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COALGEBRAS, BRAIDINGS, AND DISTRIBUTIVE LAWS

To Aurelio Carboni on his 60th birthday

STEFANO KASANGIAN, STEPHEN LACK, AND ENRICO M. VITALE

Abstract. _{We show, for a monad} T_{, that coalgebra structures on a} T_{-algebra can} be described in terms of “braidings”, provided that the monad is equipped with an invertible distributive law satisfying the Yang-Baxter equation.

1. Introduction

The aim of this note is to provide an equivalent description of T∗

-coalgebra structures on a T_{-algebra, for} T _{a monad — equipped with a special kind of distributive law — on a}

category C_,and T∗

the comonad induced by the adjunction

C L

T

//_Alg(T₎

RT

oo LT⊣RT

Our interest for such coalgebras is motivated mainly by classical descent theory: let f:R →S be a morphism of commutative unital rings, and consider the induced functor

f! : R-mod→S-mod

defined by f!(N) = N ⊗RS (where S is seen as an R-module by restriction of scalars). The descent problem forf consists in recognizing when anS-module is of the formf!(N) for someR-moduleN.A classical theorem [6, 2, 7], which establishes a deep link between descent theory and the theory of (co)monads, asserts that, if f is faithfully flat, then an S-module is of the formf!(N) if and only if it is equipped with a T∗

-coalgebra structure, where T∗

is the comonad onS-mod induced by the adjunction

R-mod f! //S-mod

f∗

oo f!⊣f∗

and f∗

is the restriction of scalars functor. (For this reason, a T∗

-coalgebra structure on

anS-moduleM is sometimes called adescent datumforM.) It is also well-known that, for

Second author supported by the Australian Research Council; third author by FNRS grant 1.5.116.01. Received by the editors 2003-12-11 and, in revised form, 2004-10-01.

Published on 2004-12-05.

2000 Mathematics Subject Classification: 18C15, 18C20, 18D10, 16B50.

Key words and phrases: Descent data, monads, distributive laws, Yang-Baxter equation. c

Stefano Kasangian, Stephen Lack, and Enrico M. Vitale, 2004. Permission to copy for private use granted.

any morphism f: R → S of commutative unital rings, there are several equivalent ways of describing what a T∗

-coalgebra structure for anS-module is, and a natural problem is to lift to a categorical level these other descriptions of T∗

-coalgebra structures.

In a recent paper [11], Menini and Stefan, extending results by Nuss [12] on non-commutative rings, replace the situation

R-mod

f!

//_S-mod

f∗

oo f!⊣f∗

by

C L

T

//_Alg(T)

RT

oo LT⊣RT

where T _{is a monad on an arbitrary category} C _{(indeed, even in the non-commutative}

situation, f∗

: S-mod → R-mod is a monadic functor, so that S-mod is equivalent to

the category of algebras for the monad on R-mod induced by the adjunction f! ⊣ f∗

). In this context, they prove that, if the monad T _{is equipped with a “compatible flip”}

K: T2 ⇒ T2, then to give a T∗

-coalgebra structure on a T_{-algebra X is equivalent to}

giving a “symmetry” on X, that is an involution T X → T X satisfying some suitable conditions.

Unfortunately, the following natural example, which is a direct generalization of the classical case of commutative rings, does not fit into their general context: let C _{be a}

braided monoidal category and letS be a monoid in C_{, then the braidingcS,S: S⊗S→}

S⊗S induces a natural isomorphismK:T2 ⇒T2 on the monadT =− ⊗S: C_→C_,but

this natural isomorphism is not a flip unless the braiding is a symmetry and the monoid is commutative. In this note we adapt the notions of “compatible flip” and “symmetry” to encompass the previous example, as well as another example coming from the theory of bialgebras.

In Section 2 we introduce the notion of BD-law on a monad T _{as a special case of}

distributive law in the sense of Beck [1]. Using a BD-lawK, we can defineK-braidingson aT_{-algebra, and we want to show thatK-braidings correspond bijectively to}T∗

-coalgebra structures. There are two different methods: in Section 3 we useK-braidings to define a category Brd(T_{, K) equipped with a forgetful functorV : Brd(}T_{, K)→Alg(}T_{).We show,}

using the Beck criterion [10], that V is comonadic; and that the corresponding comonad isT∗

,so that Brd(T_{, K) is isomorphic to Coalg(}T∗

).In Section 4 we give a different proof based as far as possible on general facts about monads and distributive laws. This second proof is quite long, but it seems to us of some interest, since it shows that the bijection between K-braidings and T∗

2. BD-laws on a monad

To begin, we fix notation. A monadT _{on a category} C _{is a triple}

T_{= (T:} C_→C_{, m:T}2 _{⇒T, e: IdC⇒T)}

consisting of a functor T, and natural transformations m and e making the diagrams

T eT +3

commute. Given two monads T _and S _{on the same category} C_{, a distributive law of} T

over S_{is a natural transformation K: T S⇒ST such that the diagrams}

T T e +3

We refer to [1, 3] for more details on monads and distributive laws. When the natural transformation K is an isomorphism, the definition of distributive law can be simplified, as in the following lemma:

2.1. Lemma. _{Consider two monads} T _and S _{on a category} C_{, and a natural}

isomor-phism K: T S ⇒ST. Then (8) implies (6) and (9) implies (7). Moreover, K satisfies (8)

Proof. _{We prove that (9) implies (7); the proof of the other implication is similar, and}

the rest of the statement is obvious. The proof is contained in the following diagram, in which unlabelled regions commute by naturality:

S eS +3 (B) K satisfies the Yang-Baxter equation:

T3 KT +3

(D) K is a distributive law (that is, it satisfies equations (6–9)). A BCD-law on T _{is a BD-law such that}

(C) the monad T_{is K-commutative:}

T2 K +3

A BD-law or BCD-law is said to be invertible if the natural transformationK is so.

2.3. Remark. _{Once again, as stated in Lemma 2.1 for the distributivity conditions,}

a natural isomorphism K: T2 ⇒T2 satisfies conditions (B) or (C) iff K−1 _does.

2.4. Remark. _If C _{is an arbitrary monoidal category, and} T _{is a monoid in} C_{, one}

can define a BD-law or BCD-law on T _{as above. A monoid equipped with an invertible}

2.5. Remark. _{If K is involutive — that is, K}2 _{= 1 — each of the conditions (8) and}

(9) implies the other. This fact, together with Lemma 2.1, means that compatible flips in the sense of Menini and Stefan [11] are precisely the involutive BCD-laws.

2.6. Example. _Let C _{= (}C_{,⊗, I, . . .) be a monoidal category and S = (S, mS, eS) a}

monoid in C_{. The monoidS induces a monad} T_on C _{in the following way:}

• T =− ⊗S: C_→C

• mX = 1⊗mS: X⊗S⊗S→X⊗S

• eX = 1⊗eS:X ≃X⊗I →X⊗S

2.6.1 _If C _{is braided, with braiding c = {cX,Y: X ⊗ Y → Y ⊗X}, then there is}

an invertible BD-law K on T _{defined by KX = 1⊗cS,S: X ⊗S ⊗S → X ⊗S ⊗ S.}

In this case, condition (B) is precisely the Yang-Baxter equation; by naturality of the braiding, conditions (8) and (9) reduce to the following equations, which hold in any braided monoidal category:

•

This BD-law is a BCD-law precisely when the monoid S is commutative; it is involutive if and only if cS,S is so, in particular if the braiding cis a symmetry.

2.6.2 _{Let K be a field and take as} C _{the category of K-vector spaces. Let H be a}

cobraided bialgebra with universal form r: H⊗H → I; there is a natural isomorphism

of H-comodules cr

V,W: V ⊗W →W ⊗V defined by

V ⊗W τ //_{W ⊗V} ∆⊗∆//_{H⊗W ⊗H⊗V} 1⊗τ⊗1//_{H⊗H⊗W ⊗V} r⊗1⊗1//_{W ⊗V}

where ∆ is the coaction and τ is the standard twist: see [9]. If S is any H-comodule algebra (in particular, one can takeS =H), then we have an invertible BD-law K onT

defined by KX = 1⊗cr

S,S: X⊗S⊗S → X ⊗S⊗S. Indeed, conditions (8-10) follow from [9, Proposition VIII.5.2], using the fact that the multiplicationmS: S⊗S →S is a homomorphism of H-comodules.

2.7. Example. _{If f: R → S is a morphism of unital rings, with R commutative,}

we can specialize Example 2.6.1 by taking C _{= R-mod, so that K is defined by the}

standard twist S ⊗S → S⊗S. This is possible also if R is not commutative, provided its image lies in the centre of S : taking now C _{= R-R-bimod, the standard twist on S}

can be defined and gives once again an invertible BD-law onT_{. If we drop the centrality}

of the category of bimodules to define a different BD-law. In fact, if C _{is an additive}

category and T_{is any monad on} C_{, then there is an involutive BCD-law K on}T _defined

by K = (eT +T e)·m−T2_{; see [11]. This case generalizes results on non-commutative}

rings established in [4, 12].

3. Coalgebras and braidings

For the reader’s convenience, let us recall how the definition of coalgebra for a comonad specializes when the comonad is of the form T∗

.

3.1. Definition. _Let T _{= (T, m, e) be a monad on a category} C _{and consider the}

comonad T∗

on Alg(T_{) induced by the adjunction}

C L

T

//_Alg(T)

RT

oo LT⊣RT

A T∗

-coalgebra structure on a T_{-algebra (X, x: T X → X) is a morphism r: X → T X}

such that

T X T r //

x

(12)

T2X

mX

X _r //_{T X}

X r //

1

''

O O O O O O O O O O O O O

O T X

x

(13)

X

X r //

r

(14)

T X

T r

T X _{T eX}//_T2_X

We denote by T∗

-coalg(X, x) the set ofT∗

-coalgebra structures on a T_{-algebra (X, x).}

3.2. Remark. _{For allX ∈}C_{,the morphismT eX: T X →T}2_{X is a}T∗

-coalgebra struc-ture on the free T_{-algebra L}T

X = (T X, mX). This is the (object part of the) canonical comparison functor C_→Coalg(T∗

).

3.3. Definition. _LetT_{= (T, m, e) be a monad on a category}C_{and let K: T}2 _{⇒ T}2

be a BD-law. AK-braidingon aT_{-algebra (X, x: T X →X) is a morphismc: T X →T X}

such that

T X c //

(15)

T X

x

X eX

OO

1 //X

T2X KX //

T c

T2X T c //

(16)

T2X

KX

T2X _KX //T2X _{T c} //T2X

T2X KX //

T x

T2X T c //

(17)

T2X

mX

T X _c //_{T X}

We denote by K-Brd(X, x) the set of K-braidings on a T_{-algebra (X, x).}

3.4. Remark. _{If K is invertible, condition (17) means that cis a morphism}

c: T(X, x)→T∗

(X, x)

3.5. Remark. _{We shall see in Proposition 3.9 that ifK is invertible or involutive then}

the same is true of any K-braiding, whence by Corollary 3.11 it will follow that if K is an involutive BCD-law, thenK-braidings are precisely symmetries in the sense of Menini and Stefan [11].

3.6. Remark. _{IfK is a BD-law on}T_{,thenKX is aK-braiding onL}T

X,for allX ∈C_.

Indeed, conditions (16) and (17) correspond respectively to conditions (10) and (9), while condition (15) is the pasting of (1) and (6). In the bijection stated in Corollary 3.8,KX corresponds to the T∗

-coalgebra structure T eX of Remark 3.2.

If T _{is a monad on a category} C _{and K: T}2 _{⇒ T}2 _{is a BD-law, we write Brd(}T_{, K)}

for the category having pairs

(X, x)∈Alg(T_{), c ∈K-Brd(X, x)}

as objects. An arrowf: (X, x), c → (X′

, x′

), c′

in Brd(T_{, K) is an arrow between the}

underlying T_{-algebras such that the diagram}

T X T f //

c

T X′

c′

T X _{T f} //_{T X}′

commutes.

We are ready to state our main result.

3.7. Theorem. _Let T_{= (T, m, e) be a monad on a category} C _{and let K: T}2 _{⇒ T}2

be a BD-law on T_{. The forgetful functor}

V : Brd(T_{, K)→Alg(}T₎

is comonadic, and the corresponding comonad on Alg(T_{) is the comonad} T∗

induced by the adjunction LT

⊣RT

between C _{and Alg(}T_).

Proof. _{We show thatV has a left adjoint and then apply Beck’s theorem. The free T}

-algebra functorLT

:C_→Alg(T_{) factorizes as L}T

=V J, whereJ :C_→Brd(T_{, K) is the}

functor sending an objectX ofC_{to the algebra (T X, mX) equipped with theK-braiding}

KX : T2X → T2X. The counit ǫ :LT

RT

→ 1 may be seen as a natural transformation

V J RT

= LT

RT

→ 1. For an object (X, x, c) of Brd(T_{, K), write β : X → T X for the}

composite

X eX //_{T X} c //_{T X}

and observe that, by commutativity of

T X _{eT X}//

x

T eX

**

T2_X

T x

KX //T2X

(17)

T c

//_T2_X

mX

and

T X _{eT X}//

c

T eX

**

T2X

T c

KX //T

2_X T c _//

(16)

T2X

KX

T X eT X//

T eX

44

T2X KX //_T2_X

T c //T

2_X

this makes β a map in Brd(T_{, K) from (X, x, c) to (T X, mX, KX). This gives the}

com-ponent at (X, x, c) of a natural transformationβ : 1→J RT

V.

The triangle equation ǫV.V β = 1 is precisely equation (15), while the other triangle equation J RT

ǫ.βJ RT

= 1 follows easily from the definitions of BD-law and of T_-algebra.

Thus there is an adjunction V ⊣ J RT, which clearly induces the same comonad as

LT

⊣RT

.

It remains to verify the Beck condition. Let f, g : (X, x, c)→ (Z, z, c′

) be morphisms in Brd(T_{, K), and let}

(W, w) i //_{(X, x)}

u

ii

f

//

g //(Z, z) v

\\

be a split equalizer diagram in Alg(T_{), with ui = 1, iu = gv, and f v = 1. Since T i is}

(split) monic, the only possibility for aK-braidingc′′

:T W →T W on (W, w) compatible with i is given by

T W T i //_{T X} c //_{T X} T u //_{T W}

and we need only check that this does indeed give a braiding; the fact that the resulting diagram is an equalizer in Brd(T_{, K) is then obvious. Thus we must check equations}

(15,16,17); instead, we allow the reader to contemplate the following diagrams at his or her leisure:

T W T i //_{T X} c //_{T X} T u //

x

T W

w

W eW

OO

i _//

1

33

X 1 //

eX

OO

T2X T

and the functor

Ψ : Brd(T_{, K)→Coalg(}T∗

1. The functor Ψ is an isomorphism of categories;

2. The map Ψ(X,x) is bijective.

Proof. _{Since the forgetful functor V : Brd(}T_{, K)→Alg(}T_{) is comonadic, the induced}

comparison functor Ψ : Brd(T_{, K)→Coalg(}T∗

3.9. Proposition. _{For a BD-law K : T}2 _{→ T}2 _{we have the following facts about a}

K-braiding c:T X →T X on an algebra (X, x): 1. The diagrams

T2_X T c //

4. If K is a BCD-law then the diagram

T X c //

Proof. _{In each case the proof goes as follows. Modify the definition of K-braiding so}

that the extra condition is assumed part of the structure, then check that the modified category Brd(T_{, K)}′

is still comonadic via the same comonad. This involves (i) proving that the cofree objects (T X, mX, KX) have the required property, and (ii) proving that in the split equalizer diagram in the proof of Theorem 3.7, the induced morphism c′′

=

T u.c.T i : T W → T W satisfies the condition if c: T X → T X does so. In each case (i)

is entirely straightforward: for example, the fact that the cofree objects satisfy (18) is precisely (9). We therefore check only (ii).

1. If c satisfies (18), then so does c′′

by commutativity of the diagram

while (19) is obtained by pasting together (17) and (18).

2 and 3. If cis invertible, then a straightforward calculation shows that T u.c−1_{.T i is}

inverse to T u.c.T i.

4. If xc=xthen the diagram

T X

x

!!

D D D D D D D D

c

//_{T X}

T u

##

G G G G G G G G

x

}}

zzzz zzzz

T W

T i_wwww;;

w w w w

w

##

G G G G G G G

G X

u

!!

D D D D D D D

D T W

w

{{

wwww wwww

W

i _zzz==

z z z z z

1 //W

commutes and so wc′′

=w.

3.10. Remark. _{IfK is a distributive law, condition (18) means thatcis a morphism}

c: T∗

(X, x)→T(X, x)

in Alg(T_{), where}T _{is the lifting of} T _{to Alg(}T_{) induced by the distributive law K.}

In fact we can use the Proposition to give two alternative formulations of the definition. One of them will be used in the following section, the other to make the connection with the “symmetries” of [11].

3.11. Corollary. _{In the definition of K-braiding, condition (17) can be replaced by}

(19), while if K is a BCD-law then (15) can be replaced by (20).

Proof. _{We have seen that for a K-braiding (19) always holds; conversely (17) follows}

easily from (19) by composing with eT X : T X → T2X. Similarly, if K is a BCD-law then (20) holds for any K-braiding, while (15) follows from (20) by composing with

eX :X →T X.

IfK is an involutive BCD-law on a monadT_{, then a symmetry on a}T_{-algebra (X, x),}

was defined in [11] to be an involutionc:T X →T X satisfying (16, 17, 20); combining the proposition and the corollary one now sees as promised that this is precisely aK-braiding.

4. Another proof

We proved our main theorem in the previous section; here we provide an alternative proof, which may be of interest to some readers. It exhibits the bijection which is the object part of the isomorphism Φ of Corollary 3.8 as being part of the natural bijection (between hom-sets) of an adjunction.

4.1. Proposition. _LetT_{= (T, m, e)be a monad on a category}C_{and let K: T}2 _⇒T2

be an invertible BD-law on T_{. For (X, x) a} T_{-algebra, we have a bijection}

K-Brd(X, x)∼=T∗

-coalg(X, x)

given by

Ψ(X,x):K-Brd(X, x)→T∗-coalg(X, x)

(c: T X →T X)→(r: X eX //_{T X} c //_{T X )}

Ψ−1 (X,x):T

∗

-coalg(X, x)→K-Brd(X, x)

(r: X →T X)→(c: T X T r //_T2_X KX //_T2_X T x //_{T X )}

As explained above, the proposition can be deduced from Corollary 3.8; our alternative proof occupies the remainder of the paper.

Let T _and S _{be monads on a category} C_{. A morphism of monads is a natural}

trans-formationϕ: S_⇒T_{such that}

S2 ϕ

2

+3

m

(21)

T2

m

S _ϕ +3_T

IdC

e

+3

e

"*

N N N N N N N N N N N N

N N N N N N N N N N N

N S

ϕ

(22)

T

Following [1], such a morphism ϕ induces a functor ϕ∗

: Alg(T_{) → Alg(}S_{) defined by}

ϕ∗

(X, x: T X → X) = (X, ϕX ·x: SX → T X → X). Moreover, ϕ∗

has a left adjoint

ϕ! : Alg(S_)→Alg(T_{) sending an} S_{-algebra (Y, y) to the objectϕ!(Y, y) in the coequalizer}

T SY T y //

T ϕY

##

G G G G G G G G

G T Y

q

//_{ϕ!(Y, y)}

T2Y

mY

<<

y y y y y y y y y

in Alg(T_{), provided that such a coequalizer exists. Indeed, if f: (Y, y) → ϕ}∗

(X, x) is a morphism of S_{-algebras, then x.T f: T Y →X coequalizes T y and mY.T ϕY. Conversely,}

fromg:ϕ!(Y, y)→(X, x), we get g.q.eY : (Y, y)→ϕ∗

(X, x).

IfK: T S⇒ST is a distributive law ofT_overS_{,we can consider the composite monad} ST _onC_{. Explicitly,}

ST_{= (ST:} C_→C_{, ST ST} SKT +3_S2_T2 mm _{+3ST , IdC} ee _{+3ST )}

Moreover, there are two morphisms of monads as in

Now, for any T_{-algebra (X, x: T X →X) and} S_{-algebra (Y, y: SY →Y), the adjunction}

ǫ2!⊣ǫ∗2 gives a natural bijection

Ψ : Alg(ST_{)[ǫ2!(X, x), ǫ1!(Y, y)]} ≃ // _Alg(T_{)[(X, x), ǫ}∗

2ǫ1!(Y, y)]

The bijection of Proposition 4.1 will turn out to be a particular case of this natural bijection. To see this, we give an explicit description of Ψ and of the hom-sets involved.

First of all, let us recall from [1] that the coequalizer defining ǫ₂!(X, x) always exists. In fact, it is given by the solid part of the following diagram in Alg(ST₎

ST T X ST x //

SmX //ST X

Sx

//

SeT X

}}

SX SeX



with action on SX given by

ST SX SKX //_S2_{T X} S

2 x

//_S2_X mX //_SX

Indeed, one easily verifies that the dotted arrows satisfy the equations for a split coequal-izer [2]. Recall also that if f is a morphism of ST_{-algebras coequalizing ST x and SmX,}

then the induced ST_{-algebra map out ofSX isf.SeX.}

Unfortunately, the existence of the coequalizer definingǫ₁!(Y, y) is not automatic. We need the following lemma, which makes sense because of Lemma 2.1:

4.2. Lemma. _{Consider two monads} S _and T _{on a category} C_{, and let K: T S ⇒ ST}

be an invertible distributive law of T _over S_{. Consider the composite monad} ST _induced

byK and the composite monad TS _{induced by K}−1_{. Then K: TS⇒ST is a morphism of}

monads, and it satisfies the following equations

TS

K

S

η2 _}}:B

} } } } }

} } } } } } }

ǫ1

$

A A A A A A A

A A A A A A

A T

η1

\d_AA

AA_AA A AA_AA

AA_A

ǫ2

z}}}} }}} }}}}

}}}

ST

where η1 =T e and η2 =eS.

Proof. _{The equations ǫ1 = Kη2 and ǫ2 = Kη1 are conditions (6) and (7), and they}

to the following equation

which can be proved using the distributivity equations three times. Indeed, the distribu-tivity admits the following graphical representation (this is condition (8))

• •

and then ǫ1!(Y, y) can be described by the following coequalizer in Alg(ST)

(K−1₎∗

with action on T Y given by

ST2Y SmY //ST Y K−

1_Y

//_{T SY} T y //_{T Y.}

We are now ready to describe the bijection Ψ.

4.3. Lemma. _{Consider two monads} S_and T_{on a category} C_{,and letK: T S⇒ST be}

(i) The hom-set Alg(S_{)[(Y, y), ǫ}∗

1(ǫ2!(X, x))] is the set of morphisms r: Y → SX such

that

SY Sr //

y

(12)

S2X

mX

Y _r //_SX

commutes.

(ii) The hom-setAlg(ST_{)[ǫ1!(Y, y), ǫ2!(X, x)] is the set of morphismsc: T Y →SX such}

that

T2SY

T2

y

T KY_//

T ST Y T Sc //_{T SSX}

T mX

T2Y

mY

(19) _{T SX}

KX

T Y _c //_{SX ST X}

Sx

oo

commutes.

(iii) The natural bijection

Ψ : Alg(ST_{)[ǫ1!(Y, y), ǫ2!(X, x)]} ≃ // _Alg(S_{)[(Y, y), ǫ}∗

1(ǫ2!(X, x))]

induced by the adjunction ǫ1!⊣ǫ∗1 is given by

Ψ(c) : Y eY //_{T Y} c //_{SX Ψ}−1_{(r) :}

T Y T r //_{T SX} KX //_{ST X} Sx //_SX.

Proof. _{Following the description ofϕ!⊣ϕ}∗

given at the beginning of this section, we have Ψ(c) and Ψ−₁

(r) given respectively by

Y eY //_{T Y} eT Y //_{ST Y} K−1 //_{T SY} T y //_{T Y} c //_SX

T Y eT Y //_{ST Y} ST r//_{ST SX} SKX//_{SST X} nT X //_{ST X} Sx //_SX

which are easily seen to reduce to formulas given in (iii).

For r: Y →SX, to be a morphism of S_{-algebras means that}

SY Sr //

y

S2X SeSX //ST SX

SKX

Y _r //_{SX S}2_X

mX

oo _S2_{T X}

S2

x

oo

For c: T Y →SX,to be a morphism of ST_{-algebras means commutativity of}

This is the best we can do working at this level of generality. ¿From now on, we assume

S₌T _{and (X, x) = (Y, y). We now combine the next two lemmas with the previous one}

to complete the proof of Proposition 4.1.

4.4. Lemma. _Let T _{be a monad on a category} C_{,and let K:T}2 _⇒T2 _{be an invertible}

distributive law on T_{. Fix a} T_{-algebra (X, x). In the bijection of Lemma 4.3, the map}

r: X →T X satisfies condition (13) iff c= Ψ−₁

(r) :T X →T X satisfies condition (15).

Proof. _{Condition (15) says precisely that Ψ(c) satisfies (13).}

(16)⇒(14) :

The proof of Proposition 4.1 is now complete.

References

[1] J. Beck, Distributive laws, in Seminar on Triples and Categorical Homology Theory

(Lecture Notes in Mathematics 80), pp. 119–140, Springer, Berlin, 1969.

[2] F. Borceux, Handbook of categorical algebra I, Cambridge University Press, 1994. [3] F. Borceux, Handbook of categorical algebra II, Cambridge University Press, 1994. [4] M. Cipolla, Discesa fedelmente piatta dei moduli, Rend. Circ. Mat. Palermo 25:43–

46, 1976.

[5] A. Davydov, Quasicommutative monoids, Lecture at Australian Category Seminar, 21 January 2004.

[6] A. Grothendieck, Technique de descente et th´eor`emes d’existence en g´eom´etrie alg´ebrique, I : G´en´eralit´es, descente par morphismes fid`elements plats, S´eminaire Bourbaki 190 (1959-1960).

[7] G. Janelidze and W. Tholen, Facets of descent III: monadic descent for rings and algebras, preprint, 2004.

[8] A. Joyal and R. Street, Geometry of tensor calculus I, Adv. Math.88:55–112, 1991. [9] C. Kassel, Quantum groups (Graduate Texts in Mathematics 155) Springer-Verlag,

New York, 1995.

[11] C. Menini and D. Stefan, Descent theory and Amitsur cohomology of triples, J. Algebra 266:261–304, 2003.

[12] P. Nuss, Noncommutative descent and non-abelian cohomology,K-Theory 12:23–74, 1997.

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