Directory UMM :Journals:Journal_of_mathematics:TAC:
Theory and Applications of Categories, Vol. 5, No. 5, 1999, pp. 91–147.
DISTRIBUTIVE LAWS FOR PSEUDOMONADS
F. MARMOLEJO
Transmitted by Ross Street
ABSTRACT. We define distributive laws between pseudomonads in a Gray-category
A, as the classical two triangles and the two pentagons but commuting only up to
isomorphism. These isomorphisms must satisfy nine coherence conditions. We also
define the Gray-category PSM(A) of pseudomonads in A, and define a lifting to be a
pseudomonad in PSM(A). We define what is a pseudomonad with compatible structure
with respect to two given pseudomonads. We show how to obtain a pseudomonad with
compatible structure from a distributive law, how to get a lifting from a pseudomonad
with compatible structure, and how to obtain a distributive law from a lifting. We show
that one triangle suffices to define a distributive law in case that one of the pseudomonads
is a (co-)KZ-doctrine and the other a KZ-doctrine.
1. Introduction
Distributive laws for monads were introduced by J. Beck in [2]. As pointed out by
G. M. Kelly in [7], strict distributive laws for higher dimensional monads are rare. We
need then a study of pseudo-distributive laws. The first step in this direction is quite
easy: just replace the two commutative triangles, and the two commutative pentagons of
[2] by appropriate invertible cells. The problem is to determine what coherence conditions
to impose on these invertible cells. We should point out that, in [7], there is a step in this
direction, keeping commutativity on the nose on the triangles and one of the pentagons,
and asking for commutativity up to isomorphism in the remaining pentagon, plus five
coherence conditions. The structure obtained from such a distributive law between two
strict 2-monads is not, in general, a strict 2-monad, and since that article deals exclusively
with strict 2-monads, what is obtained is a reflection result.
In this paper, instead of working with 2-monads we work with the more general pseudomonads. We will see that the structure obtained from a distributive law between
pseudomonads is a pseudomonad. We define a distributive law between pseudomonads
as we said above, that is to say, asking for commutativity up to isomorphism of the two
triangles and the two pentagons. We propose nine coherence conditions for these isomorphisms. See section 4 below. We observe that the coherence conditions of [7] and the ones
proposed in this paper coincide if in our setting we ask for commutativity on the nose of
the two triangles and one of the pentagons. Thus, the examples of distributive laws given
there are examples here as well.
But why exactly these nine coherence conditions?
Received by the editors 1998 June 24 and, in revised form, 1998 November 22.
Published on 1999 March 18.
1991 Mathematics Subject Classification: 18C15, 18D05, 18D20.
Key words and phrases: Pseudomonads, distributive laws, KZ-doctrines, Gray-categories.
c F. Marmolejo 1999. Permission to copy for private use granted.
91
Theory and Applications of Categories, Vol. 5, No. 5
92
A more conceptual approach to distributive laws for monads is given by R. Street in
[11]. It is shown that for a 2-category C, a distributive law is the same thing as a monad
in the 2-category MND(C), whose objects are monads in C. In this paper we introduce the
corresponding structure PSM(A), of pseudomonads for a corresponding three dimensional
structure A, see section 7.
In M. Barr and C. Wells’ book [1], exercise (Dl) asks to prove that, for monads,
a distributive law, a lifting of one monad structure to the algebras of the other, and a
monad with compatible structure with the two given monads, are essentially the same
thing. For pseudomonads we have already mentioned distributive laws. We define a lifting
as a pseudomonad in PSM(A) in section 8. In section 6, we define what a pseudomonad
whose structure is compatible with two given pseudomonads is.
We show how to obtain a composite pseudomonad with compatible structure from
a distributive law between pseudomonads, how to obtain a lifting from a pseudomonad
with compatible structure, and, closing the cycle, how to obtain a distributive law from
a lifting.
We see then, that the nine coherence conditions can be shown to hold if we define a
distributive law from a lifting. In turn, these coherence conditions allow us to define a
lifting from a distributive law between pseudomonads.
The situation for distributive laws between KZ-doctrines and (co-)KZ-doctrines is
a lot simpler. We show that either one of the triangles commuting up to isomorphism
(satisfying coherence conditions) is enough to obtain a distributive law. One such example
is the following. It is well known that adding free (finite) coproducts to categories is a KZdoctrine over Cat, and adding free (finite) products is a co-KZ-doctrine. There is a more
or less obvious distributive law of the co-KZ-doctrine over the KZ-doctrine. Observe
however that even if we arrange for these KZ-doctrine and co-KZ-doctrine to produce
strict pseudomonads, the distributive law obtained is not strict.
This article is possible thanks to the definition of tricategories given in [6]. It is
simplified by the fact that a tricategory is triequivalent to a Gray-category, a fact proved
in the same paper. We thus work in the framework of Gray-categories, as in [6], continuing
the development of the formal theory of pseudomonads started in [9].
This paper is organized as follows:
In section 2 we provide a brief description of the framework that we use, namely that
of Gray-categories. For more details we refer the reader to [6, 5].
In section 3 we recall the definition and some properties of pseudomonads given in
[9], the definition uses the definition of pseudomonoid given in [3]. We also define the
change of base 2-functors, change of base strong transformations and the change of base
modifications that we will need in later sections. Change of base turns out to be a Graynatural transformation.
In section 4 we define distributive laws for pseudomonads by replacing commutativity
on the nose by commutativity up to isomorphism. We give here the nine coherence
conditions that these isomorphisms should satisfy.
The first step to obtain compatible structures is to define a composite pseudomonad
Theory and Applications of Categories, Vol. 5, No. 5
93
from a distributive law. This is what we do in section 5.
In section 6 we define what a pseudomonad with compatible structure is with respect
to given pseudomonads. Furthermore, we exhibit the structure that makes compatible
the composite pseudomonad defined in the previous section.
We introduce the Gray-category PSM(A) in section 7, to define, in section 8, a lifting
as a pseudomonad in the Gray-category PSM(A).
In section 9 we show how to construct a pseudomonad in PSM(A), from given pseudomonads with compatible structure. In the following section, we go from a pseudomonad
in PSM(A) to a distributive law.
Section 11 deals with distributive laws of (co-)KZ-doctrines over KZ-doctrines. We
refer the reader to [9] for the definition and properties we use of KZ-doctrines, but see [8] as
well. We show that one triangle, plus coherence, is enough to produce a distributive law.
Compare with [10], where it is shown that one of the triangles suffices for a distributive
law between idempotent monads. The case of KZ-doctrines over KZ-doctrines is formally
very similar. In this latter case, we show that the composite pseudomonad is again a
KZ-doctrine.
I would like to thank the referee for helping improve the readability of this paper, and
for suggesting condition (12), after which all the conditions of section 6 were modeled.
2. Gray-categories
As in [9] we will work with a Gray-category A, where Gray is the symmetric monoidal
closed category whose underlying category is 2-Cat with tensor product as in [6]. A Graycategory is a category enriched in the category Gray as in [4]. We will briefly spell out
what this means, and we refer the reader to [6] and [4] for more details.
A Gray-category A has objects A, B, C, . . . . For every pair of objects A, B of A, A
has a 2-category A(A, B). Given another object C in A, A has a 2-functor A(C, B) ⊗
A(A, B) → A(A, C). This 2-functor corresponds to a cubical functor M : A(B, C) ×
A(A, B) → A(A, C). We will denote M by juxtaposition, M (G, F ) = GF for F ∈
A(A, B) and G ∈ A(B, C). Given f : F → F ′ in A(A, B) and g : G → G′ in A(B, C) we
will denote the invertible 2-cell Mg,f by
GF
Gf
GF ′
gF
/ G′ F
gf
gF ′
G′ f
/ G′ F ′ .
What the definition of being cubical means for M is the following: Given ϕ : f → f ′ :
F → F ′ , and f ′′ : F ′ → F ′′ in A(A, B), and γ : g → g ′ : G → G′ , and g ′′ : G′ → G′′
in A(B, C), we have that ( )F : A(B, C) → A(A, C) and G( ) : A(A, B) → A(A, C)
are 2-functors, ( )f : ( )F → ( )F ′ and g( ) : G( ) → G′ ( ) are strong transformations,
( )ϕ : ( )f → ( )f ′ and γ( ) : g( ) → g ′ ( ) are modifications, and the following three
Theory and Applications of Categories, Vol. 5, No. 5
94
equations are satisfied
gF
γF * ′
4GF
GF
g′ F
sk _
__
_ Gf
gf′
Gf ′
GF ′
gF ′′
gF
GF
/ G′ F
G′ f
gf ′′
Gf ′′
G′ f ′ ks __ G′ f
G′ ϕ
+
′ ′
′
γF 3 G F ,
gF ′
g′ F ′
=
/ G′ F ′
gF ′
/ G′ F
gf ′
GF ′
/ G′ F
gf
Gf
gF
GF
Gf ′
/ G′ F ′
g′ F ′
gF
GF
GF ′′
G′ f
Gϕ
GF ′
=
gf ′′ ◦f
G(f ′′ ◦f )
G′ f ′′
/ G′ F ′′ ,
/ G′ F ′′
G′ (f ′′ ◦f )
GF ′′
gF ′′
=
GF
and
GF
Gf
GF ′
gF
gf
gF ′
g ′′ F
/ G′ F
gf′′
G′ f
/ G′ F ′
/ G′′ F
g ′′ F ′
G′′ f
(g ′′ ◦g)F
G′′ f
Gf
/ G′′ F ′
/ G′′ F
GF ′
(g′′ ◦g)f
/ G′′ F ′ ,
(g ′′ ◦g)F ′
and if either f or g is an identity, then gf is an identity 2-cell. Now, for every object A of
A, there is a distinguished object 1A . The triangle in the definition of enriched categories
means that the action of multiplying by 1A is trivial. Now, the pentagon means that for
another object D in A, and κ : k → k ′ : K → K ′ in A(C, D) the following equations hold:
(KG)F = K(GF ),
(KG)f = K(Gf ),
(Kg)F = K(gF ),
(kG)F = k(GF ),
(KG)ϕ = K(Gϕ),
(Kγ)F = K(γF ),
(κG)F = κ(GF ),
(Kg)f = K(gf ),
(kG)f = kGf , and (kg )F = kgF .
We will use these properties freely, without further mention.
3. Pseudomonads
For the convenience of the reader we will recall here the definition of a pseudomonad in
a Gray-category A, for more details we refer the reader to [9]. We adopt the definition of
pseudomonoid given in [3].
Theory and Applications of Categories, Vol. 5, No. 5
95
3.1. Definition. A pseudomonad D on an object K of a Gray-category A is a pseudomonoid in the Gray monoid A(K, K).
We give now, in elementary terms, what this means. A pseudomonad D as above
consists of an object D in A(K, K), and 1-cells d : 1K → D, and m : DD → D and
invertible 2-cells
dD
Dd
Dm /
DDD
DD
D DD / DD Yao : z D
:::
DD ~ β
η zzz
DD
zz
IdD DDm
! }zz IdD
µ
mD
D
DD
m
/ D,
m
such that the following two equations are satisfied:
DDm
DDDD
EE
/
DDD
EE
mDD
EEDmD Dµ
EE
E" ⇐=
DDD
DDDEE
µD
mD
E⇐=
EE
E"
Dm
EEDm
EE
E"
/ DD
µ
m
⇐=
mD
DD
/
DDD
JJ
JJDm
JJ
JJ
$
mD
m−1
m
DD
⇐=
µ
/
⇐=
DDDLL Dm DDJJ
=
LLL
LL
mD LL%
µ
JJm
J
(1)
m
⇐= JJJ$
DD m / D,
DDDJ
DD F
FFm
FF
#
DdD/
µ
⇓ ;D
DD
DDDH
HH
xx
xxm
H$
x
mD
vv
DDm
mDD
/D
m
Dmvv:
DDDD
DdD tt:
ttt Dβ ⇓
JJDm
JJ
$
/ DD m / D.
DD J
JJJ ηD ⇓ tt:
J
tttmD
DdD $
=
(2)
DDD
DD
It is shown in [9] that the following three equations hold for any pseudomonad D:
DD
JJJm
:
JJ
t
t
t IdD β ⇓ J$
d /
/ D
D JJ
η ⇓ tt:
JJJ
t
tttm
Dd J$
< D DD
DDdD
D"
z
zz −1
/
1KDD dd ⇓ DD m D,
<
DD
z
D
zz
d D" zz Dd
d zzz
dDttt
1K
=
DD
DDEE
dDD
/
DDD
Dm
EE βD
EE
EE⇐ mD µ
⇐=
IdDD EE
"
DD
m
(3)
D
dDD /
DD
/
DD
dm
m
m
/ D
=
DDD
⇐=
D EE
Dm
/ DD
EE dD
EE β
EE⇐ m
IdD EEE
E"
D,
(4)
Theory and Applications of Categories, Vol. 5, No. 5
96
DDEE
EE
EEIdDD
E
Dη EEEE
"
⇐
/
DDD
DD
DDd
Dm
µ
mD
DD
=
m
⇐=
/ D
m
m
/D
EE
EE
EEIdD
md
DDd
Dd η EE
⇐=
⇐ EEE"
DDD mD / DD m / D.
DD
(5)
We recall as well the 2-categories of algebras for a pseudomonad D. Let X be another
object in A. An object in the 2-category D-AlgX consists of an object X in A(X , K),
together with a 1-cell x : DX → X and invertible 2-cells
dX
X DD / DX
DD ~
IdX
DDX
DD ψ x
DD
"
/ DX
χ
mX
X
Dx
DX
x
x
/ X,
such that the following two equations are satisfied
DDDX
EE
DDx
/
DDX
EE
mDX
EEDmX Dχ
EE
E" ⇐=
DDX
DDXEE
Dx
EEDx
EE
E"
/ DX
χ
x
µX
mX
E⇐=
EE
E
mX "
⇐=
DX
=
/
DDX
JJ
JJDx
JJ
JJ
$
mX
m−1
x
DX
⇐=
χ
/
⇐=
DDXLL Dx DXJJ
LLL
LL
mX LL%
(6)
x
JJ x
JJ
⇐= JJ$
/ X,
DX
χ
x
DDXJ
DX F
FFxF
F#
DdX/
χ ⇓
DX
DDXH
;X
HH
xx
x
x
H$
xx
mX
vv
DDx
mDX
/X
x
Dx vv:
DDDX
=
DdX tt:
ttt Dψ ⇓
JJDx
JJ
$
/ DX x / X.
DX J
JJJ ηX ⇓ tt:
J
tttmX
DdX $
(7)
DDX
DX
It is shown in [9] that for every object (ψ, χ) in D-AlgX , the following equality holds:
dDX /
DX II
DDX
Dx
II βX
II
mX χ
I
Id II$
DX
x
/ DX
x
/X
= DX
x
dDX /
DDX
dx
dX
Dx
/ DX
X II
II
II ψ x
II
Id II$
X.
(8)
Theory and Applications of Categories, Vol. 5, No. 5
97
A 1-cell (h, ρ) : (ψ, χ) → (ψ ′ , χ′ ) in D-AlgX consists of a 1-cell h : X → X ′ in A(X , K),
together with an invertible 2-cell
Dh
DX
/ DX ′
ρ
x
X
x′
/ X ′,
h
that satisfies the following two equations:
Dh
dX
X DD / DX
DD ~
Id
DD ψ x
DD
"
/ DX ′
ρ
X
=
x′
Dh
dX ′
X ′ EE
/X
(9)
/ DX
dh
h
h
dX
X
/ DX ′
EE ~ ψ′
EE
x′
Id EE"
X ′,
DDXEE
DDh
mX
EEDx
EE
E"
/
′
DDX
EE
Dρ
⇐=
DX
DXEE
χ
Dh
EEDx′
EE
E"
/DX ′
ρ
x
E⇐=
EE
E"
mX
m−1
=
LLL
x LLL%
/ ′
X
h
JJDx′
JJ
JJ
$
mX ′
h
DX ′
⇐=
χ′
DXLL Dh / DX ′JJ ⇐=
′
L
X
/
′
DDX
JJ
x′
⇐=
x
DDh
DDX
(10)
JJx′
x
JJ
J
⇐=
J$
/X ′.
X
ρ
h
A 2-cell ξ : (h, ρ) → (h′ , ρ′ ) : (ψ, χ) → (ψ ′ , χ′ ) is a 2-cell ξ : h → h′ such that the following
condition is satisfied:
Dh
DX
*
Dξ 4 DX ′
Dh′
x
X
ρ′
h′
x′
/ X′
= DX
Dh
/ DX ′
ρ
x
X
h
(11)
x
)
ξ 5 X ′ .
h′
Given another object Z of A, and K ∈ A(Z, X ), we can define a change of base
: D-AlgX → D-AlgZ . If ξ : (h, ρ) → (h′ , ρ′ ) : (ψ, χ) → (ψ ′ , χ′ ) is in D-AlgX ,
2-functor K
is ξK : (hK, ρK) → (h′ K, ρ′ K) : (ψK, χK) → (ψ ′ K, χ′ K). If
then its image under K
→ K
′ such that
k : K → K ′ then we define the strong transformation
k : K
k(ψ,χ) =
−1
−1
′
′
(ψ,χ) = Xκ defines
(Xk, xk ) and k(h,ρ) = hk . If κ : k → k : K → K in A(Z, X ), then κ
′
a modification κ
: k → k . We have actually defined a Gray-functor D-Alg : Aop → Gray.
For every object Z, we have an obvious forgetful 2-functor D-AlgZ → A(Z, K). These
2-functors define a forgetful Gray-natural transformation Φ : D-Alg → A( , K).
Theory and Applications of Categories, Vol. 5, No. 5
98
4. Distributive laws
Let D = (D, d, m, βD , ηD , µD ) and U = (U, u, n, βU , ηU , µU ) be pseudomonads on the same
object K of the Gray-category A. A distributive law of U over D consists of a 1-cell
r : U D → DU in A(K, K), together with invertible 2-cells
U
U
= DFF
= DFF
zz
FFr
FF
F
ω1 F#
/ DU
uD zz
D
Ur
UUD
zz
zz
Du
rU
/ U DU
UD
U
/ DU U
ω3
nD
r
{{
FFr
FF
F
ω2 F#
/ DU
U d {{
{{
{{
dU
r
UD
O
Dn
/ DU
O
4444 ω4
mU
/ DDU
Um
/ DU
U DD
/ DU D
Dr
rD
subject to the following coherence conditions:
(coh 1)
u
/ U K Ud / U D
KK
GGu} ssss
KK
GGd−1
KK y zzzz r
u
G
K 2
d GG
dU KKω
%
#
/ DU
D
1 GG
=
=U
zz
z
z
z
1 DD
DD
d !
u
Du
II U d
II
I$
ud U D
LL r
u:
uu
ω1 LLL%
u
u uD
/ DU.
D
Du
(coh 2)
U uD /
UUD
44 nn
44 ηs{ D−1
44U
44
4
nD
Id 44
44
44
44
U D4
UD
Ur
/ U DU
=
U uD /
UUD
HH
HH U ω1 U r
H
U Du HH$
U D HH
rU
px hh
Dn
ω3
−1
ru
DU u /
DU HH
/ DU
r
U DU
r
DU U
rU
DU U
HH −1
HHDηU Dn
H
Id HH$
DU.
(coh 3)
UUd
U U LL / U U D
LLL
U ω2
LL
Ur
U dU LLL&
U DU
dU U
n
ω2 U
rU
"
DU U
U
d
n
dU
Dn
/ DU
=
UU
n
UUd/
U U DJJ
n−1
d
Ud /
nD
JJ U r
JJ
JJ
J%
U II
UD
U DU
II ω 888
ω3
II 2
88
rU
II
88
II
II
8
r
88 DU U
II
dU III 88
II 8
II 88 Dn
II 8
$
DU.
Theory and Applications of Categories, Vol. 5, No. 5
99
(coh 4)
U U r/
= U U U DU U/ rU U DU
U U DU
AA
DD
55
DD
AA
DDU rU
55
AA
DD
A
55
U rU
A
DD
55
!
nU D U nD 5
U
DU
U
AA
nU D
nDU
55
U
DU
AU
AA U Dn
AA
55
AA
AAU Dn
| U ω3
55
AA
AA
U
~ n−1
ω rU
~
r
AA
3U
/ U DU
U
U
D
UUD
55
Ur
U U DD U r / U DUDD DU UAU U DU
55
DD
AA
rU
DD
55 u} ssss
DD
AA| −1
DDDnU
55 µU D
DD
AArn rU
DD
55
D
AA
DU
n
rU
D
nD
DD
DU U
D!
nD 5
D
55
DD
DU
U
DU
U
D
55
A
u} rrrr ω3Dn
nD DD
AA
5
DD ~
A
A|ADµU Dn
DD ω3
/ DU
DD
UD
Dn AAA
r
D!
/ DU.
UD
U U U5 D
r
(coh 5)
DD QQQ
uDD
/ U DD
QQQ
QQQ
rD
Q ~
DuD QQQω( 1 D
DU D
~
Dω1
DDu
Um
/ UD
} ω4
r
=
GG
GG
m GGG
#
DDu
Dr
%
/ DU
DDU
mU
uDD
DDGG
DDU
/ U DD
Um
um
D GG uD / U D
GG
G G ω1 r
mu
GG
Du G#
/ DU.
mU
(coh 6)
U U m/
U U DD
nDD
nD
=
U U DDD
n−1
m
DD U r
DD
DD
"
U DDGGU m / U DD
D
G
U DUDD
DD rU
DD
DD
!
DU U
DD ~ ω3
DD
DD
DD r
Dn
DU DDD DDDD
DD ~ ω4 DD
DD
DD
DD
Dr DD"
!
/ DU
DDU
GG
G
rD GG#
mU
UUm
/ UUD
33
33
33
33
3U3 r
U DU DD
U DD
D
33
z
DDU Dr
z
3
DD
zz
uuuu
~
v
z
DD
z
U ω4 33
"
|zz rU D
/ U DU
ks __
rD ks __ DU U D
U
DDU
−1
DD
ω3 D
U mU
DD rr
zz
{{
DD
zz
{{
{
z
{ DnD DU r DD
z rDU
{
z
|z
"
}{
ks ____
rU
DU D
DU DU
Dω3
zz
zz
z
~
ω4 U
z
|zz DrU
/
Dr
DDU U mU U
DU U
zz
{{
z
{
z
{
~ mn
{{
zz
}{{ DDn
|zz Dn
/ DU.
DDU
U U DD
DD
{{
{{
{
{
}{{ nDD
mU
DDU rD
DD
DD
"
Theory and Applications of Categories, Vol. 5, No. 5
100
(coh 7)
Id
U D QQQ
U dD
−1
U βD
/ U DD
&
/ U DU
Um
QQQ
QQQ
rD
Q ~
dU D QQQω( 2 D
DU D
r
~ dr
DU
dDU
= idr .
r
~ ω4
Dr
mU
/ DU
8
Um
%
/ UD
/ DDU
β U
D
Id
(coh 8)
Id
U Dd
UD
U ηD
/ U DD
~ r−1 rD
d
/ DU D
r
DU QQQ DU d
QQQ
QQQ
Dr
Q ~
DdU QQDω
Q( 2
DDU
= idr .
r
} ω4
−1
ηD U
/ DU
9
mU
Id
(coh 9)
U Dm /
U DDNN
PPP
NNNU m
PPP
NN
rDD
P
| U µD NNNN
U mD PPP(
'
DU DD
U DD U m / U D
U DDDPP
U Dm
= U DDD
rDD
DU DD
DU m
/ U DD
NNN
NNNU m
NNN
{ r−1 rD
NN&
m
/ DU D
UD
DrD
DrD
DDU DPP
DDr
| ω4 D
PPP
PPP
P
mU D PPP(
DDDUPP
DDU D
rD
r
mU
y zzzzω4
/ DU
y {{{{ω4
r
/ DDU
NNN
PPDmU
NNNmU
PPP
{
NNN
PPP
{
{
{
y
NN&
µD U
PP'
mDU
/ DU.
DDU
DDDUPP
DU D
PPP
PPP
Dr
P |
r
mDU PPPm
(
DDU
{ Dω4
DDr
Dr
mU
Observe that if the pseudomonads are strict (β, η and µ are identities), and ω1 , ω2
and ω3 are identities, then we obtain the coherence conditions of the “mild” extensions
of the classical distributive laws given in [7].
Theory and Applications of Categories, Vol. 5, No. 5
101
5. The composite pseudomonad given by a distributive law
Assume we have a distributive law of U over D as in section 4. The first question is how
to produce a composite pseudomonad from U, D, and the distributive law. This is what
we do in this section.
Define V = (V, v, p, βV , ηV , µV ) as follows: V = DU ∈ A(K, K); v is defined as the
composite
1K
u
/U
dU
/ DU ;
p is the composite
DU DU
DrU
/ DDU U DDn / DDU mU / DU ;
βV is defined to be the pasting
DU DU
rr9 O MMMMMDrU
MMM
rr
MM&
rrr d−1
__uDU
3
+
Dω
U
1
U
U> DU
DDU
LLL
q8
}}
DuDU qqq
LLDDn
}
LLL
q
}
q
}
q
DDβ
LL%
q
U
}
DDuU
q
}
q
uDU }}
/ DDU
}
II
hhh3 DDU
}}
h
h
h
}
II mU
}
hhh
h
II
h
}
h
h
h
}
II
h
h
}
hh dDU
I$
βD U
}}hhhhhhh
/ DU ;
DU
dU DUrrr
Id
ηV as the pasting
Id
Id
/
/ DU ;
l5 DU VVVVVV
:
l
l
v
V
l
HH
V
v
l
V
VVVV
v
ηD U
DηU
ll
HH
v
l
l
V
v
V
l
H
ll Dn
vv
DU u HH
DdU VVVVVV
#
vv mU
lll
V*
DU UFFXXXXXX
DDU
−1
r9
Ddn
XXXXX
r
FF
r
XXXXX
FF
rr
XXXXX
FF
rrrDDn
XXXXX
DdU U
r
FF
r
+
FF
U
F
Dω2 U −1 DDU
DU dU FF
qq8
FF
q
q
FF
qqq
FF
qqq DrU
"
DU HH
DU DU
Theory and Applications of Categories, Vol. 5, No. 5
102
and µV as the pasting
DU DrU
/ DU DDU U DU DDn / DU DDU DU mU
Dr−1
Dr−1
DrDU U
DrDU
rU
Dn
/ DDU DU U
/ DDU DU
DU DU DU
DrU DU
DDU U DU
DDU rU
DDrn−1
DDrU U
DDDU U U
DDU DU
DDrU
m−1
rU
mU DU
DU DU
mDU U
/ DDU U
DrU
DDDµU
DDDn
/ DDDU
DDDn
m−1
Dn
Dω4 U
/ DDU U
DDrU
/ DDDU U
DDDU n
DDω3 U
DDDnU
/ DDDU U
DrU
DDU Dn
DDnDU
/ DU DU
DmU U
Dmn
DmU
/ DDU
µD U
mDU
/ DDU
DDn
DDn
mU
mU
/ DU.
5.1. Theorem. V = (V, v, p, βV , ηV , µV ), as defined above, is a pseudomonad on the
object K.
Proof. Observe that the pasting of µV and ηV V −1 is
/
/ RR
/
EE
RRR
EE
R
RRDω
EE
RRR2 U DU
EE
−1
RRR
R)
Dη
EE U DU
EE
EE
EE
EE
" RDdnDU
RRR
RRR
RRR
RR
RRR)
−1
DrrU
/
−1
DrDn
/
DDrn−1
DDω3 U
/
m−1
rU
/
ηD U DU −1
+
/
/
DDDµU
/
m−1
Dn
/
/
Dω4 U
Dmn
µD U
/
/
/ .
We must show that this pasting equals p ◦ V βV . Substitute the pasting of DdnDU and
DηU DU −1 by the pasting of Dd−1
U uDU and DDηU DU . Then use (coh 2). With the help of
(2) show that
/
−1
DDruU
DDDηU
=
/
−1
DDrn
/
U −1
/
5
lll
lll
l
l
l
lll
lll
RRR
RRR
RRR
RR
DDU DβU RRRR)
/
DDDµU
-
DDrU
/
DDDn
,
Theory and Applications of Categories, Vol. 5, No. 5
103
and make the substitution. Make the substitution
/
/
Dω2 U DU
−1
DrrU
+
=
/: J
:
tt JJJ
tt
t
t
tt DU drU ttt DU dDn JJJ
JJ
tt
tt
tt
/$
/ Ht
/
HH
H
HH −1 HH −1 Dω2 DU
H HDdU rU H HDdU Dn
HH
HH
H$
H$
/
/+ ,
=
: J
tt JJJ
tDU
t
ω 1 U JJJ
t
JJ
tt
tt
/$
/
−1
DrDn
/
/
followed by the substitutions
8 N
ppp NNNNN
p
p
NNN
ppp
NN&
−1
pppks _
__
_
DdU rU
NNN
N
8
−1
N
p
NNN
DdU uDUppp
NNN
N
p
DDU
ω
U
NNN
N
p
1
N
p
N
p
NN&
NN&
−1
ppp
/ NDd
NNNU Dn
N
DDU DβU NNNN
NN&
.
JJ
JJ
JJ
J
JJ. $
DU DβU
DdU DU
,
and
/
/
ηD U DU −1
m−1
rU
+
/
/
=
/
m−1
Dn
DdrU
/
/
DdDn
/
ηD DU −1
+ ,
and
/
DdDn
ηD
=
/
Dmn
/
DU −1
/
/
5
lll
lll
l
l
l
lll
lll
RRR
RRR
RRR
R
Dβ
DU
U RRRR
D
R)/
µD U
-
DDn
mU
.
Use (coh 7), and finish with the substitution
*
DU DβU
=
/
/
PPP
PPP
PDU
PPPω1 U
PPP
(
DU drU
/
DU dDn
/
8
ppp
p
p
ppp
ppp
/
/
p8 NN
NNN
p
DU d−1
uDU
NNN
pppDU Dω1 U
p
p
N
p
ppp
DU DDβU
NN&
/
-
Theory and Applications of Categories, Vol. 5, No. 5
104
In regards to the other condition, observe that the pasting of V µV , µV V and µV is:
/ L
/ L
/ L
LLL
LLL
LLL
LLL
LLL
L
L
LLL
L
L
L
L
L
LL%
LLL
−1 LL
−1 LL
DU DrrU
%/ DU DrDn
%/
LL
−1
LLDr
L
L
L
L
L
L
LLLrU DU
LLL
LLL DU Dω4 U LLLL
LLL
LLL
LLL
LLL
L
LLL
L%
L% DU DDrn−1LLL%
L%
LLL
/ L
/ L
LLL
LLL
LLL
L
LLL
LLL
LLL
LLL
LLL
LLL
L
L
L
L
−1
L
LL DU DDω3 U LL DU DDDµU LL
LLDr
DU Dmn LL%
%
%/
%/
LLDnDU
/ L
LLL
LLL
LLL
LLL
LLL
LLL
LLL
LLL
LLL
LLL
LLL
LLL
L
L
L
L
−1
LLDDr
LL%
LL%
−1 LL
DU µD U LL%
LLL nDU LL%
DU m−1
DU
m
%
rU
Dn
/
/
/
U
LLL
LLDDω
LLL 3
LLL
LLL
LLL
L%
%
−1
−1
DrrU
DrDn
LLLm−1
LDω
LLL
LLL4 U DU
rU
DU
LLL
LLL
L
LLL
LLL
LLL DDDµU DU
L%
%
LLL
/
/
LLL
LLL
LLL
LLL
Dω4 U
−1
DDrn
LLL
L
LLL −1 L% Dm
/
/
LLmLDnDU LLL nDU
LLL
LLL
DDDµU
DDω3 U
Dmn
LLL
LL
%
%
/
/
/
LLµ
LLDLU DU
LLL
µD U
m−1
m−1
rU
Dn
L%
/
/
/
Where we have only put the name of the corresponding 2-cell in each parallelogram. To
show that this pasting is equal to the pasting of pp , µV and µV we do the following. First
make the substitution
=
/ NN
/ NN
NNN
NN
NNN
NNN
NNNDU m−1 NNNNDU m−1 NNNNN
NN& rU
NN&
NN& Dn
/
/
__
_
NNN ks _
Dω
NNN4 U DU
NNN
NN&
/
−1
D
DrDn
/ OO
ks __
OOO
OOO
OOODω4 DU
OOO
OOO
OOO Dm−1 OOO Dm−1 OOOO
OOO U rU OOO U Dn OOO
'/ .
'
'/
−1
DrDn
/
−1
DrrU
/
−1
Dr
DrU
/
−1
DDr
rU
/
/ OO
OOO
OOO
−1
OOO
Dr
DDn
O'
/
Then make the substitutions
/ L
/ L
LLL
LL
LLL
LLL
L
LLLDm−1 LLLLL
LLLDm−1
L% U rU
LL% U Dn
LL%
/
/
−1
DDrn
LLL ks __
Dm
LLLnDU
LLL
L%
/
DDDµU
DDω3 U
/
/
=
/ L
LLL
LLL
LLL
/ L DmrU %
LLL
DDDω3 U
LL
L
DDDDµU
L
DmDn LL%
/
/
LLL
LLL
LLL
LLL
LL
LLL
−1 LL
L
LLL Dm−1
Dm
L
L% rU
LL% Dn
LLL%
/
/
/
−1
DDDrn
Theory and Applications of Categories, Vol. 5, No. 5
105
and
/ NN
/ NN
NNN
NNN
NNN
NNN
NNN
−1 NN
NNN
DU
DDr
N
n NNN
NN&
NNN
N&
/ NN
NNN
NNN
NNN
NNN
NNN
NNN
NNN
−1
NNNDrDnDU NNDU
DDDµU NNNN
DDω3 U NNDU
N
N
N
NN&
N
NNN
N&
N&
/
/
NNN
NNN
N
NNN
−1
−1
−1 NNN
DrDDn
DrDrU
NNDDr
NNN nDU NNNN
/
&
NNN
/
NNN
−1
−1
NNN
DDrrU
DDrDn
NNN
/
&
/
=
/
/ OO
OOO
OOO
OOO
O'
−1
/
/ OO DrDrU
OOO
O
OOO
OOO
−1
−1
DDr
DDrU
O
OOO
OOO
U Dn
rU
OOO
O
O
/
/
'
'
−1
−1
OOO DrDDn
OOO
OOO DDrrU
OOO
OOO
OOO
OOO
OOO
OOO DDDU r−1 OOO
OOO
OOO
n
OOO
OO
OOO
OOO/'
'
−1 O'
OOO
DDr
O
O
OOO Dn
OOO
OOO
OO
OO
OOO
DDDU
DµU OOOO
OOO ω3 U OOOODDDU
OO'
O
O'/
O'
/ .
−1
DrDU
rU
−1
DrDU
Dn
Now use (coh 4). Proceed with the following substitutions
=
/ M
MMM
MMM
MMM
MMM
M
MDDDU
MMM
Dµ
MMM
U
MMM
M
M
&/
&
−1
DDDr
MMM nU
MMM
−1
M
MM
DDDrn
MMM
DDDDµ UM&
/
MMM
U
MMM
DDDDµU
M
MM
MMM
M&
/
/ N
NNN
NNN
−1
DDDrU
NNN
n
−1 NNN
DDDr
n
&
/ N
NNN
N
−1
NN N
DDDDrn
NNN
DDDDµUNN&
/
MMM
LLL
MMM
LLL
L
L
M
M
M
LLDDDDµ
U
MMM
LLL
M&
&
/ ,
followed by
/ PP
PPP
PPP
PPP
PPP
P PDDDU
PPP rn−1
PPP
PP(
PP(
/
DDDω3 U U
=
DDDn−1
Dn
−1
DDDrU
n
PPP
PPP
PPP
PPP
P(
/
−1
DDDDrn
/
/ OO
OOO
OOO
OOO
O'
/ ODDDω
NNN
OOO 3 U
NNN
OOO
−1
N NDDDr
OOO
NNN n
OO'
N'
/ ,
Theory and Applications of Categories, Vol. 5, No. 5
106
and then
/ NN
/ NN
NNN
NNN
NN
NNN
N
NNN Dm−1 NNN Dm−1 NNNNN
rU
Dn
N
N
NN&
N
N&
/
/
__
_ &
NNN ks _
NµNDNU DU
−1
−1
mrU
mDn
NNN
NN&
/
/
=
/
/ NN
NNN
NNN
NN
__ N'
/ M
s
k
MMM
MMM µ DU
MMM
MD
MMM m−1 MMMM
m−1
rU
Dn
MM&
M
M&
/
/ ,
m−1
DrU
m−1
DDn
/
MMM
MMM
MMM
MM&
followed by the substitution
/ NN
/ NN
NNN
NNN
NN
NNN
N
NNNDDDrn−1NNNNN
NNN
NN&
NN&
NNN
/ NN
NNN
NNN
NNN
N
N
N
NNN
N
N
NNN DDDω U NN DDDDµNNNN
NNNm−1
NNN
N
3
U
DnDU
N
N
NN&
NNN
N&
NN&
/
/
NNN
NNN
−1
−1
m
m
NNN
DrU
DDn
NN&
/
/
=
/
/ OO
OOO
OOO
OOO
O'
/
/
−1
m
OOO
OOO
OOO DrU
OOO
OOO
OOO
OOO
OOO
OOO DDr−1 OOO
OOO
OOO
n
O
O
OOO
OOO
OO'
OO'
'
−1
OOO
/
m
O
O
OOO DDn
OOO
OOO
O
O
OOO
O
O
OOODDω3 U OOOOO DDDµU OOOOO
O'
O/'
O'
/ .
m−1
DU rU
m−1
DU Dn
Use (coh 9) to show that
/ OO
OOO
OO
OOO
OOO DU µD UOOOOO
OO'
OO'
/
=
OOO Dω4 DU
OOO
OOO
DmrU OO'
OOO
OOO
OOO
OO'
Dm
Dn
OOO
OOO
OO
µ
DU OOO'
D
OOO
OOO
OOO
OO'
/
Dmn
/
µD U
DDω4 U
/
4U
/ OO Dω
OOO
OOO
DDmn
Dmn OOO'
/
OOO
OOO
mmU
OO
µD U OOO'
/
OOO
OOO
O
OOO
OOO µD U OOOOO
OO'
OO'
/
Dω4 U
/ OO
OOO
OOO
OOO
/
'
−1
DrmU
Theory and Applications of Categories, Vol. 5, No. 5
107
and make the substitution. Next the substitution
/ NN
/ NN
NNN
NN
NN
NNN
−1 NN
−1 NN
NNDU
NNN DrrU NNDU
NNNDrDn NNNNN
−1
&/
&
&/
DU
NNN DrrU
−1
−1
NNN
DrDU rU
DrDU
Dn
NNN
NN&
/
/
−1
DDrU
rU
DDω3 U DU
NNN
NNN
NNN
N
−1 N&
DU
NNN mrU
NNN
NNN
NN&
/
/
/
/
/
/
m−1
DU rU
/
−1
DrU
DrU
−1
DDrU
Dn
DDDn−1
rU
=
DDDn−1
Dn
m−1
DU Dn
/
DDn−1
DrU
DDn−1
DDn
/ L DDω3 DU
LLL
LL
L
LLL
m−1
&
/
/
rDU
KK
KK
KKK
KK
KK
K
K
KK
K
−1 KK
−1 KK
KK DrrU
KKK
KK DrDn
KK
K
%
/%
/%
/
/ L
LLL
LLL
LLL
−1
/ LDrrDU &
LLL
LL
L
LLL
&
−1
DrU
DDn
m−1
U DrU
m−1
U DDn
,
followed by the substitution
/ M
/ N
NNN
NNN
MMM
−1
NNN
DrU
NNN
M
mU
M
NNN
M
NNN
M
−1
NNN
DrrDU NN& DU Dω4 U MMMM
/
MMM
NNN
NNN
LLL
LLL
NNN
N
M
NNN
LLL
LLL
N
NNN
MMM
−1 NNN
N
LLL
LLL
NNN
NNN DrrU
MMM
NNN
−1
LL&
LL& DrDrU
N
NNN
M&
& DDU ω4 U
&
/
NNN
N
N
=
NNN
MMM
LLL
LLL
NNN
NNN
N
MMM
LLL
LLL
NNN
NNN
NN
LLL DU Dmn MMM
LLL
N
N
−1 NN
N
N
N&
N& DrDn
N'
−1
LL&
LL& DrDDn
MM&
/
NNN
MMM
/
NNN
N
M
N
MMM
−1
NNN
DrmU
MMMDDU mn NNNNN
NNN
N&
M
NN&
&
/ .
/
−1
Substitute the pasting of DDU mn , DDω4 U , DDrDn
and DDDrn−1 by the pasting of
−1
DDω4 U U , DDrn and DDmU n . Then observe that the pasting of DDmU n , DDmn and
DDDDµU equals the pasting of DDDµU , DDmnU and DDmn . Use (coh 6). To finish
the proof, make the substitution
/
NNN
NNN
NNN
−1
mrDU NN&
LLL
LLL
LLL
LL&
/
MMM
MMM
m−1
U mU
MMM
MMM
DDω4 U
/
MMM
NNN
LLL
NNN
MMM
LLL
NNN
MMM
LLL
NN&
−1
MM&
mDrU LL&
/ M
=
LLL
LLL
MMM
LLL
LLL
LLL DDmn MMMMM
LLL
LL&
LL& m−1
MM&
DDn
/
NNN
NNN
m−1
mU
NNN
NN&
/
NNN
NNN
NNN
NNN
NNN
NNN
NN
NNN
NNN m−1 NNNN
rU
NN&
NNN
Dω4 U
NNN
NNN
NNN
NNN
NNN
NNN
NN&
NN&
/
MMM
MMM
MMM
MM&
NNN
NNN
NNN
−1
mDn NN'
NNN
NN
Dmn NNNN
NN&
/ .
Theory and Applications of Categories, Vol. 5, No. 5
108
6. Compatible pseudomonad structures
We consider now the question of when can a pseudomonad be considered as the composite
of two pseudomonads.
Let D, U be pseudomonads on the same object K of the Gray-category A. Given
another pseudomonad V = (V, v, p, βV , ηV , µV ) on the same object K, we say that V is
compatible with the pseudomonads D and U if V = DU and there are invertible 2-cells:
U
B
DISTRIBUTIVE LAWS FOR PSEUDOMONADS
F. MARMOLEJO
Transmitted by Ross Street
ABSTRACT. We define distributive laws between pseudomonads in a Gray-category
A, as the classical two triangles and the two pentagons but commuting only up to
isomorphism. These isomorphisms must satisfy nine coherence conditions. We also
define the Gray-category PSM(A) of pseudomonads in A, and define a lifting to be a
pseudomonad in PSM(A). We define what is a pseudomonad with compatible structure
with respect to two given pseudomonads. We show how to obtain a pseudomonad with
compatible structure from a distributive law, how to get a lifting from a pseudomonad
with compatible structure, and how to obtain a distributive law from a lifting. We show
that one triangle suffices to define a distributive law in case that one of the pseudomonads
is a (co-)KZ-doctrine and the other a KZ-doctrine.
1. Introduction
Distributive laws for monads were introduced by J. Beck in [2]. As pointed out by
G. M. Kelly in [7], strict distributive laws for higher dimensional monads are rare. We
need then a study of pseudo-distributive laws. The first step in this direction is quite
easy: just replace the two commutative triangles, and the two commutative pentagons of
[2] by appropriate invertible cells. The problem is to determine what coherence conditions
to impose on these invertible cells. We should point out that, in [7], there is a step in this
direction, keeping commutativity on the nose on the triangles and one of the pentagons,
and asking for commutativity up to isomorphism in the remaining pentagon, plus five
coherence conditions. The structure obtained from such a distributive law between two
strict 2-monads is not, in general, a strict 2-monad, and since that article deals exclusively
with strict 2-monads, what is obtained is a reflection result.
In this paper, instead of working with 2-monads we work with the more general pseudomonads. We will see that the structure obtained from a distributive law between
pseudomonads is a pseudomonad. We define a distributive law between pseudomonads
as we said above, that is to say, asking for commutativity up to isomorphism of the two
triangles and the two pentagons. We propose nine coherence conditions for these isomorphisms. See section 4 below. We observe that the coherence conditions of [7] and the ones
proposed in this paper coincide if in our setting we ask for commutativity on the nose of
the two triangles and one of the pentagons. Thus, the examples of distributive laws given
there are examples here as well.
But why exactly these nine coherence conditions?
Received by the editors 1998 June 24 and, in revised form, 1998 November 22.
Published on 1999 March 18.
1991 Mathematics Subject Classification: 18C15, 18D05, 18D20.
Key words and phrases: Pseudomonads, distributive laws, KZ-doctrines, Gray-categories.
c F. Marmolejo 1999. Permission to copy for private use granted.
91
Theory and Applications of Categories, Vol. 5, No. 5
92
A more conceptual approach to distributive laws for monads is given by R. Street in
[11]. It is shown that for a 2-category C, a distributive law is the same thing as a monad
in the 2-category MND(C), whose objects are monads in C. In this paper we introduce the
corresponding structure PSM(A), of pseudomonads for a corresponding three dimensional
structure A, see section 7.
In M. Barr and C. Wells’ book [1], exercise (Dl) asks to prove that, for monads,
a distributive law, a lifting of one monad structure to the algebras of the other, and a
monad with compatible structure with the two given monads, are essentially the same
thing. For pseudomonads we have already mentioned distributive laws. We define a lifting
as a pseudomonad in PSM(A) in section 8. In section 6, we define what a pseudomonad
whose structure is compatible with two given pseudomonads is.
We show how to obtain a composite pseudomonad with compatible structure from
a distributive law between pseudomonads, how to obtain a lifting from a pseudomonad
with compatible structure, and, closing the cycle, how to obtain a distributive law from
a lifting.
We see then, that the nine coherence conditions can be shown to hold if we define a
distributive law from a lifting. In turn, these coherence conditions allow us to define a
lifting from a distributive law between pseudomonads.
The situation for distributive laws between KZ-doctrines and (co-)KZ-doctrines is
a lot simpler. We show that either one of the triangles commuting up to isomorphism
(satisfying coherence conditions) is enough to obtain a distributive law. One such example
is the following. It is well known that adding free (finite) coproducts to categories is a KZdoctrine over Cat, and adding free (finite) products is a co-KZ-doctrine. There is a more
or less obvious distributive law of the co-KZ-doctrine over the KZ-doctrine. Observe
however that even if we arrange for these KZ-doctrine and co-KZ-doctrine to produce
strict pseudomonads, the distributive law obtained is not strict.
This article is possible thanks to the definition of tricategories given in [6]. It is
simplified by the fact that a tricategory is triequivalent to a Gray-category, a fact proved
in the same paper. We thus work in the framework of Gray-categories, as in [6], continuing
the development of the formal theory of pseudomonads started in [9].
This paper is organized as follows:
In section 2 we provide a brief description of the framework that we use, namely that
of Gray-categories. For more details we refer the reader to [6, 5].
In section 3 we recall the definition and some properties of pseudomonads given in
[9], the definition uses the definition of pseudomonoid given in [3]. We also define the
change of base 2-functors, change of base strong transformations and the change of base
modifications that we will need in later sections. Change of base turns out to be a Graynatural transformation.
In section 4 we define distributive laws for pseudomonads by replacing commutativity
on the nose by commutativity up to isomorphism. We give here the nine coherence
conditions that these isomorphisms should satisfy.
The first step to obtain compatible structures is to define a composite pseudomonad
Theory and Applications of Categories, Vol. 5, No. 5
93
from a distributive law. This is what we do in section 5.
In section 6 we define what a pseudomonad with compatible structure is with respect
to given pseudomonads. Furthermore, we exhibit the structure that makes compatible
the composite pseudomonad defined in the previous section.
We introduce the Gray-category PSM(A) in section 7, to define, in section 8, a lifting
as a pseudomonad in the Gray-category PSM(A).
In section 9 we show how to construct a pseudomonad in PSM(A), from given pseudomonads with compatible structure. In the following section, we go from a pseudomonad
in PSM(A) to a distributive law.
Section 11 deals with distributive laws of (co-)KZ-doctrines over KZ-doctrines. We
refer the reader to [9] for the definition and properties we use of KZ-doctrines, but see [8] as
well. We show that one triangle, plus coherence, is enough to produce a distributive law.
Compare with [10], where it is shown that one of the triangles suffices for a distributive
law between idempotent monads. The case of KZ-doctrines over KZ-doctrines is formally
very similar. In this latter case, we show that the composite pseudomonad is again a
KZ-doctrine.
I would like to thank the referee for helping improve the readability of this paper, and
for suggesting condition (12), after which all the conditions of section 6 were modeled.
2. Gray-categories
As in [9] we will work with a Gray-category A, where Gray is the symmetric monoidal
closed category whose underlying category is 2-Cat with tensor product as in [6]. A Graycategory is a category enriched in the category Gray as in [4]. We will briefly spell out
what this means, and we refer the reader to [6] and [4] for more details.
A Gray-category A has objects A, B, C, . . . . For every pair of objects A, B of A, A
has a 2-category A(A, B). Given another object C in A, A has a 2-functor A(C, B) ⊗
A(A, B) → A(A, C). This 2-functor corresponds to a cubical functor M : A(B, C) ×
A(A, B) → A(A, C). We will denote M by juxtaposition, M (G, F ) = GF for F ∈
A(A, B) and G ∈ A(B, C). Given f : F → F ′ in A(A, B) and g : G → G′ in A(B, C) we
will denote the invertible 2-cell Mg,f by
GF
Gf
GF ′
gF
/ G′ F
gf
gF ′
G′ f
/ G′ F ′ .
What the definition of being cubical means for M is the following: Given ϕ : f → f ′ :
F → F ′ , and f ′′ : F ′ → F ′′ in A(A, B), and γ : g → g ′ : G → G′ , and g ′′ : G′ → G′′
in A(B, C), we have that ( )F : A(B, C) → A(A, C) and G( ) : A(A, B) → A(A, C)
are 2-functors, ( )f : ( )F → ( )F ′ and g( ) : G( ) → G′ ( ) are strong transformations,
( )ϕ : ( )f → ( )f ′ and γ( ) : g( ) → g ′ ( ) are modifications, and the following three
Theory and Applications of Categories, Vol. 5, No. 5
94
equations are satisfied
gF
γF * ′
4GF
GF
g′ F
sk _
__
_ Gf
gf′
Gf ′
GF ′
gF ′′
gF
GF
/ G′ F
G′ f
gf ′′
Gf ′′
G′ f ′ ks __ G′ f
G′ ϕ
+
′ ′
′
γF 3 G F ,
gF ′
g′ F ′
=
/ G′ F ′
gF ′
/ G′ F
gf ′
GF ′
/ G′ F
gf
Gf
gF
GF
Gf ′
/ G′ F ′
g′ F ′
gF
GF
GF ′′
G′ f
Gϕ
GF ′
=
gf ′′ ◦f
G(f ′′ ◦f )
G′ f ′′
/ G′ F ′′ ,
/ G′ F ′′
G′ (f ′′ ◦f )
GF ′′
gF ′′
=
GF
and
GF
Gf
GF ′
gF
gf
gF ′
g ′′ F
/ G′ F
gf′′
G′ f
/ G′ F ′
/ G′′ F
g ′′ F ′
G′′ f
(g ′′ ◦g)F
G′′ f
Gf
/ G′′ F ′
/ G′′ F
GF ′
(g′′ ◦g)f
/ G′′ F ′ ,
(g ′′ ◦g)F ′
and if either f or g is an identity, then gf is an identity 2-cell. Now, for every object A of
A, there is a distinguished object 1A . The triangle in the definition of enriched categories
means that the action of multiplying by 1A is trivial. Now, the pentagon means that for
another object D in A, and κ : k → k ′ : K → K ′ in A(C, D) the following equations hold:
(KG)F = K(GF ),
(KG)f = K(Gf ),
(Kg)F = K(gF ),
(kG)F = k(GF ),
(KG)ϕ = K(Gϕ),
(Kγ)F = K(γF ),
(κG)F = κ(GF ),
(Kg)f = K(gf ),
(kG)f = kGf , and (kg )F = kgF .
We will use these properties freely, without further mention.
3. Pseudomonads
For the convenience of the reader we will recall here the definition of a pseudomonad in
a Gray-category A, for more details we refer the reader to [9]. We adopt the definition of
pseudomonoid given in [3].
Theory and Applications of Categories, Vol. 5, No. 5
95
3.1. Definition. A pseudomonad D on an object K of a Gray-category A is a pseudomonoid in the Gray monoid A(K, K).
We give now, in elementary terms, what this means. A pseudomonad D as above
consists of an object D in A(K, K), and 1-cells d : 1K → D, and m : DD → D and
invertible 2-cells
dD
Dd
Dm /
DDD
DD
D DD / DD Yao : z D
:::
DD ~ β
η zzz
DD
zz
IdD DDm
! }zz IdD
µ
mD
D
DD
m
/ D,
m
such that the following two equations are satisfied:
DDm
DDDD
EE
/
DDD
EE
mDD
EEDmD Dµ
EE
E" ⇐=
DDD
DDDEE
µD
mD
E⇐=
EE
E"
Dm
EEDm
EE
E"
/ DD
µ
m
⇐=
mD
DD
/
DDD
JJ
JJDm
JJ
JJ
$
mD
m−1
m
DD
⇐=
µ
/
⇐=
DDDLL Dm DDJJ
=
LLL
LL
mD LL%
µ
JJm
J
(1)
m
⇐= JJJ$
DD m / D,
DDDJ
DD F
FFm
FF
#
DdD/
µ
⇓ ;D
DD
DDDH
HH
xx
xxm
H$
x
mD
vv
DDm
mDD
/D
m
Dmvv:
DDDD
DdD tt:
ttt Dβ ⇓
JJDm
JJ
$
/ DD m / D.
DD J
JJJ ηD ⇓ tt:
J
tttmD
DdD $
=
(2)
DDD
DD
It is shown in [9] that the following three equations hold for any pseudomonad D:
DD
JJJm
:
JJ
t
t
t IdD β ⇓ J$
d /
/ D
D JJ
η ⇓ tt:
JJJ
t
tttm
Dd J$
< D DD
DDdD
D"
z
zz −1
/
1KDD dd ⇓ DD m D,
<
DD
z
D
zz
d D" zz Dd
d zzz
dDttt
1K
=
DD
DDEE
dDD
/
DDD
Dm
EE βD
EE
EE⇐ mD µ
⇐=
IdDD EE
"
DD
m
(3)
D
dDD /
DD
/
DD
dm
m
m
/ D
=
DDD
⇐=
D EE
Dm
/ DD
EE dD
EE β
EE⇐ m
IdD EEE
E"
D,
(4)
Theory and Applications of Categories, Vol. 5, No. 5
96
DDEE
EE
EEIdDD
E
Dη EEEE
"
⇐
/
DDD
DD
DDd
Dm
µ
mD
DD
=
m
⇐=
/ D
m
m
/D
EE
EE
EEIdD
md
DDd
Dd η EE
⇐=
⇐ EEE"
DDD mD / DD m / D.
DD
(5)
We recall as well the 2-categories of algebras for a pseudomonad D. Let X be another
object in A. An object in the 2-category D-AlgX consists of an object X in A(X , K),
together with a 1-cell x : DX → X and invertible 2-cells
dX
X DD / DX
DD ~
IdX
DDX
DD ψ x
DD
"
/ DX
χ
mX
X
Dx
DX
x
x
/ X,
such that the following two equations are satisfied
DDDX
EE
DDx
/
DDX
EE
mDX
EEDmX Dχ
EE
E" ⇐=
DDX
DDXEE
Dx
EEDx
EE
E"
/ DX
χ
x
µX
mX
E⇐=
EE
E
mX "
⇐=
DX
=
/
DDX
JJ
JJDx
JJ
JJ
$
mX
m−1
x
DX
⇐=
χ
/
⇐=
DDXLL Dx DXJJ
LLL
LL
mX LL%
(6)
x
JJ x
JJ
⇐= JJ$
/ X,
DX
χ
x
DDXJ
DX F
FFxF
F#
DdX/
χ ⇓
DX
DDXH
;X
HH
xx
x
x
H$
xx
mX
vv
DDx
mDX
/X
x
Dx vv:
DDDX
=
DdX tt:
ttt Dψ ⇓
JJDx
JJ
$
/ DX x / X.
DX J
JJJ ηX ⇓ tt:
J
tttmX
DdX $
(7)
DDX
DX
It is shown in [9] that for every object (ψ, χ) in D-AlgX , the following equality holds:
dDX /
DX II
DDX
Dx
II βX
II
mX χ
I
Id II$
DX
x
/ DX
x
/X
= DX
x
dDX /
DDX
dx
dX
Dx
/ DX
X II
II
II ψ x
II
Id II$
X.
(8)
Theory and Applications of Categories, Vol. 5, No. 5
97
A 1-cell (h, ρ) : (ψ, χ) → (ψ ′ , χ′ ) in D-AlgX consists of a 1-cell h : X → X ′ in A(X , K),
together with an invertible 2-cell
Dh
DX
/ DX ′
ρ
x
X
x′
/ X ′,
h
that satisfies the following two equations:
Dh
dX
X DD / DX
DD ~
Id
DD ψ x
DD
"
/ DX ′
ρ
X
=
x′
Dh
dX ′
X ′ EE
/X
(9)
/ DX
dh
h
h
dX
X
/ DX ′
EE ~ ψ′
EE
x′
Id EE"
X ′,
DDXEE
DDh
mX
EEDx
EE
E"
/
′
DDX
EE
Dρ
⇐=
DX
DXEE
χ
Dh
EEDx′
EE
E"
/DX ′
ρ
x
E⇐=
EE
E"
mX
m−1
=
LLL
x LLL%
/ ′
X
h
JJDx′
JJ
JJ
$
mX ′
h
DX ′
⇐=
χ′
DXLL Dh / DX ′JJ ⇐=
′
L
X
/
′
DDX
JJ
x′
⇐=
x
DDh
DDX
(10)
JJx′
x
JJ
J
⇐=
J$
/X ′.
X
ρ
h
A 2-cell ξ : (h, ρ) → (h′ , ρ′ ) : (ψ, χ) → (ψ ′ , χ′ ) is a 2-cell ξ : h → h′ such that the following
condition is satisfied:
Dh
DX
*
Dξ 4 DX ′
Dh′
x
X
ρ′
h′
x′
/ X′
= DX
Dh
/ DX ′
ρ
x
X
h
(11)
x
)
ξ 5 X ′ .
h′
Given another object Z of A, and K ∈ A(Z, X ), we can define a change of base
: D-AlgX → D-AlgZ . If ξ : (h, ρ) → (h′ , ρ′ ) : (ψ, χ) → (ψ ′ , χ′ ) is in D-AlgX ,
2-functor K
is ξK : (hK, ρK) → (h′ K, ρ′ K) : (ψK, χK) → (ψ ′ K, χ′ K). If
then its image under K
→ K
′ such that
k : K → K ′ then we define the strong transformation
k : K
k(ψ,χ) =
−1
−1
′
′
(ψ,χ) = Xκ defines
(Xk, xk ) and k(h,ρ) = hk . If κ : k → k : K → K in A(Z, X ), then κ
′
a modification κ
: k → k . We have actually defined a Gray-functor D-Alg : Aop → Gray.
For every object Z, we have an obvious forgetful 2-functor D-AlgZ → A(Z, K). These
2-functors define a forgetful Gray-natural transformation Φ : D-Alg → A( , K).
Theory and Applications of Categories, Vol. 5, No. 5
98
4. Distributive laws
Let D = (D, d, m, βD , ηD , µD ) and U = (U, u, n, βU , ηU , µU ) be pseudomonads on the same
object K of the Gray-category A. A distributive law of U over D consists of a 1-cell
r : U D → DU in A(K, K), together with invertible 2-cells
U
U
= DFF
= DFF
zz
FFr
FF
F
ω1 F#
/ DU
uD zz
D
Ur
UUD
zz
zz
Du
rU
/ U DU
UD
U
/ DU U
ω3
nD
r
{{
FFr
FF
F
ω2 F#
/ DU
U d {{
{{
{{
dU
r
UD
O
Dn
/ DU
O
4444 ω4
mU
/ DDU
Um
/ DU
U DD
/ DU D
Dr
rD
subject to the following coherence conditions:
(coh 1)
u
/ U K Ud / U D
KK
GGu} ssss
KK
GGd−1
KK y zzzz r
u
G
K 2
d GG
dU KKω
%
#
/ DU
D
1 GG
=
=U
zz
z
z
z
1 DD
DD
d !
u
Du
II U d
II
I$
ud U D
LL r
u:
uu
ω1 LLL%
u
u uD
/ DU.
D
Du
(coh 2)
U uD /
UUD
44 nn
44 ηs{ D−1
44U
44
4
nD
Id 44
44
44
44
U D4
UD
Ur
/ U DU
=
U uD /
UUD
HH
HH U ω1 U r
H
U Du HH$
U D HH
rU
px hh
Dn
ω3
−1
ru
DU u /
DU HH
/ DU
r
U DU
r
DU U
rU
DU U
HH −1
HHDηU Dn
H
Id HH$
DU.
(coh 3)
UUd
U U LL / U U D
LLL
U ω2
LL
Ur
U dU LLL&
U DU
dU U
n
ω2 U
rU
"
DU U
U
d
n
dU
Dn
/ DU
=
UU
n
UUd/
U U DJJ
n−1
d
Ud /
nD
JJ U r
JJ
JJ
J%
U II
UD
U DU
II ω 888
ω3
II 2
88
rU
II
88
II
II
8
r
88 DU U
II
dU III 88
II 8
II 88 Dn
II 8
$
DU.
Theory and Applications of Categories, Vol. 5, No. 5
99
(coh 4)
U U r/
= U U U DU U/ rU U DU
U U DU
AA
DD
55
DD
AA
DDU rU
55
AA
DD
A
55
U rU
A
DD
55
!
nU D U nD 5
U
DU
U
AA
nU D
nDU
55
U
DU
AU
AA U Dn
AA
55
AA
AAU Dn
| U ω3
55
AA
AA
U
~ n−1
ω rU
~
r
AA
3U
/ U DU
U
U
D
UUD
55
Ur
U U DD U r / U DUDD DU UAU U DU
55
DD
AA
rU
DD
55 u} ssss
DD
AA| −1
DDDnU
55 µU D
DD
AArn rU
DD
55
D
AA
DU
n
rU
D
nD
DD
DU U
D!
nD 5
D
55
DD
DU
U
DU
U
D
55
A
u} rrrr ω3Dn
nD DD
AA
5
DD ~
A
A|ADµU Dn
DD ω3
/ DU
DD
UD
Dn AAA
r
D!
/ DU.
UD
U U U5 D
r
(coh 5)
DD QQQ
uDD
/ U DD
QQQ
QQQ
rD
Q ~
DuD QQQω( 1 D
DU D
~
Dω1
DDu
Um
/ UD
} ω4
r
=
GG
GG
m GGG
#
DDu
Dr
%
/ DU
DDU
mU
uDD
DDGG
DDU
/ U DD
Um
um
D GG uD / U D
GG
G G ω1 r
mu
GG
Du G#
/ DU.
mU
(coh 6)
U U m/
U U DD
nDD
nD
=
U U DDD
n−1
m
DD U r
DD
DD
"
U DDGGU m / U DD
D
G
U DUDD
DD rU
DD
DD
!
DU U
DD ~ ω3
DD
DD
DD r
Dn
DU DDD DDDD
DD ~ ω4 DD
DD
DD
DD
Dr DD"
!
/ DU
DDU
GG
G
rD GG#
mU
UUm
/ UUD
33
33
33
33
3U3 r
U DU DD
U DD
D
33
z
DDU Dr
z
3
DD
zz
uuuu
~
v
z
DD
z
U ω4 33
"
|zz rU D
/ U DU
ks __
rD ks __ DU U D
U
DDU
−1
DD
ω3 D
U mU
DD rr
zz
{{
DD
zz
{{
{
z
{ DnD DU r DD
z rDU
{
z
|z
"
}{
ks ____
rU
DU D
DU DU
Dω3
zz
zz
z
~
ω4 U
z
|zz DrU
/
Dr
DDU U mU U
DU U
zz
{{
z
{
z
{
~ mn
{{
zz
}{{ DDn
|zz Dn
/ DU.
DDU
U U DD
DD
{{
{{
{
{
}{{ nDD
mU
DDU rD
DD
DD
"
Theory and Applications of Categories, Vol. 5, No. 5
100
(coh 7)
Id
U D QQQ
U dD
−1
U βD
/ U DD
&
/ U DU
Um
QQQ
QQQ
rD
Q ~
dU D QQQω( 2 D
DU D
r
~ dr
DU
dDU
= idr .
r
~ ω4
Dr
mU
/ DU
8
Um
%
/ UD
/ DDU
β U
D
Id
(coh 8)
Id
U Dd
UD
U ηD
/ U DD
~ r−1 rD
d
/ DU D
r
DU QQQ DU d
QQQ
QQQ
Dr
Q ~
DdU QQDω
Q( 2
DDU
= idr .
r
} ω4
−1
ηD U
/ DU
9
mU
Id
(coh 9)
U Dm /
U DDNN
PPP
NNNU m
PPP
NN
rDD
P
| U µD NNNN
U mD PPP(
'
DU DD
U DD U m / U D
U DDDPP
U Dm
= U DDD
rDD
DU DD
DU m
/ U DD
NNN
NNNU m
NNN
{ r−1 rD
NN&
m
/ DU D
UD
DrD
DrD
DDU DPP
DDr
| ω4 D
PPP
PPP
P
mU D PPP(
DDDUPP
DDU D
rD
r
mU
y zzzzω4
/ DU
y {{{{ω4
r
/ DDU
NNN
PPDmU
NNNmU
PPP
{
NNN
PPP
{
{
{
y
NN&
µD U
PP'
mDU
/ DU.
DDU
DDDUPP
DU D
PPP
PPP
Dr
P |
r
mDU PPPm
(
DDU
{ Dω4
DDr
Dr
mU
Observe that if the pseudomonads are strict (β, η and µ are identities), and ω1 , ω2
and ω3 are identities, then we obtain the coherence conditions of the “mild” extensions
of the classical distributive laws given in [7].
Theory and Applications of Categories, Vol. 5, No. 5
101
5. The composite pseudomonad given by a distributive law
Assume we have a distributive law of U over D as in section 4. The first question is how
to produce a composite pseudomonad from U, D, and the distributive law. This is what
we do in this section.
Define V = (V, v, p, βV , ηV , µV ) as follows: V = DU ∈ A(K, K); v is defined as the
composite
1K
u
/U
dU
/ DU ;
p is the composite
DU DU
DrU
/ DDU U DDn / DDU mU / DU ;
βV is defined to be the pasting
DU DU
rr9 O MMMMMDrU
MMM
rr
MM&
rrr d−1
__uDU
3
+
Dω
U
1
U
U> DU
DDU
LLL
q8
}}
DuDU qqq
LLDDn
}
LLL
q
}
q
}
q
DDβ
LL%
q
U
}
DDuU
q
}
q
uDU }}
/ DDU
}
II
hhh3 DDU
}}
h
h
h
}
II mU
}
hhh
h
II
h
}
h
h
h
}
II
h
h
}
hh dDU
I$
βD U
}}hhhhhhh
/ DU ;
DU
dU DUrrr
Id
ηV as the pasting
Id
Id
/
/ DU ;
l5 DU VVVVVV
:
l
l
v
V
l
HH
V
v
l
V
VVVV
v
ηD U
DηU
ll
HH
v
l
l
V
v
V
l
H
ll Dn
vv
DU u HH
DdU VVVVVV
#
vv mU
lll
V*
DU UFFXXXXXX
DDU
−1
r9
Ddn
XXXXX
r
FF
r
XXXXX
FF
rr
XXXXX
FF
rrrDDn
XXXXX
DdU U
r
FF
r
+
FF
U
F
Dω2 U −1 DDU
DU dU FF
qq8
FF
q
q
FF
qqq
FF
qqq DrU
"
DU HH
DU DU
Theory and Applications of Categories, Vol. 5, No. 5
102
and µV as the pasting
DU DrU
/ DU DDU U DU DDn / DU DDU DU mU
Dr−1
Dr−1
DrDU U
DrDU
rU
Dn
/ DDU DU U
/ DDU DU
DU DU DU
DrU DU
DDU U DU
DDU rU
DDrn−1
DDrU U
DDDU U U
DDU DU
DDrU
m−1
rU
mU DU
DU DU
mDU U
/ DDU U
DrU
DDDµU
DDDn
/ DDDU
DDDn
m−1
Dn
Dω4 U
/ DDU U
DDrU
/ DDDU U
DDDU n
DDω3 U
DDDnU
/ DDDU U
DrU
DDU Dn
DDnDU
/ DU DU
DmU U
Dmn
DmU
/ DDU
µD U
mDU
/ DDU
DDn
DDn
mU
mU
/ DU.
5.1. Theorem. V = (V, v, p, βV , ηV , µV ), as defined above, is a pseudomonad on the
object K.
Proof. Observe that the pasting of µV and ηV V −1 is
/
/ RR
/
EE
RRR
EE
R
RRDω
EE
RRR2 U DU
EE
−1
RRR
R)
Dη
EE U DU
EE
EE
EE
EE
" RDdnDU
RRR
RRR
RRR
RR
RRR)
−1
DrrU
/
−1
DrDn
/
DDrn−1
DDω3 U
/
m−1
rU
/
ηD U DU −1
+
/
/
DDDµU
/
m−1
Dn
/
/
Dω4 U
Dmn
µD U
/
/
/ .
We must show that this pasting equals p ◦ V βV . Substitute the pasting of DdnDU and
DηU DU −1 by the pasting of Dd−1
U uDU and DDηU DU . Then use (coh 2). With the help of
(2) show that
/
−1
DDruU
DDDηU
=
/
−1
DDrn
/
U −1
/
5
lll
lll
l
l
l
lll
lll
RRR
RRR
RRR
RR
DDU DβU RRRR)
/
DDDµU
-
DDrU
/
DDDn
,
Theory and Applications of Categories, Vol. 5, No. 5
103
and make the substitution. Make the substitution
/
/
Dω2 U DU
−1
DrrU
+
=
/: J
:
tt JJJ
tt
t
t
tt DU drU ttt DU dDn JJJ
JJ
tt
tt
tt
/$
/ Ht
/
HH
H
HH −1 HH −1 Dω2 DU
H HDdU rU H HDdU Dn
HH
HH
H$
H$
/
/+ ,
=
: J
tt JJJ
tDU
t
ω 1 U JJJ
t
JJ
tt
tt
/$
/
−1
DrDn
/
/
followed by the substitutions
8 N
ppp NNNNN
p
p
NNN
ppp
NN&
−1
pppks _
__
_
DdU rU
NNN
N
8
−1
N
p
NNN
DdU uDUppp
NNN
N
p
DDU
ω
U
NNN
N
p
1
N
p
N
p
NN&
NN&
−1
ppp
/ NDd
NNNU Dn
N
DDU DβU NNNN
NN&
.
JJ
JJ
JJ
J
JJ. $
DU DβU
DdU DU
,
and
/
/
ηD U DU −1
m−1
rU
+
/
/
=
/
m−1
Dn
DdrU
/
/
DdDn
/
ηD DU −1
+ ,
and
/
DdDn
ηD
=
/
Dmn
/
DU −1
/
/
5
lll
lll
l
l
l
lll
lll
RRR
RRR
RRR
R
Dβ
DU
U RRRR
D
R)/
µD U
-
DDn
mU
.
Use (coh 7), and finish with the substitution
*
DU DβU
=
/
/
PPP
PPP
PDU
PPPω1 U
PPP
(
DU drU
/
DU dDn
/
8
ppp
p
p
ppp
ppp
/
/
p8 NN
NNN
p
DU d−1
uDU
NNN
pppDU Dω1 U
p
p
N
p
ppp
DU DDβU
NN&
/
-
Theory and Applications of Categories, Vol. 5, No. 5
104
In regards to the other condition, observe that the pasting of V µV , µV V and µV is:
/ L
/ L
/ L
LLL
LLL
LLL
LLL
LLL
L
L
LLL
L
L
L
L
L
LL%
LLL
−1 LL
−1 LL
DU DrrU
%/ DU DrDn
%/
LL
−1
LLDr
L
L
L
L
L
L
LLLrU DU
LLL
LLL DU Dω4 U LLLL
LLL
LLL
LLL
LLL
L
LLL
L%
L% DU DDrn−1LLL%
L%
LLL
/ L
/ L
LLL
LLL
LLL
L
LLL
LLL
LLL
LLL
LLL
LLL
L
L
L
L
−1
L
LL DU DDω3 U LL DU DDDµU LL
LLDr
DU Dmn LL%
%
%/
%/
LLDnDU
/ L
LLL
LLL
LLL
LLL
LLL
LLL
LLL
LLL
LLL
LLL
LLL
LLL
L
L
L
L
−1
LLDDr
LL%
LL%
−1 LL
DU µD U LL%
LLL nDU LL%
DU m−1
DU
m
%
rU
Dn
/
/
/
U
LLL
LLDDω
LLL 3
LLL
LLL
LLL
L%
%
−1
−1
DrrU
DrDn
LLLm−1
LDω
LLL
LLL4 U DU
rU
DU
LLL
LLL
L
LLL
LLL
LLL DDDµU DU
L%
%
LLL
/
/
LLL
LLL
LLL
LLL
Dω4 U
−1
DDrn
LLL
L
LLL −1 L% Dm
/
/
LLmLDnDU LLL nDU
LLL
LLL
DDDµU
DDω3 U
Dmn
LLL
LL
%
%
/
/
/
LLµ
LLDLU DU
LLL
µD U
m−1
m−1
rU
Dn
L%
/
/
/
Where we have only put the name of the corresponding 2-cell in each parallelogram. To
show that this pasting is equal to the pasting of pp , µV and µV we do the following. First
make the substitution
=
/ NN
/ NN
NNN
NN
NNN
NNN
NNNDU m−1 NNNNDU m−1 NNNNN
NN& rU
NN&
NN& Dn
/
/
__
_
NNN ks _
Dω
NNN4 U DU
NNN
NN&
/
−1
D
DrDn
/ OO
ks __
OOO
OOO
OOODω4 DU
OOO
OOO
OOO Dm−1 OOO Dm−1 OOOO
OOO U rU OOO U Dn OOO
'/ .
'
'/
−1
DrDn
/
−1
DrrU
/
−1
Dr
DrU
/
−1
DDr
rU
/
/ OO
OOO
OOO
−1
OOO
Dr
DDn
O'
/
Then make the substitutions
/ L
/ L
LLL
LL
LLL
LLL
L
LLLDm−1 LLLLL
LLLDm−1
L% U rU
LL% U Dn
LL%
/
/
−1
DDrn
LLL ks __
Dm
LLLnDU
LLL
L%
/
DDDµU
DDω3 U
/
/
=
/ L
LLL
LLL
LLL
/ L DmrU %
LLL
DDDω3 U
LL
L
DDDDµU
L
DmDn LL%
/
/
LLL
LLL
LLL
LLL
LL
LLL
−1 LL
L
LLL Dm−1
Dm
L
L% rU
LL% Dn
LLL%
/
/
/
−1
DDDrn
Theory and Applications of Categories, Vol. 5, No. 5
105
and
/ NN
/ NN
NNN
NNN
NNN
NNN
NNN
−1 NN
NNN
DU
DDr
N
n NNN
NN&
NNN
N&
/ NN
NNN
NNN
NNN
NNN
NNN
NNN
NNN
−1
NNNDrDnDU NNDU
DDDµU NNNN
DDω3 U NNDU
N
N
N
NN&
N
NNN
N&
N&
/
/
NNN
NNN
N
NNN
−1
−1
−1 NNN
DrDDn
DrDrU
NNDDr
NNN nDU NNNN
/
&
NNN
/
NNN
−1
−1
NNN
DDrrU
DDrDn
NNN
/
&
/
=
/
/ OO
OOO
OOO
OOO
O'
−1
/
/ OO DrDrU
OOO
O
OOO
OOO
−1
−1
DDr
DDrU
O
OOO
OOO
U Dn
rU
OOO
O
O
/
/
'
'
−1
−1
OOO DrDDn
OOO
OOO DDrrU
OOO
OOO
OOO
OOO
OOO
OOO DDDU r−1 OOO
OOO
OOO
n
OOO
OO
OOO
OOO/'
'
−1 O'
OOO
DDr
O
O
OOO Dn
OOO
OOO
OO
OO
OOO
DDDU
DµU OOOO
OOO ω3 U OOOODDDU
OO'
O
O'/
O'
/ .
−1
DrDU
rU
−1
DrDU
Dn
Now use (coh 4). Proceed with the following substitutions
=
/ M
MMM
MMM
MMM
MMM
M
MDDDU
MMM
Dµ
MMM
U
MMM
M
M
&/
&
−1
DDDr
MMM nU
MMM
−1
M
MM
DDDrn
MMM
DDDDµ UM&
/
MMM
U
MMM
DDDDµU
M
MM
MMM
M&
/
/ N
NNN
NNN
−1
DDDrU
NNN
n
−1 NNN
DDDr
n
&
/ N
NNN
N
−1
NN N
DDDDrn
NNN
DDDDµUNN&
/
MMM
LLL
MMM
LLL
L
L
M
M
M
LLDDDDµ
U
MMM
LLL
M&
&
/ ,
followed by
/ PP
PPP
PPP
PPP
PPP
P PDDDU
PPP rn−1
PPP
PP(
PP(
/
DDDω3 U U
=
DDDn−1
Dn
−1
DDDrU
n
PPP
PPP
PPP
PPP
P(
/
−1
DDDDrn
/
/ OO
OOO
OOO
OOO
O'
/ ODDDω
NNN
OOO 3 U
NNN
OOO
−1
N NDDDr
OOO
NNN n
OO'
N'
/ ,
Theory and Applications of Categories, Vol. 5, No. 5
106
and then
/ NN
/ NN
NNN
NNN
NN
NNN
N
NNN Dm−1 NNN Dm−1 NNNNN
rU
Dn
N
N
NN&
N
N&
/
/
__
_ &
NNN ks _
NµNDNU DU
−1
−1
mrU
mDn
NNN
NN&
/
/
=
/
/ NN
NNN
NNN
NN
__ N'
/ M
s
k
MMM
MMM µ DU
MMM
MD
MMM m−1 MMMM
m−1
rU
Dn
MM&
M
M&
/
/ ,
m−1
DrU
m−1
DDn
/
MMM
MMM
MMM
MM&
followed by the substitution
/ NN
/ NN
NNN
NNN
NN
NNN
N
NNNDDDrn−1NNNNN
NNN
NN&
NN&
NNN
/ NN
NNN
NNN
NNN
N
N
N
NNN
N
N
NNN DDDω U NN DDDDµNNNN
NNNm−1
NNN
N
3
U
DnDU
N
N
NN&
NNN
N&
NN&
/
/
NNN
NNN
−1
−1
m
m
NNN
DrU
DDn
NN&
/
/
=
/
/ OO
OOO
OOO
OOO
O'
/
/
−1
m
OOO
OOO
OOO DrU
OOO
OOO
OOO
OOO
OOO
OOO DDr−1 OOO
OOO
OOO
n
O
O
OOO
OOO
OO'
OO'
'
−1
OOO
/
m
O
O
OOO DDn
OOO
OOO
O
O
OOO
O
O
OOODDω3 U OOOOO DDDµU OOOOO
O'
O/'
O'
/ .
m−1
DU rU
m−1
DU Dn
Use (coh 9) to show that
/ OO
OOO
OO
OOO
OOO DU µD UOOOOO
OO'
OO'
/
=
OOO Dω4 DU
OOO
OOO
DmrU OO'
OOO
OOO
OOO
OO'
Dm
Dn
OOO
OOO
OO
µ
DU OOO'
D
OOO
OOO
OOO
OO'
/
Dmn
/
µD U
DDω4 U
/
4U
/ OO Dω
OOO
OOO
DDmn
Dmn OOO'
/
OOO
OOO
mmU
OO
µD U OOO'
/
OOO
OOO
O
OOO
OOO µD U OOOOO
OO'
OO'
/
Dω4 U
/ OO
OOO
OOO
OOO
/
'
−1
DrmU
Theory and Applications of Categories, Vol. 5, No. 5
107
and make the substitution. Next the substitution
/ NN
/ NN
NNN
NN
NN
NNN
−1 NN
−1 NN
NNDU
NNN DrrU NNDU
NNNDrDn NNNNN
−1
&/
&
&/
DU
NNN DrrU
−1
−1
NNN
DrDU rU
DrDU
Dn
NNN
NN&
/
/
−1
DDrU
rU
DDω3 U DU
NNN
NNN
NNN
N
−1 N&
DU
NNN mrU
NNN
NNN
NN&
/
/
/
/
/
/
m−1
DU rU
/
−1
DrU
DrU
−1
DDrU
Dn
DDDn−1
rU
=
DDDn−1
Dn
m−1
DU Dn
/
DDn−1
DrU
DDn−1
DDn
/ L DDω3 DU
LLL
LL
L
LLL
m−1
&
/
/
rDU
KK
KK
KKK
KK
KK
K
K
KK
K
−1 KK
−1 KK
KK DrrU
KKK
KK DrDn
KK
K
%
/%
/%
/
/ L
LLL
LLL
LLL
−1
/ LDrrDU &
LLL
LL
L
LLL
&
−1
DrU
DDn
m−1
U DrU
m−1
U DDn
,
followed by the substitution
/ M
/ N
NNN
NNN
MMM
−1
NNN
DrU
NNN
M
mU
M
NNN
M
NNN
M
−1
NNN
DrrDU NN& DU Dω4 U MMMM
/
MMM
NNN
NNN
LLL
LLL
NNN
N
M
NNN
LLL
LLL
N
NNN
MMM
−1 NNN
N
LLL
LLL
NNN
NNN DrrU
MMM
NNN
−1
LL&
LL& DrDrU
N
NNN
M&
& DDU ω4 U
&
/
NNN
N
N
=
NNN
MMM
LLL
LLL
NNN
NNN
N
MMM
LLL
LLL
NNN
NNN
NN
LLL DU Dmn MMM
LLL
N
N
−1 NN
N
N
N&
N& DrDn
N'
−1
LL&
LL& DrDDn
MM&
/
NNN
MMM
/
NNN
N
M
N
MMM
−1
NNN
DrmU
MMMDDU mn NNNNN
NNN
N&
M
NN&
&
/ .
/
−1
Substitute the pasting of DDU mn , DDω4 U , DDrDn
and DDDrn−1 by the pasting of
−1
DDω4 U U , DDrn and DDmU n . Then observe that the pasting of DDmU n , DDmn and
DDDDµU equals the pasting of DDDµU , DDmnU and DDmn . Use (coh 6). To finish
the proof, make the substitution
/
NNN
NNN
NNN
−1
mrDU NN&
LLL
LLL
LLL
LL&
/
MMM
MMM
m−1
U mU
MMM
MMM
DDω4 U
/
MMM
NNN
LLL
NNN
MMM
LLL
NNN
MMM
LLL
NN&
−1
MM&
mDrU LL&
/ M
=
LLL
LLL
MMM
LLL
LLL
LLL DDmn MMMMM
LLL
LL&
LL& m−1
MM&
DDn
/
NNN
NNN
m−1
mU
NNN
NN&
/
NNN
NNN
NNN
NNN
NNN
NNN
NN
NNN
NNN m−1 NNNN
rU
NN&
NNN
Dω4 U
NNN
NNN
NNN
NNN
NNN
NNN
NN&
NN&
/
MMM
MMM
MMM
MM&
NNN
NNN
NNN
−1
mDn NN'
NNN
NN
Dmn NNNN
NN&
/ .
Theory and Applications of Categories, Vol. 5, No. 5
108
6. Compatible pseudomonad structures
We consider now the question of when can a pseudomonad be considered as the composite
of two pseudomonads.
Let D, U be pseudomonads on the same object K of the Gray-category A. Given
another pseudomonad V = (V, v, p, βV , ηV , µV ) on the same object K, we say that V is
compatible with the pseudomonads D and U if V = DU and there are invertible 2-cells:
U
B