′ −→ M is quasi-inverse to F , then G can be equipped with thestructure of a monoidal functor such that F ◦ G and G ◦ F are isomorphic to the identityfunctors of M′ and M, respectively, as monoidal functors.

3.5. Let M be a monoidal category and 1π−→ e a closed idempotent in M. It is clear

that eM is a right tensor ideal in M. Moreover, there is a natural monoidal functor

Φ : eMe −→ EndM(eM) (3.2)

taking every object m of eMe to the pair (Fm, Rm) = (m⊗ ·, αm,·,·), i.e.,

Fm(X) = m⊗X, RmX,Y = αm,X,Y : (m⊗X)⊗ Y

'−→ m⊗ (X ⊗ Y ).

The monoidal structure on Φ is defined by Φ2(a, b) = α−1a,b,· and Φ0 = (λπ

· )−1, i.e.,

Φ2(a, b) : a⊗ (b⊗X)α−1

a,b,X−−−→ (a⊗ b)⊗X and Φ0 : X(λπ

X)−1

−−−−→ e⊗X.

Theorem 3.7. With the notation and assumptions as above, the monoidal functor(Φ, Φ2, Φ0) is an equivalence of monoidal categories.

In the special case (e, π) = (1, id1), we recover Theorem 3.6.

Proof. As remarked above, it is enough to show that Φ is an equivalence of the underlyingcategories. We construct an explicit quasi-inverse of Φ. Let us define a functor

Ψ : EndM(eM) −→ eMe

by Ψ(F, R) = F (e). The functorial collection of isomorphisms

ρπm : m⊗ e

'−→ m, m ∈ eMe,

constructed above yields an isomorphism of functors

ρπ : Ψ ◦ Φ'−→ IdeMe .

It remains to construct an isomorphism

Φ ◦Ψ'−→ IdEndM(eM) .

For each pair (F, R) ∈ EndM(eM) we have, by construction,

Φ(Ψ(F, R)) =(F F (e), RF (e)

),

and we need to produce a functorial isomorphism between (F, R) and(F F (e), RF (e)

)in order

to complete the proof of the theorem. For each m ∈ eM, we have a functorial isomorphism

θm : F (e)⊗m'−→ F (m)


given by the composition

F (λm) ◦ F((π ⊗ idm)−1

)◦Re,m : F (e)⊗m

'−→ F (e⊗m)'−→ F (1⊗m)

'−→ F (m).

The collection of isomorphisms θm defines an isomorphism of functors

θ : F F (e) '−→ F,

which itself depends functorially on the pair (F, R). Hence it remains to check that θ iscompatible with R and RF (e) = αF (e),·,· in the obvious sense. This is equivalent to provingthe commutativity of the diagram

F F (e)(X)⊗ YR

F (e)X,Y //

θX⊗idY

��

F F (e)(X ⊗ Y )

θX⊗Y

��F (X)⊗ Y

RX,Y

// F (X ⊗ Y )

To show that it commutes, we expand it as follows:(F (e)⊗X)⊗ Y //

��

F (e)⊗ (X ⊗ Y )

��

F (e⊗X)⊗ Y

))RRRRRRRRRRRRRR

��

F(e⊗ (X ⊗ Y )

)uukkkkkkkkkkkkkk

��

F((e⊗X)⊗ Y

)

��

F (1⊗X)⊗ Y

))RRRRRRRRRRRRRR

��

F(1⊗ (X ⊗ Y )

)uukkkkkkkkkkkkkk

��

F((1⊗X)⊗ Y

)))SSSSSSSSSSSSSS

F (X)⊗ Y // F (X ⊗ Y )

All arrows in this diagram are the obvious ones. It is easy to check that the diagram abovecommutes using the pentagon axiom for (F, R), the naturality of α, and the naturality ofR. The (straightforward) details are left to the reader. �


3.6. Let C be a category and D ⊆ C a subcategory. We recall ([McL98], §IV.3) that D issaid to be reflective (resp., coreflective) if the inclusion functor D ↪→ C admits a left (resp.,right) adjoint.

Lemma 3.8. If e is a closed idempotent in a monoidal category M, then the strictly fullsubcategories eM, Me and eMe of M are reflective.

The following result is a partial converse to this lemma:

Theorem 3.9. Assume that M is a monoidal category which is left closed, i.e., for any pairof objects X, Y of M, there exists an object Hom`(X, Y ) ∈M and a functorial isomorphism

HomM(Z ⊗X, Y )'−→ HomM(Z, Hom`(X, Y ))

for all objects Z of M. Then the map e 7→ eM provides a bijection between

(i) the set of isomorphism classes of closed idempotents in M, and(ii) the set of reflective right tensor ideals M′ ⊆M which have the property that if Y ∈M′

and X ∈M, then Hom`(X, Y ) ∈M′.

This theorem has an obvious analogue for right closed monoidal categories and left tensorideals. We also note that if M is left rigid (this property is stronger than being left closed),then the second condition in (ii) holds automatically for right tensor ideals, since in this case

we have a natural isomorphism Hom`(X, Y )'−→ Y ⊗X∗ for any two objects X, Y ∈M.

4. Idempotents in braided monoidal categories

4.1. As in Section 1, we begin by giving a precise definition of the type of categories wework with. See [McL98], §XI.1 for more details.

Definition 4.1. A strictly associative braided monoidal category, to be abbreviated BMC,is a 6-tuple M = (M,⊗,1, λ, ρ, β), where (M,⊗,1, λ, ρ) is a strictly associative monoidalcategory in the sense of Definition 1.2 and β is a bifunctorial collection of isomorphisms

βa,b : a⊗ b'−→ b⊗ a ∀ a, b ∈M,

called the braiding of M, satisfying the axioms

(i) for every a ∈M, the triangle

a⊗ 1βa,1 //

ρa""FFFFFFFF 1⊗ a

λa||xxxxxxxx

a

commutes, so that, in particular, β1,1 = id1⊗1; and


(ii) the following two diagrams, which result from the “hexagon axiom” for a usual braidedmonoidal category, but which reduce to triangles due to the strict associativity assump-tion, commute for all a, b, c ∈M:

a⊗ b⊗ cida ⊗βb,c

wwppppppppppp βa⊗b,c

''NNNNNNNNNNN

a⊗ c⊗ bβa,c⊗idb

// c⊗ a⊗ b

(4.1)

anda⊗ b⊗ c

βa,b⊗idc

wwppppppppppp βa,b⊗c

''NNNNNNNNNNN

a⊗ c⊗ bidb ⊗βa,c

// c⊗ a⊗ b

(4.2)

4.2. From now on until the end of this section, we fix a BMC M and study closed idem-potents in M.

Proposition 4.2. If e is a closed idempotent in M, then

(a) βe,e = ide⊗e, and(b) for every object m ∈M, we have βm,e ◦ βe,m = ide⊗m.

Theorem 4.3. If e is a closed idempotent in M, the natural functor

M−→ eMe, X 7→ e⊗X ⊗ e

can be upgraded to a braided monoidal functor.

5. Idempotents in triangulated monoidal categories

5.1. Open idempotents. As mentioned in the introduction, there is a notion dual to thatof a closed idempotent:

Definition 5.1. An open idempotent in a monoidal category M is an arrow eπ−→ 1 in M

such that the two induced arrows,

e⊗ eide ⊗π−−−→ e⊗ 1 and e⊗ e

π⊗ide−−−→ 1⊗ e,

are isomorphisms. An object e of M is an open idempotent if there exists an arrow eπ−→ 1

which is an open idempotent.

It is clear that if M is a monoidal category, then the opposite category Mop inherits anatural monoidal structure, and we have an obvious bijection between open idempotents inM and closed idempotents in Mop. Thus the results of Sections 1, 3 and 4 have obviousanalogues for open idempotents whose formulation is left to the reader. In particular, if eis an open idempotent in M, the strictly full subcategories eM, Me and eMe of M aredefined in the same way as for closed idempotents. More interesting is the fact that if M isa triangulated monoidal category, then there is a bijection between the set of isomorphism


classes of closed idempotents in M and the set of isomorphism classes of open idempotentsin the same category M, see Proposition 5.3 below.

5.2. Locally closed idempotents. To motivate our definition of a locally closed idempo-tent in a triangulated monoidal category, let us first consider the abelian situation. Let Xbe a topological space and M the category of sheaves of complex vector spaces on X withthe usual tensor product. If ξ : Y ↪→ X is the inclusion of a locally closed subspace, wecan consider the constant sheaf CY with stalks C on Y and form ξ!CY ∈ M. To study theproperties of this sheaf, let us write Y = U ∩ Z, where U is open in X and Z is closed inX. We assume moreover that X = U ∪Z; this is always possible to achieve, and under thiscondition each of U , Z determines the other one uniquely. We have a diagram of inclusions

X

U/ �

j>>~~~~~~~

Z/ O

i``@@@@@@@

Y/ Oi′

``@@@@@@@ / � j′

>>~~~~~~~

ξ

OO

which by adjunction induces a diagram

j!CU� � //

��

CX

��ξ!CY

� � // i!CZ

that is both cartesian and cocartesian (this uses the assumption that X = U∪Z). Moreover,the left (resp., right) vertical arrow is a closed idempotent in the category of sheaves ofcomplex vector spaces on X (resp., on U), and the top (resp., bottom) horizontal arrow isan open idempotent in the category of sheaves of complex vector spaces on X (resp., onZ). This motivates two possible definitions of a locally closed idempotent in an arbitrarymonoidal category:

Definition 5.2. A locally closed idempotents in a monoidal category M is either

(i) an open idempotent u −→ e in the category eMe corresponding to a closed idempotent1 −→ e in M, or

(ii) a closed idempotent v −→ u in the category vMv corresponding to an open idempotentv −→ 1 in M.

In a triangulated monoidal category these two definitions turn out to be equivalent, seeTheorem 5.4 below. In the rest of the section we explain the definition of a triangulatedmonoidal category and prove the two results we have mentioned.


5.3. Main results.

Proposition 5.3. Let M be a triangulated monoidal category. If eπ−→ 1 is an open

idempotent in M and

eπ−→ 1

π′−→ e′ −→ e[1]

is a distinguished triangle obtained from π, then 1π′−→ e′ is a closed idempotent in M.

Moreover, the map π 7→ π′ defined this way yields a bijection between the set of isomorphismclasses of open idempotents in M and that of closed idempotents in M.

Theorem 5.4. Let M be a triangulated monoidal category. An object of M is a locallyclosed idempotent in the sense of Definition 5.2(i) if and only if it is a locally closed idem-potent in the sense of Definition 5.2(ii).

References

[Ad73] J.F. Adams, Idempotent functors in homotopy theory, in: “Manifolds—Tokyo 1973 (Proc. Internat.Conf., Tokyo, 1973),” pp. 247–253. Univ. Tokyo Press, Tokyo, 1975.

[BD06] M. Boyarchenko and V. Drinfeld, A motivated introduction to character sheaves and the orbit methodfor unipotent groups in positive characteristic, Preprint, September 2006, arXiv: math.RT/0609769

[DFH1] A. Deleanu, A. Frei and P. Hilton, Idempotent triples and completion, Math. Z. 143 (1975), 91–104.[DFH2] A. Deleanu, A. Frei and P. Hilton, Generalized Adams completion, Cahiers Topologie Geom.

Differentielle 15 (1974), 61–82.[McL98] S. MacLane, “Categories for the working mathematician”, 2nd ed. Graduate Texts in Mathematics,

5. Springer-Verlag, New York, 1998.

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Idempotents in Monoidal Categories