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 Gϕ
GF GF Gf GF gF G f ks __ G f
G ϕ
+ γF 3 G F ,
gF GF gF
GF
/ G F
G f
gf / G F
gf g F / G F gF gF
GF
Gf / G F
gf
Gf
G f
/ G F g F gF
GF
=
=
G(f ◦f )
G f / G F ,
/ G F G (f ◦f )
gf ◦f
GF gF =
GF
and
GF
Gf
GF gF
 gf
gF g F
/ G F
G f
/ G F / G F
G f
gf
/ G F Gf
g F (g ◦g)F
/ 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 /
DD
DDD
D DD / DD Yao : z D
:::
DD ~ β
η zzz
DD
zz
IdD DDm
! }zz IdD
mD
D
µ
DD
m
/ D,
m
such that the following two equations are satisfied:
/
DDD
EE
DDm
DDDD
EE
mDD
EEDmD Dµ
EE
E" ⇐=
DDD
DDDEE
µD
mD
E⇐=
EE
E" Dm
EEDm
EE
E"
/ DD
µ
m
⇐=
mD
DD
m
Dmvv:
DDDD
JJDm
JJ
JJ
$
mD
m−1
m
mDD
DD
⇐= µ
/
⇐=
DDDLL Dm DDJJ
=
LLL
LL
mD LL%
/D
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
/
DDD
JJ
DDm
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
⇐=
DD
m
mD
/D
EE
EE
EEIdD
md
DDd
Dd η EE
⇐=
⇐ EEE"
DDD mD / DD m / D.
m
DD
=
m
/ D
(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
X DD / DX
DD ~ dX
IdX
DDX
DD ψ x
DD
" mX
X
Dx
/ DX
χ
DX
x
x
/ X,
such that the following two equations are satisfied
DDDX
EE
/
DDX
EE
DDx
mDX
EEDmX Dχ
EE
E" ⇐=
DDX
DDXEE
Dx
EEDx
EE
E"
/ DX
χ
x
µX
mX
E⇐=
EE
E mX "
⇐=
DX
x
Dx vv:
=
DDx
/
DDX
JJ
JJDx
JJ
JJ
$
mX
m−1
x
mDX
DX
⇐= χ
/
⇐=
DDXLL Dx DXJJ
LLL
LL
mX LL%
/X
(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
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
X II
DX
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
/ DX Dh
DX
 ρ
x
X
x
/ X ,
h
that satisfies the following two equations:
X DD / DX
DD ~ dX
Id
Dh
/ DX  ρ
DD ψ x
DD
" X
=
x
h
/X
h
/ DX
dX
X
 dh
Dh
dX (9)
/ DX EE ~ ψ
EE
x
Id EE" X EE
X ,
DDXEE
DDh
mX
EEDx
EE
E"
/
DDX
EE
Dρ
⇐=
DX
DXEE
χ
x
E⇐=
EE
E" Dh
EEDx
EE
E"
/DX ρ
X
h
DX ⇐=
χ
DXLL Dh / DX JJ ⇐=
L
=
LLL
x LLL%
/ X
h
JJDx
JJ
JJ
$
mX m−1
mX
x
⇐=
x
/
DDX
JJ
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
x
*
Dξ 4 DX Dh
X
ρ
h
x
/ X
= DX
x
X
Dh
/ DX  ρ
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(ψ,χ) =
k : K → K then we define the strong transformation −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
nD
zz
zz
Du
/ U DU
rU
UD
U
/ DU U
ω3
r
{{
FFr
FF
F
 ω2 F#
/ DU
U d {{
{{
{{
dU
Dn
/ DU
O
r
UD
O
4444 ω4
mU
/ DDU
Um
/ DU
U DD
/ DU D
Dr
rD
subject to the following coherence conditions:
(coh 1)
/ 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
=
=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
ω3
r
U DU
r
DU U
−1
ru
DU u /
Dn
/ DU
DU HH
rU
DU U
HH −1
HHDηU Dn
H
Id HH$ DU.
(coh 3)
/ UUD
U U LL
LLL U ω2
LL
Ur
U dU LLL& U DU
UUd
dU U
n
ω2 U
rU
" DU U
U
dn
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
ω
3
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
UUD
U
U
D
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
rU
DU
n
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
/ U DD
uDD
QQQ
QQQ rD
Q ~
DuD QQQω( 1 D DU D
~ Dω1
DDu
%
/ UD
Um
} ω4
Dr
DDU
r
/ DU
mU
=
DDGG
uDD
GG
GG
m GGG
#
DDu
DDU
/ U DD
 um D GG uD / U D
GG G G ω1 r
 mu
GG
Du G# / DU.
Um
mU
(coh 6)
U U m/
U U DD
DD U r
DD
DD
"
nDD
nD
=
U U DDD
 n−1
m
U DDG U m / U DD
GG
D
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
/ UUD
33
33
33
33
3U3 r
U DD
U DU 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
D
{
z
}{{ DnD DU r D"
|zz rDU
____
s
k
DU D
DU DU
rU
Dω3
zz
z
z
~ ω4 U
zz
|zz DrU
/ DU U
Dr
DDU U mU U
z
{{
zz
z
{{
{
z
~ mn
{
z
}{{ DDn
|zz Dn
/
DDU
DU.
U U DD
DD
{{
{{
{
{
}{{ nDD
mU
UUm
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
QQQ
QQQ rD
Q ~
dU D QQQω( 2 D Um
&
/ U DU
DU D
r
~ dr
DU
dDU
= idr .
r
~ ω4
Dr
/ DDU
β U
D
mU
/ DU
8
Um
%
/ UD
Id
(coh 8)
Id
U Dd
UD
r
U ηD
/ U DD
~ r−1 rD
d
/ DU D
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
DrD
DDr
rDD
DU DD
DrD
DDU DPP
| ω4 D
PPP
PPP
P
mU D PPP(
DDDUPP
DU m
/ U DD
NNN
NNNU m



NNN
{ r−1 rD
NN&
m
/ DU D
UD
DDU D
rD
r
DU D
DDr
Dr
mU
y zzzzω4 / DU
{ Dω4
y {{{{ω4
r
/ DDU
NNN
PPDmU
NNNmU
PPP
{
NNN
PPP
{
{
{
y
NN& µD U
PP'
mDU
/ DU.
DDU
DDDUPP
PPP
PPP
Dr
P |
r
mDU PPPm
(
DDU
U Dm
= U DDD
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> DU
U
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
/
/ DU ;
l5 DU VVVVVV
:
l
l
v
V
l
HH
V
v
l
V
VVVV
v
DηU
ηD U
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
Id
DU DU
Theory and Applications of Categories, Vol. 5, No. 5
102
and µV as the pasting
/ DU DDU U DU DDn / DU DDU DU mU
Dr−1
Dr−1
DrDU U
DrDU
rU
Dn
/ DDU DU U
/ DDU DU
DU DrU
DU DU DU
DrU DU
DDU U DU
DDU rU
DDU Dn
DDDU U U
DDnDU
DDU DU
DDrU
m−1
rU
DU DU
mDU U
/ DDU U
DrU
DDDµU
DDDn
/ DDDU
DDDn
m−1
Dn
DmU U
Dmn
DmU
mU
DDn
/ DDU
µD U
mDU
/ DDU
DDn
DrU
Dω4 U
/ DDU U
DDrU
/ DDDU U
DDDU n
DDω3 U
DDDnU
/ DDDU U
mU DU
DDrn−1
DDrU U
/ DU DU
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
/
/ U −1
- −1
DDrn
DDDµU
=
/ / 5
lll
lll
l
l
l
lll
lll
RRR
RRR
RRR
RR
DDU DβU RRRR)
/
,
DDrU
DDDn
Theory and Applications of Categories, Vol. 5, No. 5
103
and make the substitution. Make the substitution
/
/
Dω2 U DU
+ =
/: J
:
tt
tt JJJ
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
DrrU
/ −1
DrDn
/ followed by the substitutions
8 N
ppp NNNNN
p
p
NNN
ppp
NN&
−1
pppks _
__
_
DdU rU
N
NNN
8
−1
N
p
NNN DdU uDUppp
NNN
N
p
DDU
ω
U
N
NNN
p
1
N
p
N
p
NN& NN&
−1
ppp
/ NDd
NNNU Dn
N
DDU DβU NNNN
NN& .
JJ
JJ
JJ
JJ
J$
.
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
- µD U
=
/ / 5
lll
lll
l
l
l
lll
lll
RRR
RRR
RRR
R
Dβ
DU
U RRRR
D
R)/
.
DDn
mU
Use (coh 7), and finish with the substitution
/
PPP
PPP PDU
PPPω1 U
PPP
( /
* DU DβU
=
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
NNN
NN
NNN
NNNDU m−1 NNNNDU m−1 NNNNN
NN& rU NN& Dn NN&
/
/
__
_
NNN ks _
Dω
NNN4 U DU
NNN
NN& /
=
−1
Dr
DrU
/ −1
/ OO
OOO
OOO
−1
OOO
Dr
DDn
O'
/
−1
DDr
Dn
/ OO
/ OO
ks __
OOO
OOODω4 DU
O
OOO
OOO
OOO Dm−1 OOO Dm−1 OOOO
OOO U rU OOO U Dn OOO
'/ .
'
'/
DDrrU
−1
DrDn
/ −1
DrrU
/ Then make the substitutions
/ L
/ L
LLL
LLL
LL
LLL
L
LLLDm−1 LLLLL
LLLDm−1
LL% U Dn
LL%
L% U rU
/
/
LLL ks __
Dm
LLLnDU
LLL
L% / DDDµU
DDω3 U
−1
DDrn
/ / =
/
/ L
LLL
LLL
LLL
/ L DmrU %
LLL
DDDω3 U
LL L
DDDDµU
L
DmDn LL% /
/
LLL
LLL
LLL
LLL
LLL
LLL
LLL Dm−1
L
LLL
LLLDm−1
Dn
L% rU L% %/
/
−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
DDω3 U NNDU
DDDµU NNNN
N
N
N
N
NN&
NNN
N&
N&
/
/
NNN NNN
N
NNN
−1
−1
−1 NNN
DrDrU
DrDDn
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
DDrU
DDr
O
OOO
OOO
rU
U Dn
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
NNN
DDDDrn
NNN
DDDDµUNN& /
MMM
LLL
MMM
LLL
LL
MMM
LLDDDDµ
U
MMM
LLL
M& &
/ ,
followed by
/ PP
PPP
PPP
PPP PPP
P PDDDU
PPP rn−1
PPP
PP(
PP(
/
DDDω3 U U
PPP
PPP
PPP
PPP
P( −1
DDDrU
n
−1
DDDDrn
/ / =
DDDn−1
Dn
/ 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
N
NNN
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'
/ OO
OOO
OOO
OOO
/
'
−1
DrmU
DDω4 U
Dω4 U
Dmn
µD U
/ / / 4U
/ OO Dω
OOO OOO
DDmn
Dmn OOO' /
OOO OOO mmU
OO
µD U OOO' /
OOO
OOO OOO
O OOO µD U OOOOO
OO'
OO' /
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& / DDDn−1
rU
−1
DrU
DrU
/ DDDn−1
Dn
/ / m−1
DU Dn
/ / −1
DDrU
Dn
/ m−1
DU rU
/
=
/ L
LLL
LLL
LLL
−1
/ LDrrDU &
LLL
LL L
LLL
& −1
DrU
DDn
DDn−1
DrU
DDn−1
DDn
m−1
U DrU
m−1
U 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
%/
%
/%
/ ,
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 <<<
< dU
u θ1 |||| <<<
z
/ DU,
1K
DU
DU
DU
DU
JJ
FF
s9
w;
s
JJ p
w
F
s
p
w
F
DU dU w
DuDUss
JJ
F
w
F
s
JJ
w
w
ss
θ2  FF"
θ3 J%
ww
ss
/
/ DU,
DU, DDU
DU U
Dn
mU
v
subject to coherence conditions. To describe the coherence conditions introduce the following pastings:
θ4
=
/
DU
5
DU θ−1 kkkDU
2 kkk
kkk DU p
DU Dn
DU DU UTTT
TTTT
TT
DU DU dU T)
DU DU DU
p
pDU
pU
µV
p−1
dU DU DU
SSS
5
kkk
SSS
kkk
θ
k
k
p SSSS)
k
k DU dU
2
/ DU,
DU U
Dn
and
θ5
=
mU DU
DDU DUTTT
TTTT
TT
DuDU DU T)
/ DU DU
5
k
k
k
kk
k
k
k
k pDU
θ3 DU −1
DU DU DU
Dp
p
DU p
µ−1
Dup
V
DU
DU
5
S
SSS
jj
j
j
S
j
S
j
p SSSS)
jjjjDuDU
θ3
/ DU.
DDU
mU
The coherence conditions are:
/ DU,
mm6 F
m
m
m
mmm
mm99m
m
m
m 99 θ2
5 −1
DU
DU
DU u 55 DU θ1
9
s
55
s
ss
55
ss
s
s
5
ss DU dU
Dn
DU5 TTTT
TTTT
55
TDU
TTTvT
55
TTTT
55
T)
5
DU U
Id
ηV
(12)
p
=
DηU ,
Theory and Applications of Categories, Vol. 5, No. 5
(13)
DU
D ULL
DuU
109
LLLDU dU
LLL
LL%
Dn
−1
DudU
DU
9 DUQQQ ~ −1
ss
QQQθ2
ss
QQ
s
ssDuDU
p QQQQQ
s
s
~
Q( s
θ3
DdU /
/ DU
DU
DDU
7
mU
−1
ηD U
=
DβU ,
Id
(14)
DDU
LL
D
LLLDuDU
mU
LL
duDU LLL%
dDU
/ DU DU
U
DU
QQQ ~ θ−1
dU DU ;
v;
QQQ 3
v
v
θ
DU
1
v
QQ
v
v
p QQQQQ
v
Q( vv uDU
~ β
V
1 DU
DU
=
βD U,
Id
/ DU U
kk5
−1
k
k
θ
U
3
k
k
kkkk pU
mU U
DDU U TTT
TTTT
TT
DuDU U T)
DU DU U
DuDn
DDn
DU Dn
(15)
=
DU θ−1 Dn
4
mn
DU
DU
SSSS
5
j
SSS
jjjj
j
j
j
p SSSS)
jj DuDU
θ3
/ DU
DDU
mU
DnU
DU U U SS
SSS
SSS
DU dU U SS)
/
l5 DU U
l
l
l
lll
lll pU
DU DU U
DU dn
DU n
(16)
θ2 U −1
DU Dn
θ4−1
=
Dµ−1
U
Dn
DU DU SS
SSSS
jjj5
j
j
S
j
p SSSS)
jjjjDU dU
θ2
/ DU
DU U
Dn
/
k5 DDU
−1
k
k
Dθ
k
3
kk
kkkk Dp
DmU
DDDU TT
TTTT
TT
DDuDU TT)
DDU DU
mDU
m−1
uDU
mU DU θ−1
5
(17)
=
µD U
mU
DU DU SS
SSSS
jjj4
j
j
S
θ
j
p SSSS)
jjjjDuDU
3
/ DU
DDU
mU
Some other equations we will need later are contained in the following:
Theory and Applications of Categories, Vol. 5, No. 5
110
6.1. Proposition. Assume we have invertible 2-cells θ1 , θ2 , and θ3 , satisfying the coherence conditions. Then
(Dn, θ4 ) : (βV U, µV U ) → (βV , µV )
(18)
is a 1-cell in V-AlgK , and the following are 2-cells in V-AlgK :
/ (βV , µV )
p7
p
DηU ppppp
p
ppp (Dn,θ4 )
Id
(βV , µV )N
NNN
NNN
NNN
N'
(DU u,p−1
u )
(19)
(βV U, µV U )
(βV U, µV U ) (Dn,θ
4)
V
(DU n,p−1
)3
n gg
gggg
(βV U U, µV U U )W
WWWWW
+
(DnU,θ4 U )
We also have that
DµU
VVVVV
*
(β , µ ),
h4 V V
hhhhh
(20)
(βV U, µV U ) (Dn,θ4 )
(p, θ5−1 ) : (βD U DU, µD U DU ) → (βD U, µD U )
(21)
is a 1-cell in D-AlgK , and the following are 2-cells in D-AlgK :
(DU p,m−1
Up)
d2
ddddddd
(βD U DU, µD U DUX)
µV
(βD U DU DU, µD U DU DUZ )
ZZZZZZZ
,
−1
(pDU,θ5 DU
)
−1
)
XX(p,θ
XXX5 X,
(β U, µD U ),
f2 D
ffffff
(βD U DU, µD U DU )
(p,θ5−1 )
(βD U5 DU, µD U DU )
SSS
kk
SSS(p,θ5−1 )
(DU dU,m−1
U dU
kk)kk
k
kkkk
k
k
k
k
(βD U U, µD U U )
θ2
(Dn,m−1
n )
(DU Dn,m−1
)2
U Dnd
dddddd
ZZZZZZZ
ZZ,
(pU,θ5 U −1 )
(23)
SSS
SSS
SS)
/ (βD U, µD U ),
(βD U DU, µD U DU )
θ4
(βD U DU U, µD U DU U )
(βD U U, µD U U )
−1
)
XXX(p,θ
XXX5 X,
(β U, µD U ),
ff2 D
fffff
(Dn,m−1
n )
Furthermore, if we define σ1 as,
U@ DU QQ
QdU DU
θ DU QQQQ(
1
uDU g3 DU DUOOO p
gg ggg
OOO
g
g
g
g
OO'
gggggggg vDU
βV /
DU
Id
(22)
DU,
(24)
Theory and Applications of Categories, Vol. 5, No. 5
111
σ2 as
PPP
PPP
P
dU U DU PPP(
d−1
U dU DU
dnDU
U DU
dU DU
/ U DU
dU DU DU DU dU DU/
DU DU DU
RRR θ DU
2
RRR
pDU
R
DnDU RRRR
(
/ DU DU
DU U DURR
nDU
Up
/ U DU DU
U dU DU
U U DUPP
d−1
Up
DU p
dU DU
/ DU DU
µV
p
p
/ DU,
γ as
/ U DU
U U LL
LLL
LL
dU DU
d−1
U dU
dU U LLL&
DU dU
n
DU U OO / DU DU
U dU
OOO θ2
OOO
p
OOO
' / DU.
Dn
dn
U
dU
then (σ1 , σ2 ) is an object in U-AlgK , and (dU, γ) : (βU , µU ) → (σ1 , σ2 ) is a 1-cell in U-AlgK .
Proof. We show first that (Dn, θ4 ) is a 1-cell in V-AlgK . To show that (Dn, θ4 ) satisfies
(9), start on the left hand side and make the substitutions:
/
JJ
JJ βV U
JJ
JJ
J$ /
µV
/9 ,
tt
t
t
vDU dU
tt
tt
t
/
/ /
/
JJ
JJβV DU
JJ
JJ
J$ / O
O
=
p−1
dU
=
/ O
O
vp
/ βV DU
/ ,
t9
t
t
tt
tt
t
/ t
βV
and
/
θ2
$ =
vDn .
vDU dU
/ DU θ2−1
vp
/ v
To show that (Dn, θ4 ) satisfies (10), start on the left hand side, cancel DU θ2 and its
inverse, make the substitution
/ NN
NNN
NNN
NNN
NNN
NNN
NNN
NN& DU p−1
dU
&/
NNN
NNN µV U
NNN
p−1
dU
NN& / / OO
OOO
OOO
p−1
OOO
DU dU
O'
/
N
M
= MMM
NNN µV DU
NNN
MMM
NNN
MMM
p−1
N' M&
dU
/ .
Theory and Applications of Categories, Vol. 5, No. 5
112
Then use (1), and conclude with
DU DU θ2−1
/
/$
p−1
DU dU
p−1
p
/ /
=
/ p−1
Dn
/>
AA
AADU θ2−1 }}}
AA
}
AA }}}
}
Next we show that DηU in (19) is a 2-cell in V-AlgK . To show that satisfies (11), start
on the left hand side of (11). Use (12) to substitute the pasting of DU DηU and DU θ2−1
for the pasting of DU βV and DU DU θ1−1 . Then make the substitution
DU DU θ1−1
/
/$
p−1
u
p−1
dU
/ / /
=
p−1
v
/ > .
@@
@@DU θ1−1 |||
@@
|
@@ |||
|
Now use (5), and then use (12) once again.
Next we show that DµU in (20) is a 2-cell of V-algebras. Start on the right hand side
of (11). Use (16) to substitute the pasting of θ2 and DµU , by the pasting of DU d−1
n , DU θ4
and θ2 U . Make the substitution
/ OO
OOO
OOO
p−1
n
OOO
O'
/
N
MMM
NNN
−1
p
MMM
NNN dU
MMM
NNN
MM& DU d−1
n
N& / ,
/ OO
OOO
OOO
OOO
OOO
OOO
OOO
OOODU DU d−1
n
O'/
'
O
= OOO p−1
OOO dU U
OOO
p−1
Dn
OO' / .
Since we already know that (Dn, θ4 ) is a V-algebra morphism, we can substitute the
pasting of µV , p−1
Dn , and θ4 , by the pasting of DU θ4 , θ4 and µV . By the definition of θ4 ,
we can substitute the pasting of µV U , p−1
dU U and θ2 U by the pasting of DU θ2 U and θ4 U .
Finally, use (16).
That (p, θ5−1 ) is a morphism of D-algebras, and that µV , θ2 , and θ4 are 2-cells of
D-algebras are similar and left to the reader.
Now we show that (σ1 , σ2 ) is an object in U-AlgK . Paste ηU DU −1 onto the left hand
side of (7), and make a substitution on it to obtain
/
/ OO
OOO
OOO
−1
OOO d−1
d
O
OOO U uDU
OOO U dU DU
OOO
OOO
'
'/
/ OOO
OOOθ2 DU
OOO
OOO
ηU DU
'1 /
d−1
Up
µV
(25)
/ / .
Theory and Applications of Categories, Vol. 5, No. 5
113
Also paste ηU DU −1 on what turns out to be the right hand side of (7), and make the
following two substitutions:
E FF
FF
FF
F#
5 FF
k
k
U θ1 DUkk
FF
k
k
k
FF
kk U βV
k
F#
k
kk
/
dU DU
=
= C
{{ CCC
CC
{{
{
CC
{
{{ DU βV
/!
u: TTTTTT
uu
TTTT
u
TTTT
uu
u
) TT
uu
TTTT −1
:
u
TTTT
v
TdTUTTdU DU
v
uu d−1
TTTT
v
u
U
uDU
v
TTTT
u
TTTT
v
u
TT) vv
T*
uu
DU
θ
DU
v
1
2
T
e
v
e
−1
T
e
TTTT dU p
eeee
vv
e
e
e
v
e
T
e
TTTT
v
eee
DU βV
vv
eeeeee
TTT) veveeeeeee
/ ,
/
/
= PPPP
PPP η DU
PPPV
µV
PPP
p
' /
/ .
Compare the pasting we arrive at with (25), and use equation (12).
Now, the left hand side of (6) in this case, is the following pasting:
/ QQ
/ QQ
QXQXQXXXX
QQQ
QQQ
QQQ XXXXXX
QQQ U d−1
QQQ
XXXXX U d−1
QQQ
Q
QQQ
U
p
U
dU
DU
Q
XXXXX
QQQ
QQQ
QQQ
XXXX+
Q/(
QQQ
/( RR
QQQ
QQQ
RRR
QQQ
QQQ
RRR
QQQ
QQQ U dU nDU
RRR
QQQ U µV
U θ2 DU
QQQ
RRR
Q
Q(
Q/(
(/
QQQ
QQQ
QQQ
QQQ
QQQ
QQQ
d−1
d−1
QQQ
Up
U dU DU
QQQ
QQ(
QQQ
/
/ QQQ
QQQ µU DU
QQQ
QQQ
QQQ θ DU
QQQ
µV
QQ2Q
QQQ
dnDU
QQ( Q( /
/ .
Make the following sequence of substitutions:
/
/
µU DU
/ dnDU
/
U µV
d−1
Up
/ µV
/ / =
=
t
tt
tt
t
t
ztt
Hµ
HHV DU −1
HH
HH
HH
$ / / =
t
tt
tt
t
t
ztt
HH
HdHU pDU
HH
HH
H$ /
DU µV
t: KKKK
tt
KKK
t
t
KKK
tt
d−1
U nDU
tt
/
/%
dnU DU
DµU DU
/ 9 ,
HH
t
HH
dnDU ttt
HH
HH
tt
H$ ttt
/ / J
JJJ
JJJ
JJJ
%
/ −1
dU p uu
u
u
DU µV
uu
uu
z
u
/ ,
/ J
JJJ
JJJ
−1
pp
JJJ
/
%
µV uuu
u
µV
uu
uu
z
u
/ ,
d−1
U DU p
Theory and Applications of Categories, Vol. 5, No. 5
/ NN
NNN
NN
NNN
NNN U d NNNNN
nDU
NN&
NN&
/
NNN −1
NdNNU nDU
NNN
d−1
U dU DU
NN& / 114
/ OO
OOO
OOO
OOO
O'
/
N
M
= MMM
NNNd−1
U
DnDU
N
MMM
N
MMM DU dnDU NNNN
N& M&
/ ,
/
d−1
U dU U DU
/ N
NNN
NNUNθ2 DU NNNNN
NNN
NNN
NN& NN&
NNN
NNNdU pDU
NNN
NN& d
U DnDU
/
=
d−1
U DU dU DU
/ NNN
NDU
NNNθ2 DU
NNN
N& ,
and
/ NN
/ NN
NNN
NNN
NNN
NNN
NNN
NNN
NNNU d
N
NNN
NN& U dU DU
NN& U d−1
Up
&/
/
NNNd−1
U DU
NUNdU
−1
NNN
d−1
U DU p
NNN dU DU dU DU
/
& / /
/ N
NNN
NNN
d−1
NNN
U U dU DU
N&
/
/
N
N
N
= NNN
NNN
NNNd−1
U
dU
DU
NNN
NNN
N
NNDU
NNN DU d−1NNNNN
NN& d−1
N&
N& Up
U dU DU
/
/
d−1
UUp
On the other hand, the right hand side of (6) in this case is the pasting
8 N
ppp NNNNN
p
p
NNN
ppp
NN&
ppp
q8
qqq
q
q
n−1
p
qqq
q
q
q
8
q
q
q
qqq
q
q
q
q
qqq
n−1
qqq
dU DU
qqq
qqq
8
qqq
q
q
qq
qqq
q
q
eeee2
qqq
eeeeee
q
e
e
e
q
e
e
qqq eeeeeeee
qeqeqeeeeee
/
/
/
d−1
U dU DU
d−1
Up
/ / MMM
MMMθ2 DU
µV
MMM
MM& d
nDU
/
/ .
q8
p7
q
p
d−1
q
p
µ
Up q
V pp
q
qqq
ppp
q
p
q
p
/ q
/ p
pp8
pp87
d−1
p
U dU DUppp
p
p
ppp θ2 DU pppp
p
ppp
pp
ppp
dnDU ppp
pp
ppp
On this pasting make the following substitutions:
/
/ N
NNN
NNN
NNN
N&
/
/
NNN
NNN
NNN
d
NNN
N
NNN nDU
NNN −1 NNNNN
NNN
NN& dU dU DU
NN& d−1
NN& Up
/
/
n−1
dU DU
/
Dn−1
dU DU
/ n−1
p
/ NN
NNN
NNN
NNN
&
θ2 DU p
Dn−1
p
p
p
pp
ppp
p
p
/ xp
/ NN
/ OO
NNN
OOO
NNN
NNN
N
OOO
NNN −1
NNN
−1
OOO
d
d
N
N
N& U U dU DU
N&
UUp
/'
/
= MMMM dnU DU
MMM
Dn−1
Dn−1
MMM
p
dU DU
M& /
/ ,
=
/ NN
/ NN
NNN
NNN
NNN
NNN
N
NN
NNN
NNN
NDU
NN& DU d−1 NNNN&
N& d−1
Up
U dU DU
/
/
θ2 U DU p
pp
pp p−1
ppp
p−1
p
p
p
dU DUppp
p
p
p
p
ppp
ppp
ppp
p
p
xppp
p
p
x
x
/
/
Theory and Applications of Categories, Vol. 5, No. 5
115
and
/
p−1
dU DU
/ NN
NNN
NNN
θ2 DU
NNN
.&
=
/ PP
APAPP
PPP
θ2 DU
AAPPPDU
PPP
AA PPPP
PPP
AA PPP
PP(
( P
AA
PPP
AA
PPµPV DU −1
AA
PP
AA
θ4 DU PPPP A
/(
(definition of θ4 ).
Compare both results, and use equation (16).
The claim about (dU, γ) is left to the reader.
Next we prove that the composite of two pseudomonads via a distributive law as
defined in section 5, is compatible.
6.2. Theorem. Assume we have a distributive law as in section 4 and let V the composite pseudomonad defined in section 5. If we define θ1 = iddU ◦u , and θ2 as the pasting
DU DUMM
MMMDrU
rr9
MMM
MM&
Dω
U
2
/ DDU U
DU dUrrr
rr
rrr
DU U
Dn
DdU U
DDn
Ddn
/ DDU
DU UUUUU
UUUU DdU
ηD U −1
UUUU
UUUU
mU
UUUU
Id
UU* DU,
and θ3 as the pasting
DU DUMM
MMMDrU
rr9
M
Dω U MMMM
&
1
/ DDU U
KK
DDuU
KK DDn
KK
DDβU
KK
KK
%
/ DDU mU / DU,
Id
DuDUrrr
rr
rrr
DDU
then the pseudomonad V is compatible with U and D.
Theory and Applications of Categories, Vol. 5, No. 5
116
Proof. Conditions (12), (13), and (14) are fairly easy and left to the reader. The remaining ones are also easy once we have shown that
θ4
=
DU DU U
DU Dn /
and
DU DU
mU DU /
=
θ5
DDU DU
DrU
Drn−1
/ DDU U
DrU U
DDnU
DDµU
DDU U
mU U
DDn
DU U
Dn
DDn
mDn
/ DDU
DDDn
/ DDU
m−1
n
DrU
mrU
/ DDU U
DDDU UmDU U
DDn
DDrU
DDU U U DDU n
DU DU
DDDU
DmU
mU
/ DU,
DDU
mDU
µD U
mU
mU
/ DU.
To show the first equality above, start with the definition of θ4 and make the following
substitutions:
hh3 NNN
hhhh
NN
h
h
h
−1 NNN
hhhh
DU
Dd
h
NNN
n
h
h
h
hNhhh
/
/&
8
NNN DU Dω U −1 ppp
2
NNN
p
−1
DrDn
NNN
ppp −1
N& pppp DrrU / NNN Dr−1
p8
p
NNN U dU
p
NNN
ppp
NN& ppppp
/
rr8
r
r
−1
r
DrdU
rrr −1
rrr Drn /8 ,
r8
p
r
p
r
p
rr
DDU d−1
n ppp
p
rrr
p
r
p
r
r
/8 pp
NNN
NNN DDU ω2 U −1 ppppp
NNN
p
NN& ppppp
=
and
8 N
ppp NNNNN
p
p
NNN
ppp −1
NN&
ppp mU dU /
−1
8
N
N
p
m
NNN
p
rU
p
p
NNN
p
m−1
Dn
NNN
ppp
Dω2 U
Vppp
&/ / VVVV
p8
p
VVVV
p
Dd
n p
VVVV
VVVV
ppp
VVVV pppp
+
=
{= DDD
DD
{{
{
DD
{{
{{ DDω2 U D/ !
QQQ
QQQ
QQQ
QQQ
QQQ
QQQ
QQQ
QQQ
DDd
QQ(
Q
n
Q(
/
QQQ
−1
m
n
QQQ
QQQ
−1
mdU
QQQ
QQ( / .
Now use (coh 3) on the pasting of DDU ω2 U −1 , DDω3 U , DDn−1
dU and DDω2 U . Next
make the substitutions:
/
DDω2 U U
) =
DDU d−1
n
/ −1
DDrn
/ DDd−1
Un
/ OO
OOO
OOO
OOO
'
DDω2 U
pp
p
p
p
ppp
xppp
/ ,
Theory and Applications of Categories, Vol. 5, No. 5
/ OO
OOO
OOO
OOO
OOO
OOO
OOO
OOO DDd−1
U
n
O'/
'
?O?OO
DDd
??OOO nU
?? OOOO
DDDµU
?? OO' ??
o/7
??
o
o
?? DDdn oooo
?? ooo
oo
117
/
/
DDµU
DDdn
=
/ ,
/ and
/
DDdn
m−1
dU
/
Dmn
/ µD U
/ ηD U −1
mU ◦ DDn ◦ DηD U U −1 .
=
/ /; Finally, use (coh 8).
The proof for θ5 is very similar, except that we need the equation:
uU D /
UUD
44 nn
44 βs{ D−1
44U
44
4
nD
Id 44
44
44
44
U D4
UD
Ur
/ U DU
rU
DU U
px hh
ω3
r
Dn
/ DU
=
UD
r
/ UUD
uU D
ur
uDU /
Ur
DU LL
U DU
LLL ω1 U
LL
rU
DuU LLL&
DU U
Id
DβU
)
Dn
DU.
To prove this last equation, observe that IdDU is isomorphic to Dn ◦ rU ◦ uDU , thus it
suffices to show that the equation holds when followed by Dn ◦ rU ◦ uDU . And, since ur
and ω3 are invertible, we can paste them below on both sides of the equation. This last
equation is not hard to prove, using (coh 4) and (coh 2). The rest of the proof is left to
the reader.
7. The Gray-category of pseudomonads in a Gray-category
In this section we define the Gray-category PSM(A) of pseudomonads on a Gray-category
A.
The objects of PSM(A) are pseudomonads in A.
Given pseudomonads D on K, and U on L, we denote the 2-category PSM(A)(D, U)
as [D, U], and we define it as follows. The objects of [D, U] are pairs (G, G0 ), where
G : U-Alg → D-Alg : Aop → Gray is a Gray-natural transformation, G0 ∈ A(L, K), such
Theory and Applications of Categories, Vol. 5, No. 5
that
Φ
/ D-Alg
G
U-Alg
A( , L)
118
G0 ( )
Φ
/ A( , K)
commutes, where Φ is the forgetful Gray-natural transformation defined in section 3. We
can consider U-Alg and D-Alg as trihomomorphisms and G as a tritransformation. A 1-cell
(G, G0 ) → (H, H0 ) in [D, U] is a pair (g, g0 ), where g : G → H is a strict trimodification,
and g0 : G0 → H0 is in A(L, K), such that
G
U-Alg
g
+
3 D-Alg
H
= U-Alg
Φ
Φ
A( , K)
G0 ( )
,
g0 ( ) 2 A( , K).
A( , L)
H0 ( )
What we mean by strict trimodification is, a trimodification in which all the invertible
modifications required in the definition ([6],pg. 25) are identities. A 2-cell (g, g0 ) → (h, h0 )
in [D, U] is a pair (γ, γ0 ), where γ : g → h is a perturbation, and γ0 : g0 → h0 is in A(L, K),
such that
U-AlgX gX ⇓
γX
+
/ ⇓ g D-Alg
X
X3
ΦX
A(X , K)
U-AlgX
=
ΦZ
+
γ0 ( )
A(Z, L) g0 ( ) ⇓ /⇓ h0 ( ) A(X
, K).
3
Vertical and horizontal compositions are the obvious ones. The rest of the operations are
defined as follows:
(H, H0 )(G, G0 ) = (HG, H0 G0 )
(H, H0 )(g, g0 ) = (Hg, H0 g0 )
(h, h0 )(G, G0 ) = (hG, h0 G0 )
(τ, τ0 )(G, G0 ) = (τ G, τ0 G0 )
(H, H0 )(σ, σ0 ) = (Hσ, H0 σ0 )
8. Liftings
8.1. Definition. A lifting is a pseudomonad in the Gray-category PSM(A).
in PSM(A) has to have domain a
In more detail, observe that a pseudomonad D
pseudomonad U = (U, u, n, βU , ηU , µU ) in A, with domain some object K of A. Observe
is of the form
that D
d), (m,
β), (
= ((D,
D), (d,
D
m), (β,
η , η), (
µ, µ)).
Theory and Applications of Categories, Vol. 5, No. 5
119
Taking the second entries, we obtain a pseudomonad D = (D, d, m, βD , ηD , µD ) in A on
X , and
the same object K. Furthermore, given an object X of A, we can define DX = D
dX = dX etcetera, to obtain a pseudomonad DX = (DX , dX , mX , βX , ηX , µX ) in Gray, on
the 2-category U-AlgX . D and the family DX X behave well with forgetful 2-functors
and change of base.
9. From a pseudomonad with compatible structure to a pseudomonad in
PSM(A)
Assume we have a pseudomonad V that is compatible with pseudomonads D and U as in
on the object U of PSM(A). Let X be an
section 6. We want to define a pseudomonad D
object of A. We begin by defining a 2-functor DX : U-AlgX → U-AlgX . Given an object
(ψ, χ) in U-AlgX , with
U XDD
{=
DD x
uX {{{
D
{
{
DDD!
{
~
{
ψ
/ X,
X
Id
UUX
nX
UX
Ux
χ
x
/ UX
x
/ X,
Define the first entry of DX (ψ, χ) as
U DX
F
NN
NNNU DuX
NNN
NNN
&
dU DX
U DU XOO
OOOdU DU X
d−1
OOO
uDX U DuX
OOO
θ1 DX '
__
/ DU DU X
DX
ks __DU
NNN
DU DuX
s9
oo7
NNNpX
sss
ooo
NNN
o
v
ssss
DuX
o
o
β
X
NN&
V
ss vDX
ooo vDU X / DU X
/
DX
DU X
II
DuX
II Dx
II
II
Dψ I$
Id
. DX,
Theory and Applications of Categories, Vol. 5, No. 5
120
and the second entry of DX (ψ, χ) as
U U DX
nDX
U DU X
U dU DU X
U Dx
/ U U DU X
/ U DU DU X U pX / U DU X
/ U DX
MMM
22
MUMMDuU X
22
U DuX
MMM
U Du−1
22
x
M&
22
ks ____
/ U DU X
Id
U DU U X
22dU U DU X
U DβU X
x
q dU DUUUDU
dU DU DU X
q
22
X
q
q
22
dU DU X
qqUqDnX
q
d−1
q
22
xq __
U DU x / DU DU X
22
nDU X
U DU X ks __ DU DU U X
d−1
DU DU x
dU DnX qq
U dU DU X
22
q
22
p
pU X
dU DUq
X
qqq
p−1
d−1
x
U pX
2
qq DU DnX q
x
DU dU DU
X
/ DU DU DU X
/ DU DU X ks __ DU U X
/ DU X
DU U DUNX
DU pX
DU x
NNN θ4 X −1 rr
r
θ
DU
X
NN 2
rr
pDU X µ X
pX
Dx
n−1
dnDU X DnDU XNNNN
Dχ
rrrDnX
V
r
DuX
N
r
' xr
/ U DU X
/ DU DU X
/ DU X
/ DX.
U U DuX
U DuX
pX
dU DU X
Dx
If (h, ρ) : (ψ, χ) → (ψ , χ ) in U-AlgX , then define the first entry of DX (h, ρ) as Dh, and
the second as the pasting
U DX
U DuX
U DU X
U Dh
/ U DX U DuX U Du−1
h
/ U DU X U DU h
dU DU X d−1
U DU h
/ DU DU X dU DU X
DU DU X DU DU h
p−1
h
pX
DU X
Dx
DX
DU h
Dρ
Dh
pX / DU X Dx
/ DX Given ξ : (h, ρ) → (h , ρ ) in U-AlgX , define DX (ξ) = Dξ.
9.1. Proposition. The above definitions make DX : U-AlgX → U-AlgX a 2-functor.
is a
: U-Alg → U-Alg as DX at every object X of A, then D
Furthermore, if we define D
D) ∈ [U, U].
Gray-transformation, and (D,
Proof. The hardest part of the proof is to show that DX (ψ, χ) is indeed an object in
U-AlgX . This is what we do, and we leave the rest to the reader. We must show that
(6) and (7) are satisfied. To show (7), pass ηU DX to the left hand side, and perform the
following sequence of substitutions:
/
ηU DX −1
+ /
/
n−1
DuX
dnDU X
/ =
/ /
/ N
rr8
rr8
rr8 NNNN
r
r
r
r
r
r
NN
r
r
r
rrrU uDuX rrrrrd−1
rrr Dη DU X −1 NNNN
U uDU Xrrr
U
&/ ,
rrr
r
/
/
Theory and Applications of Categories, Vol. 5, No. 5
/ NN
/
NNNθ2 DU X
NNN
DηU DU X −1 NNN& 2
8
rrr
r
r
rrrDU θ1 DU X
rrr
=
/
8
ppp
p
−1
p
p
dU dU DU X
pp−1
ppp dU uDU X /; 8
p
p
ppp
pppDU θ1 DU X
p
p
p
/
µV X
ηV DU X −1
0 /
=
/ =
d−1
U pX
/; / J
JJ
JJU uDuX JJJ
JJ
JJ
JJ
JJ
J$
J$
/
=
JJ
JJ
JJ
U θ1 DU X JJJ
1$
U DuX
/ NN
NNN
NNN
NNN
&
U DβU X
p
p
p
pp
ppp
p
p
xp
/;
7 ,
d−1
U vDU X
}>
}}
}
}}DU βV X
}}
r8
rrr
r
r
rr
U βV X
rrr
(see (12))
/C
,
pX
/;
,
dU DU X
/ NN
8 N
NNN
ppp NNNNN
p
N
p
N
p
N
p
NNN −1 NNNNN
p
p
U
θ
DX
1
p
U
d
&/
U DuX
/&
VpVVV
VVVV
pp8
p
VVVV
p
VVVV U vDuX pppp
VVVV
V+ ppp
/ NN
NNN
NNN
NNN
NN
NNN
NN& U Du−1 NNNN&
uX
/ NN
NNN
NNN
NNN
U DηU X −1
&
0 ,
=
/ NN
8 N
NNN
ppp NNNNN
p
NNN
p
NNN
p
p
NNN
p
−1
N
p
d
U
Dη
X
N
U
DnX
p
U
p
&/
&/
/ NN
NNN
pp8 NNNNN
p
p
N
NNN
NNN
pp
p
p
−1
−1 NN
N
p
DU
Dη
X
θ
X
N
N&
p
4
U
p
&/
/
/ ; NN
NNN
NNN
N
ηV DU X −1 NN&
/ ,
8
rrr
r
r
rrr U θ1 DU X
rrr
=
/
p8
ppp
p
p
pp−1
ppp dU vDU X 8
ppp
p
p
ppp DU βV X
ppp
121
=
/ NN
8 N
NNN
ppp NNNNN
p
NNN
p
N
p
N
p
N
p
Dχ NNN&
DηU X −1 NN&
ppp
/
/
=
JJ −1 / JJ
JJdU DU uX JJ
JJ
JJ
JJ
JJ
J$
J$
/
JJ
JJ
JJ
DU DηU X −1JJJ
%
0 ,
JJ −1 / JJ
JJ puX
JJ
JJ
JJ
JJ
JJ
J$
J$
/ J
JJ
JJ
J
DηU X −1 JJJ
%
0 ,
=
v:
vv
v
v
vv DU ψ
vv
/>
(see (19))
Dx
/ ,
Theory and Applications of Categories, Vol. 5, No. 5
and
/
/
U Du−1
uX
U Du−1
x
d−1
U DU uX
d−1
U DU x
p−1
uX
p−1
x
/ /
=
/:
U Dψ
U DuX
/ / 122
/ / /; .
dU DU X
pX
DU ψ
To get from the left hand side of (6) to the right hand side, make the following sequence
of substitutions:
/ NN
NNN
NNN
NNN
NNN
NNN
−1
NNN
NNU
& nDuX
&/
NNN
NNµNU DX
NNN
n−1
DuX
NN& / /
/
µU DU X
dnDU X
/ =
/ TTT
SSS
T
SSS
SSS U dnDU XTTTTTT
S)
T)/
d−1
U nDU X
1 θ4 DU X
/ / / NN
NNN
NNN
d−1
U DU dU DU X
NNN
/
/
&
NNN
NNN
NNN d−1
NNN
NNU pX
N
DU θ2 DU X NNNN DU µV X NNN
&/ &1
/
(definition of θ4 )
d−1
U DU pX
/
p−1
dU DU X
/ d−1
U DnDU X
/
=
/ =
d−1
U dU U DU X
d−1
U pX
d−1
U DnDU X
t: JJJ
JJ
tt
t
JJ
tt
JJ
t
tt DU θ2 DU X
/$
DµU DU X
/5 ,
6
l
k
l
k
d
l
nDU
X
k
kk
lll
kkk
llll
/ kk
/ OO
OOO
(see (16))
OOO
OOO
OOO
OOO
OOO
OO' DU d−1
'/
nDU X
θ2 U DU X p
p
p
p
p
p
θ4 DU Xpp
pp
pp
ppp
p
p
ppp
p
x
xpp
/ ,
/
/
=
dnU DU X
d−1
U dU DU X
/ ,
l6 −1
k5
lDU
k
l
k
l
d
k
nDU
ll
kk
kkX
llll
/ kk
/ NN
/ NN
NNN
NNN
NNN
NN
N
U θ2 DU X
NN& U µV X NNNN&
/
1
/ TTT
SSS
TTTT
SSS
SSSd−1
SU) nDU X TTTT)/
=
/ / NN
NNN
NNN
NNN
&
DµU DU X θ2 DU X
p
p
p
pp
ppp
p
p
/ xp
/ K
KK
KK
KK
n−1
U DuX
KK
K%
/
JJ
= HHH
µ
DU
X
J
U
JJ
HH
JJ
HH
JJ
HH n−1
DuX
$
/ % ,
/ µV DU X
θ2 DU X
/6 ,
Theory and Applications of Categories, Vol. 5, No. 5
/ NN
NNN
NN
NNN
NNN DU µ XNNNNN
V
NN&
NN&
/
NNN
µ
DU
X
NNVN
NNN
µV X
NN& / / PP
/ PP
PPP
PPP
PPP
PPP
PPP
PPP
PPP
P
PPP
PPP
PPP
−1
−1
PP(
U
d
U
d
P
P
(
(
U pX
U dU DU X
−1
/
/
PPPdU dU U DU X
PPP
−1
PPP
dU DU pX
d−1
PPP
U DU dU DU X
P( /
/ θ2 U DU X
) / K
KK
KK
KK
KK
K%
/
H
J
= HH
JJ µV X
J
HH
HH µ X JJJJ
HH V
J% $
/ ,
p−1
pX
DU d−1
U dU DU X
DU d−1
U pX
p−1
dU DU X
p−1
pX
/ J
JJ
t: JJJ
JJ
JJ
tt
t
JJ
JJ
t
t
J
t
U
d
J$ U DnX JJ$
tt U U DβU X
/
/
/ /
/ QQ
QQQ
QQQ
QQQ
QQ(
/
/
PPP
OOO
OOO
d
nDU
X
P
OOO
OOO
PPP
OOO
OOO
PPP
OOO
OOO
PPP
d−1
O' d−1
O/'
(/ ,
U pX
U dU DU X
=
=
8/ NNN
8
NNN
rrr
rrrU d−1
r
r
U DuU X
r
r
NNN
r
r
r
r
r
r
NN&
rr U DU DβU X
r
rrr
/ ,
/
/
/
d−1
U DU DuU X
d−1
U DU DnX
p−1
DuU X
p−1
DnX
=
dU DU DU X
/ / pDU X
=
/ / / ; ,
/ PP
PPP
PPP
PPP
PP'
/
O
N
= NNN
OOdO−1
OOdU
NNN
U
UX
OODU
NNN
OOO
−1
DU
d
NN' U DuU X
O' / ,
d−1
U U DuU X
DU d−1
U DuU X
p−1
DuU X
/ DU DβU X
/ OO
OOO
OOO
OOO
OOO
OOO
OOO
OOUO d−1
U
DuU
X
O'/
'
OOO −1
dOUOdU DU X
OOO
d−1
OOO
U DU DuU X
O' / /
n−1
pX
n−1
dU DU X
: J
tt JJJ
JJ
tt
t
tt U DU Dβ X JJJ
t
U
t
/$
/ K
KKK
KKK
KKK
%
Dn−1
pX θ2 DU X tt
t
tt
tt
t
y
/ , t
Dn−1
dU DU X
/ / PP
/ PP
PPP
PPP
PPP
PPP
PPP
PPP
PPP
P
PPP
PPP −1
PPP
−1
PP(
d
d
P
P
(
(
U U pX
U U dU DU X
/
/
PPP dnU DU X
PPP
PPP
Dn−1
Dn−1
pX
PPP
dU DU X
P( /
/ ) /
=
/ / / QQ
QQQ
QQQ
QQQ
QQ(
/
/
PPP d−1
OOO
OOO
PPP U dU DU X
OOO
OOO
PPP
OOO
OOO
OOO
OO
−1
−1 PPP
ODU
' dU dU DU X OO'/ DU dU pX P(/ ,
d−1
U U pX
d−1
U U dU DU X
/
θ2 DU X
/
=
/
/ 123
Dn−1
DuU X
/ OO
OOO
OOO
OOO
'
θ2 DU U X p
p
p
p
p
ppp
xppp
/ ,
Theory and Applications of Categories, Vol. 5, No. 5
/ OO
OOO
OOO
OOO
OO
OOO −1
OOdOU U DuU X OOOOO
'
'/
OOO
OOdOnDU X
OOO
Dn−1
DuU X
OOO
' / =
124
/
/
n−1
DuU X
dnDU U X
/ U DβU X
, / dU DnX
/ ,
DU DβU X
, /
OOO
OOO
OOO
OOO
OOO
O
OOO U θ4 X −1 OOOOO
'
'/
OOO −1
d
OOUODU DnX
OOO
d−1
U pX
OOO
' / / OO
OOO
OOO
OOO
OO
OOO
OOODU θ4 X −1 OOOOO
'
'/
OOO
OOOp−1
DnX
OOO
µV X
OOO
' / U DnX
/ PP
PPP
PPP
µV U X
PPP
PP'
/
O
N
= NNN
OOO θ X
OOO 4
NNN
NNN
−1 OOO
θ
X
OO' 4
NN'
/ ,
/ NN
NNN
NNN
NNN
&
U DβU X
pp
p
p
p
ppp
xppp
/ J
JJ
t: JJJ
JJ
JJ
tt
t
J
t
d
JJU DnX JJJ
tt
J
J$
t
t
$
/:
U DµU X t:
JJ
JJ
t−1
tt
t
t
JJ
t
t
JJ
tt dU DnXttt
J$ ttt
t
t
/
/ J
JJ
t: JJJ
JJ
J
tt
t
JJ θ4 X −1 JJJ
t
t
J
JJ
t
J$
tt
/$
DU DµU X t:
JJ
t:
JJ
t
t
JJ
tt θ4 X ttt
JJ
tt
t
J$ ttt
tt
/ t
=
t: KKKK
tt
KKK
t
t
KKK
U Dχ
tt
tt
/
/%
U Du−1
nX
U Du−1
x
d−1
U DU nX
d−1
U DU x
p−1
nX
p−1
x
/ / / / PP
PPP
PPP
d−1
PPP
U pU X
PP'
/
OOO
= NNNN
−1
O
d
OO U DnX
NNN
NNN DU θ X −1OOOO
OO' 4
NN'
/ ,
=
=
/ OO
OOO
OOO
OOO
OO
OOO
OO' U Du−1 OOOO'
nX
/
U DβU U X p
pp
ppp
p
p
p
p
p
U DµU Xppp
ppp
pp
xppp
/ , xp
/
t:
t: JJJ
JJ
tdt−1
tt
t
t
JJ
t
t
JJ
tt U DU nX
tt
t
t
t
$
t
/
DU DµU X
JJ
JJ
:
JJ
JJ
tt
t
JJd
J
t
JJU DnU X JJJ
tt
J$
J$ ttt
/
/
t:
t: JJJ
JJ
tt p−1 ttt
t
JJ
nX tt
tt
JJ
t
t
tt
$
tt
/
DµU X
JJ
JJ
t:
JJ
JJ
t
JJ
J
tt
JθJ4 U X −1 JJJ
tt
J$
J$ ttt
/
=
/ ,
/
/
U Du−1
Ux
U Du−1
x
d−1
U DU U x
d−1
U DU x
p−1
Ux
p−1
x
/ / / / (see (18))
/ / / 9 ,
/ HH
t
HH
HH DU χ tttt
HH
HH tttt
$ t
(see (20))
Theory and Applications of Categories, Vol. 5, No. 5
/ OO
OOO
OOO
OOO
OO
OOO
OOO DU χ OOOOO
'
'/
OOO
OOODµU X
OOO
Dχ
OOO
' / / PP
PPP
PPP
Dn−1
PPP
x
PP'
/
O
N
= NNN
OOO
OOO Dχ
NNN
OOO
NNN
Dχ
OO' NN'
/ ,
/ NN
NNN
NNN
NNN
NN
NNN
NN& U Du−1 NNNN&
Ux
/
U DβU U X p
pp
p
−1
p
dU DU U x
ppp
xppp dU DnU X / p
p
ppp
p−1
Ux
ppp −1
p
p
xp θ4 U X / p
p
p
p
p
p
−1
Dnx ppp
pp
ppp
ppp
xppp
/ xpp
/8
/
8
pp8
ppp U d−1
ppp U p−1
x ppp
p
p
U DU x
p
p
p
p
ppp
ppp
pppd−1
U DU x ppp
/ pp
/ pp
p8
ppp
−1
p
p
d
d−1
U pU X
U dU DU U X
ppp
/ / pp
/ OO
OOO
OOO
OOO
OOO
OOO
OOO
OOO DU p−1
x
O'/
'
OOO
OOOµV U X
OOO
p−1
x
OOO
' / /
θ2 DU U X
) and
125
=
/
/
8
ppp
p
−1
−1
p
dU pX
dU dU DU X
ppp
ppp d−1
U U DU x /
/8 8
pDU
p8
−1
−1 ppp
p
p
d
p
p
DU px pp
U DUp
xp
pp
p
ppp
ppp
p
p
ppp
p
p
p
p
p
p
/
/
/ PP
PPP
PPP
p−1
PPP
DU x
PP'
/
O
N
= NNN
OOO µ X
OOO V
NNN
NNN
−1 OOO
p
OO' x
NN'
/ ,
=
=
DU d−1
U DU x
/ p−1
DU x
/ / J
JJ
JJ
JJ
JJ
$
U DβU X t
t
t
tt
tt
ztt dU DnX t
tt
tt
d−1
t
U DU x
tt −1
/ zt θ4 X t
tt
tt
p−1
t
x
tt
/ zt
Dn−1
DU x
/ OO
OOO
OOO
OOO
'
θ2 DU X
p
p
p
pp
ppp
p
p
/ , xp
/ OO
OOO
OOO
OOO
OO
OOO
−1
OOO dU U DU x OOOOO
'
'/
OOO
OdOnDU
UX
OOO
Dn−1
OOO
DU x
O' / / PP
PPP
PPP
PPP
PP'
/
OOO
= NNNN
d
O
X
OOnDU
NNN
OOO
NNN
−1
NN' dU DU x OOO' / ,
/ OO
OOO
OOO
OOO
OOO
OOO
OOO
OOO U U Du−1
x
O'/
'
OOO −1
OOnODuU X
OOO
n−1
DU x
OOO
' / / PP
PPP
PPP
n−1
PPP
Dx
PP'
/
P
O
= OOO
PPP n−1
PPPDuX
OOO
PPP
OOO
PP' OO' U Du−1
x
/ .
n−1
DU x
Theory and Applications of Categories, Vol. 5, No. 5
126
We are now ready to define dX : 1 → DX , and mX : DX DX → DX . The first entry of
(dX )(ψ,χ) is dX, while the second is the pasting
/ U DX
;;
VVVVV
VUVuX
;;
VVVVV
U DuX
U duX
;;
VVVVV
V+
;;
;
U U X U dU X / U DU X
dU X ;;
;;
;;
dU U X
dU DU X
d−1
d−1
U dU X
U uX
;
DU
dU
X
/ DU DU X
DU X DU uX / DU U XQQ
Q
Q
θ
QQQ 2 X
DηU X −1 pX
QQ
DnX QQQ(
/ DU X
Id
U X; VVVVVV
x
U dX
dx
X
Dx
/ DX,
dX
and (dX )(h,ρ) = dh .
Define the first entry of (mX )(ψ,χ) as mX, and the second as the pasting
U DDX SS
U DuDX
/ U DU
U mX
SSSS
SSSS
S
U DDuX SSSS)
U DuX
U muX
/ U DU X
U DU DXSS
U DDU XSS
U mU X
SSSS U Du−1
SSSS
dU
X
SSSS
SSDDU
DuX
S
SSSS
U DuDU X
dU DU DX
S
U DU DuX SSSS)
SS)
d−1
U mU X
s
k
_
_
dU DU X
DU DU DX
U DU DU X dU DuDU X DU
DDURX
SSSS
k
R
k
RRR
SSSS d−1
kk
k
DU
mU
X
R
k
U
DU
DuX
R
k
SSS
RRR
DU θ X
pDX
dU DU DUkk
Xkk
RRR
3
DU DU DuX SSSS)
ukkkk DU DuDU X )
_
_
s
k
/ DU DU X
DU DX p−1__
DU
DU
DU
X
DU
pX
DuX
kkk
kkkk
k
k
DU DuX
k
k
ukkkk pDU X
DU DU X YYYYYY
YYYYYY
YYYYId
YYYYYY
pX
DdU DU X
µV X
YYYYYY
ηD U DU X
YYYY,
/ DU DU X
DDU DU X
RRR
mU DU X
RRR
RRR
DpX
θ5 X
pX RRRR)
/ DU X
DDU X
mU X
DDx
DDX
mX
mx
Dx
/ DX.
Define (mX )(h,ρ) = mh .
9.2. Proposition. With the above definitions, dX : 1 → DX and mX : DX DX → DX
are strong transformations. Furthermore, defining d as dX , and m
as mX at every object
Theory and Applications of Categories, Vol. 5, No. 5
127
d) : 1 → (D,
D) and (m,
D)(D,
D) → (D,
D) are
X of A, we have that, (d,
m) : (D,
1-cells in the 2-category [U, U].
Proof. The size of the diagrams obtained is what makes it hard to prove that (mX )(ψ,χ)
satisfies (10). To get from the left hand side of (10) to the right hand side make the
following three substitutions:
TTTT
TTTT
TTTT
T
)
TTTUTDu−1
TTTpX
TTTT
T
)
TTTdT−1
U DU pX
TTTT
TTT)
T
TTTTp−1
TpX
TTTT
TT)
T
−1
TTTTDdU pX
TTTT
TTT)
T
TTTT DµV X
TTTT
TTT)
/ TTT
TTTT
TTTT
T)/
U θ5 X
TTTT
TTTT
TTTT
TTTT
TTTT
TTTT
T
TTTT
TTTT
TTTT
TTTT
TTTT
U mx
TTTT
TTTT
TTTT
T
T
TT)
TTTT
T
−1 T)
/
T
T
U
Du
TTTT Dx
TTTT
TTTT
TTTT
TTTT
TTTT
TTT) TTT)
TTTT
T
TTTTd−1
TTTT
T
HHTTT
−1
U DU Dx
T
U
Du
TTTT
TTTT DuX TTTTTTT
HH TTTT
HH TTT
TTTT
TTTT
TTTT
T
HH
) )
) T
HH TTTTTT p−1
−1
T
d
TTdTUTDU DuX
U DuDU X
jjj
TTTTDx
j
HH
j
T
j
TTTT
j
HH
TTTT
T) ujjjjj
$ T
DU DβU X
−1 ) −1
DU
Du
T
j
j
x
T
j
j
p
TTTT
j
jj
DuX
TTTT
jjjj
jjjj
T) Yujjjjj
ujjjjDdU DnX T
−1
YYYYYY
j TTTDd
YYYY
TTUTTDU x
jjjj
j
j
ηD U DU X YYYYYYYYY
j
T
T
j
YYYYY,
j
T
T) ujj Dθ4 X −1 T
/
Dp−1
T
j
x
T
j
T
j
T
j
T
j
T
j
TTTT
jj
T) YujjYjj
YYYYYY
DDχ
YYYYYY
YYYYYY
YYYYY, TTTT
TTTT
TTTT
U muX
)/
TTTT d−1
X
TTUTmU
TTTT
T/) DU θ3 X
TTTT µV X
TTTT
TTTT
θ5 X
)/ mx
/ =
[[[[[[[[
[[[[[[[[
[[[[[[[[
[[[[[/ U θ3 DU X
d−1
U pDU X
/ VVV
VVVV
VVVV
VVV
U DβU X hhh+
h
h
hhh
shhhhhh
−1
/ VVV
d
d
U DnX hhh
U pX
VVVV
h
h
h
VVVV
h
VVV+ shhhhhhh
/
U Du−1
x
/ d−1
U DU x
/ V
VVVV
p−1
x
VVVV
θ4 X −1
V
V
ηD U DU DU X VVV/+ V µV DU X
/ µV X
VVVV
hhh
h
VVVV
h
h
h
Dχ
VVVV hhhh
[[
V/+
θ5 DU X
/3 shhh
[[[[[[[[
h
h/3
h
θ5 X hhhh
mx
[[[[[[[[
hhhh
h
h
h
h
h
[[[[[[[[
h
hhh
[[[- hhhhh
/ hhh
Theory and Applications of Categories, Vol. 5, No. 5
NNN
NNN
NNN
NN&
NNNd−1
NUNpDX
NNN
NNN
&
NNN µV DX
NNN
NNN
NN&
/ NN
NNN
NNN
NNN
−1
U pDuX
/&
NNN
NNN
NNN U Du−1
U DuX
NN&
/
U DβU DX p
pp
p
−1
p
dU DU U DuX
ppp
xppp
/
dU DnDXp
p
p
p
p
ppp
xppp
−1
θ4 DX −1 p NN pU DuX
NNN
p
p
p
NNN
p
NNN
ppp
p
p
xp
&
NNN
NNN
NNN
NNN
NNN Dn−1
NNN DuX
NNN
NN&
128
NNN
NNN
NNN U ηD U DU X
UVDu−1 NN&
VVdU
NNN
VVDU
VVVXV
N θ3 DU X
VVVV NU
VVVNVNNNN
VVVVN&
V
+
VVVV
VVVVd−1
U
DU
dU
DU
X
VVVV
VVVV
VVVV +
p−1
dU DU X
/
NNN
NNN
NNN DdU dU DU X
NN&
/
NNN
NNN2 DU X
Dθ
NNN
NN&
DdnDU X
/
/ N
NNN
NNN
NNN
NNN
NNN
U µV X
NNN
NNN
N&
/
/8* <
p <
p
p
<
p
<
p
<<
ppp
<<
ppp
<<
d−1
U pX
−1
<<
dU pDU X
<<
<
/ /
NNN
NNN
NNN
µV DU X
ηD U DU DU
X
NN&
/ NN
NNN
NNN
NNN
θ
DU
X
5
/& =
/
/
d−1
U DU DU DuX
/ p−1
DU DuX
/ p−1
DuX
/ NN
8/
NNNηD U DU X
ppp
p
NNN
p
NNN ppppp
& p
d−1
U DU pX
/
/ d−1
U pX
/ p−1
pX
/ NN
µ X
NNN V
NNN
NNN
µV X
&/ and
g3 OOO
ggggg
OOO
g
g
g
g
OOO
gg
g
g
g
g
gg
U U muX OOO
'/
gOgggg
/ WOOWWW
NNN
 OOOO
OOOWWWWW
NNN

W
O
O
W

W
O
O
W
NN
WWWWW
OOO
OOO

−1
W
O
O

WWWW+ U D−1 NNNNN
OU' U DuDuX
O/'


N
U
mU
W
OOOU dU DuDU X WWWW X
OO

WWWWW NNNNN
OOO
 OOOO



WWWWW NNN
OOO
OOO


OOO
OOOU d−1
 U DU θ3 X WWWWWNWNW
n−1
N+'/


DuDX
'
'
U
DU
DuX
−1
/
OOdOU dU DU DX
OOO


OOO
OOO

−1
OOO
OOO
d−1
U DU pX
OOO dU DU DU DuX
OOO 
/
' ' WW d
/ WWnDU
θ2 DU X
WWWDX
WWWWW
WWWWW
p−1
p−1
pX
DU DuX
WWWWW
W+* / / µV X
/ Theory and Applications of Categories, Vol. 5, No. 5
=
129
/ PP
PPP
PPP
PPP
PP(
PXPXPXXXX
n−1
mX
PPP XXXXXX
X
PPP
PPP XXXXXXXXXX
P
XXX+
/
( Q
−1
PPP
PPP
n−1
QQQ dU dU DU X
PPP
PPP DuX
QQQ
PPP
PPP
QQQ
PPP
PPP
QQQ
U
m
uX
P(
P/( ( PPP
PPP U Du−1
PPP
PPP DuX
PPP
PPP
PPP
PPP
PPP
θ2 DU X
PPP
P
PPP
P( dU DuDU X PPP(
PPdPnDU X
−1
X
PPPd−1
PP
nn XXXXXdXUXmU
PPUPDU DuX
XXXXXX PPPPP
nnn
PPP
n
X
n
XXXXX PPP
PPP
nn
XXXXXPP( ! DU θ3 X
P' wnnnn
X/+ .
The fact that the three equalities above are indeed satisfied and the rest of the proof are
left to the reader.
and µ
It is easily seen that, defining η, β,
at every X as (ηX )(ψ,χ) = ηD X, (βX )(ψ,χ) =
β), and (
η , η), (β,
µ, µ) in [U, U]. We thus
βD X, and (µX )(ψ,χ) = µD X, we obtain 2-cells (
have:
d), (m,
β), (
= ((D,
D), (d,
9.3. Theorem. D
m), (β,
η , η), (
µ, µ)) is a pseudomonad in
PSM(A), on the object U.
10. From a pseudomonad in PSM(A) to a distributive law
Assume that we have a pseudomonad in PSM(A) as in section 8. We produce a distributive law as follows. Consider DK : U-AlgK → U-AlgK . Since the diagram
U-AlgK
ΦK
DK
A(K, K)
/ U-Alg
ΦK
D( )
K
/ A(K, K)
commutes, we have that DK (βU , µU ) is of the form (σ1 , σ2 ) with
U? DU@
@@
@@s
@@

uDU



DU
U U DU
| σ1
Id
nDU
/ DU,
DK
U DU
U
U-AlgK
DK
/ U-Alg
K
U
/ U-Alg
/ U DU
σ2
Furthermore, since the diagram
U-AlgK
Us
K
s
s
/ DU.
Theory and Applications of Categories, Vol. 5, No. 5
130
is the change of base 2-functor defined in section 3, then DK (n, µU ) :
commutes, where U
DK (βU U, µU U ) → DK (βU , µU ) is of the form (Dn, σ3 ) : (σ1 U, σ2 U ) → (σ1 , σ2 ) with σ3 of
the form
U Dn /
U DU
U DU U
σ3
sU
DU U
s
/ DU.
Dn
Similarly, it can be shown that (dK )(βU ,µU ) is of the form (dU, σ4 ), with
UU
U dU /
n
σ
U
U DU
4
dU
s
/ DU.
If we apply DK to (σ1 , σ2 ) we obtain another U-algebra (σ1 , σ2 ), with
U DDU
<
FF
x
xx
uDDU
xx
x
x
xx
U U DDU
DDU
FF FsF
F
 σ1 F#
/ DDU,
nDDU
U s
/ U DDU
σ2
U DDU
s
s
/ DU.
Applying DK to (s, σ2 ) : (βU DU, µU DU ) → (σ1 , σ2 ), we obtain a morphism (Ds, τ1 ) with
U DU DU
sDU
U Ds
/ U DDU
τ1
DU DU
s
/ DDU.
Ds
We also have that (mK )(βU ,µU ) is of the form (mU, γ1 ), with
U DDU
s
DDU
U mU
/ U DU
γ1
mU
s
/ DU.
Define σ5 as the pasting
U DuDU
/ U DU
U mU
U DDU
Id
U Dσ1−1
OOO
OOO
sDU
U Ds OOO'
DU DUOO τ1 U DDU
OOO
OOO
s
Ds OOO'
U DU DU
OO
DDU
s
γ1
mU
/ DU.
Theory and Applications of Categories, Vol. 5, No. 5
131
It is with the help of σ1 to σ5 that we define the distributive law as follows:
Define r = s ◦ U Du, ω1 as the pasting
U D NN
s9
NU
NNDu
ss
NN'
s
ss
u
Du
D JJJ
U
8 DUNNN s
JJJ
NN
ppp
p
p
J
σ1 NN&
pp uDU
Du $
/ DU,
DU
uD
Id
ω2 as the pasting
U dsss9
U D NN
NU
NNDu
NN'
U
d
u pU
8 DUNNN
NN
pp
p
p
s NN&
pp U dU DU,
U U NN
σ4
NNnN
pp8
p
p
NNN
p
−1 ηU
&
ppp dU
Id
1U
s
sss
U JJJ
JJUJu
J$
ω3 as the pasting
UUD
U U Du/
U U DU
nD
Us
/ U DU U DuU/ U DU U sU / DU U
LLL
LLL
LU DβU U Dn
Id LLL% U DUKK
nDU
n−1
Du
/ U DU
UD
σ2
s
U Du
Dn
KKK σ3−1
KK
s KKK
% / DU,
and ω4 as the pasting
UD
O
U Du
/ U DU
O
//// U mu
U mU
/ U DU DU
U
DDU
QQQ
o7
U DuU−1DuDU
nn6
o
QQQsDU
n
o
n
o
n
o
Du
QQQ
n
s−1
oo
nUnDU Du
o
QQQ
n
o
n
U
DDu
Du
nn
oo
(
/
/
/ DU DU
U DD
U DU D
DU D
Um
U DuD
/ DU
O
s
sD
DU Du
//// σ5
Ds
mU
/ DDU.
The proof that we obtain a distributive law in this way is based on the behavior of the
2-cells σ1 -σ5 . Aside from the obvious conditions that come from the facts that (σ1 , σ2 ) is a
U-algebra, (Dn, σ3 ) and (dU, σ4 ) are homomorphisms of U-algebras, we have the following
conditions.
10.1. Proposition. The following are 2-cells in U-AlgK :
DηU : Id → (Dn, σ3 ) ◦ (DU u, s−1
u ).
(26)
Theory and Applications of Categories, Vol. 5, No. 5
132
DµU : (Dn, σ3 ) ◦ (DU n, s−1
n ) → (Dn, σ3 ) ◦ (DnU, σ3 U ).
dn : (Dn, σ3 ) ◦ (dU U, σ4 U ) → (dU, σ4 ) ◦ (n, µU ).
(27)
Furthermore, the following equations are satisfied:
DDULL
/ U DDU U mU / U DU
uDDU
LLLDuDU
LLL
LL%
uDuDU
U DuDU
mU
=
DU DUNNuDU DU/ U DU DU
Id
NNN N NσN1 DU sDU
NNN
' Dσ1
σ5
s
DU DU
$
DDU
Ds
DDU
mU
uDDU/
DU KK
U DDU
umU
uDU /
U mU
U DU
KKK σ1
KKK
Id KK%
s
DU,
/ DU
/ NN
NNN
NNN
NNN
NNN
NNN
NNN
& N
NNN
NNN
NNN
NNN
NN&
NNN
N??NN
NNN
NNN
??NNN
N
?? NN
NNN
?? NNN
NNN
N
?
NNN
N
?
N
−1
N
U σ5 ⇓
⇐ ?
n
NNN
NNN
N DuDU
Uo DβU DU ? N
NNN
NNN
UNN
Du−1
NN⇐
s
'/
'
NNN
NN⇐
σ2 DU
N
NNN
N
NNN
&N ⇐
N& NNN
σ3 DU −1 −1
NNN
s
s
⇐
O
e
NN& eeeeeee OO ⇐
OOO
re
NNN
OOO
NNN
OO NNN
NNN Dσ2 '
NNN
σ5
N⇐
NNN
⇐
NNN
'
/
=
/ U U DU
LL
LL
LL
−1
nDDU
nDU
LL
nmU
LL
LL
LL
/
LL
U DDUOO U mU U DULL
LL
OOO
LL
LLU s
LL
OOO
LL
L
LL
O
L
LL
U DuDU OO'
LL
LL
LL
LL
LL
L
U DU DU
LL
LL
NNN
LL
LL
NNN
LL
L
LL
NNN
LL
L
s
LL
NN'
LL
sDU
&
LL
LL
L
DU DU
U
DU
L
LLL
LL
LL σ2
LLL σ5
L
s
LL
L
LL
Ds LLL%
& / DU
DDU
U U DDU
U U mU
mU
(28)
Theory and Applications of Categories, Vol. 5, No. 5
133
/ U DU U
= U DDU UU mU U / U DU U
K
JJ
K
OOO
KKKU Dn
JJU Dn
JJ
OOO
KKK
U DuDU U
JJ
O
KK%
U mn
U DDn OO'
J$
/
U DU DUOOU U−1
DDU U mU U DU
U DU DU U
U DU
OOO U DuDn
O
OOO
sDU U
U DuDU
sDU U
sU
U DU Dn OO' U DDU OUO
U mU U
DU DU OUO
U DU DU
DU DU U
OOO s−1
OOO Dn sDU
DsU
OOO
DU
Dn
' DDU UOO
DU
DU
OOO Dσ3
OOO
Ds
σ5
DDn OOO'
DDU
U DUMM
U DuU
MMM
MM
U DdU MMM&
U DU UMM
s
σ5 U
DDU UNN mU U / DU UJJ σ3
NNN
JJ Dn
JJ
NNN
JJ
N
m
n
N
DDn
J$ N&
/ DU.
DDU
s
DsU
/ DU
mU
=
U DDU
−1
U mU
MMM U DudU
MMM
U DuDU
U DU dU MM& DU UMM
U DU DU
MMM s−1
MMM dU sDU
Dn
MM& DU
dU
DU MM
DU DU
MMM Dσ4
MMM
Ds
σ5
DdU MM&
U DU
U uDU
mU
LLLId
LLL
U DuU
−1
LL%
U DβU
/ U DU
U DU U
#
/ U DU
U Dn
sU
U DU
s
DU
/ DU,
:
vvvv
v
DdU
vv
vv mU
Id
ηD U
=
U DULL
rr
LLLU dDU
LLL
LL%
Id
s
k
s
k
_
_
_
_
U U DULL ηU DU U βD U Ur DDU
LLL
rr
LLL
rrr
r
L
r
nDU
L% yrr U mU
U uDUrrr
rr
ry rr
U DU
dU DU
dDU
/ DU
/ DU
sDU
σ4 DU
/ DU DU
Ds
Dn
s
DDU
/ U DDU
MMM
MMUMmU
U duDU U DuDU MMM
M&
/ U DU DU
U DU
σ3
s
U dDU
ds
DU U
U U DU U dU DU
nDU
(30)
U DULL
sU
DDU
mU
Id
U ηD U
(29)
σ5
s
/ DDU
MMM
MMmU
MMM
MM& βD U
0 DU
Id
s
DU,
Theory and Applications of Categories, Vol. 5, No. 5
134
U mU /
U6 DDU
U DU
n
nn
n
U DmU
n
n
U DuDU
nnn
nnn
_
_
U DDDU Uks Du−1 Un6 DU DU
mU
nn
nnn
n
U DuDDU
sDU
n
n
nnn U DU mU
U DU DDU ks s__−1 nDU
DU
6
mU nn
n
n
n
s
sDDU
σ5
nnn
nnn DU mU
DU DDU
DU DuDU
Ds
DU DU DU
DsDU
Dσ5
/ DU
DDU
8
6
mU qq
nnn q
nnn
n
qqq
DDs
µ
U
n
D
q
n
q
n
nn DmU
qq mU
/ DDU
DDDU
DDU DU
mDU
=
U mU /
U6 DDU
U DU
m
m qq8
q
U DmUmmmm
q
q
m
U µD U qqqqU mU
mmm
q
mmm
Ul DDDU U mDU / U DDU
l
l
U DuDDU
U DDuDU
lll
U DuDU
lll
l
l
U muDU vll
__
_
ks _
U DU DDU
U DDU DUU mU DU / U DU DU
RRR U Du−1
RRR DuDU
RRR
s
sDDU
U DuDU DU
RR
U DU DuDU RR(
__
DU DDURR sks−1
U DU DU DU
RRRDuDU
RRR
R
sDU DU σ5 DU
sDU
DU DuDU RRRR(
(31)
DU DU DU
DsDU
DDU DU
DDs
DDDU
DU.
8
q
qqq
Ds qqq
qqq mU
mU DU
/ DU DU
mDU
/ DDU
ms
σ5
Observation: The equations that appear in the proposition can be written as condi-
Theory and Applications of Categories, Vol. 5, No. 5
135
tions in U-AlgK . For instance, it can be shown that the pair of pastings:
ϕ
=
U DDU
B
NN
NNNU DuDU
NNN
NNN
'
uDDU uDuDU U
DU
DU
7
NNN
pp
NNsDU
uDU DUppp
NNN
p
p
p
p
NNN'
pp
DuDU
σ1 DU
/ DU DU
/ DU DU
DDU
LLL
Id
LLLDs
LLL
L%
Dσ1 Id
. DDU,
and
ϑ
U U DuDU
=
nDDU
/
U sDU
nDU DU
/
U Ds
QQQ
QQUQDuU DU
QQQ
} U Du−1
s
QQQ
Q(
ks ____
Id
U DβU DU mmmm
U DU s
m
mmm
m
m
U
DnDU
vmmm
sU DU
} s−1
/
/ U DuDU
sDU
s
sDU
| n−1
DuDU
U DuDU
| σ2 DU
/ sDU
ks __
n
n
σ3 DU −1
nnn
nn
nnnDnDU
n
n
/ vn
/ DU s
| Dσ2
Ds
/ ,
Ds
define an object in U-AlgK . Thus, the first two equations say that (mU, σ5 ) : (ϕ, ϑ) →
(σ1 , σ2 ) is a U-algebra morphism. Similarly the rest of the equations. We leave the details
for the interested reader.
Proof. The fact that DηU and DµU are 2-cells in U-AlgK follow from applying DK to ηU
and µU , respectively. The statement about dn follows from pseudonaturality of dK . As
for the equations, the proof of the last equation is fairly typical, and it is the only one we
do. We begin by introducing some notation and deducing some equations we will need.
The notation is
DK ( (βU DDU, µU DDU )
DK ((σ1 DU, σ2 DU )
DK ((σ1 , σ2 )
(s ,σ2 )
/ (σ , σ ))
1
2
(Ds,τ1 )
/ (σ , σ ))
1
(mU,γ1 )
2
/ (σ1 , σ2 ))
=
(Ds ,τ2 )
(σ1 DDU, σ2 DDU )
1
(DDs,τ3 )
/ (σ , σ ),
=
(σ1 DU, σ2 DU )
=
(DmU,γ2 ) / (σ , σ ),
(σ1 , σ2 )
1
2
and (mK )(σ1 ,σ2 ) = (mDU, γ3 ) : (σ1 , σ2 ) → (σ1 , σ2 ).
/ (σ , σ ),
1
2
2
Theory and Applications of Categories, Vol. 5, No. 5
136
Now the equations. Since
U-AlgK
DK DK
,
mK 2 U-AlgK = U-AlgK
DK
ΦK
A(K, K)
ΦK
DD( )
,
m( ) 2 A(K, K),
A(K, K)
D( )
we have that ms = (mK )(s,σ2 ) is a 2-cell
(σ1 DU, σ2 DU )
(DDs,τ3 )
(mU DU,γ1 DU )
/ (σ1 DU, σ2 DU )
ms
(σ1 , σ2 )
(Ds,τ1 )
/ (σ , σ )
1
(mDU,γ3 )
2
in U-AlgK . From this we obtain the equation
/ PP
PPP
PPP
/
PPP
PPP τ1 (
PPP m PPP
PP( s
PP( /
/ QQ
PPP
QQ
PPP
PPU( ms QQQQ(
/
PPP τ3
PPP
γ3
PP' / .
=
γ1 DU
(32)
Similarly, (µK )(βU ,µU ) is a 2-cell
(σ1 , σ2 )
(mDU,γ3 )
(σ1 , σ2 )
(DmU,γ2 )
/ (σ , σ )
1
µD U
2
(mU,γ1 )
/ (σ1 , σ2 ).
(mU,γ1 )
We thus obtain the equation
/ PP
PPP
PP
PPP
PUP( µD U PPPP(
/
PPP γ3
PPP
γ1
PP( / / QQ
QQQ
QQQ
/
OOO γ1 (
OOO
O
OOO
OOµ' D U OOOO' / ,
=
γ2
Observe that
(βU DU DU, µU DU DU )
(sDU,σ2 DU )
(σ1 DU, σ2 DU )
(U Ds,n−1
Ds )
τ1
(Ds,τ1 )
/ (βU DDU, µU DDU )
(s ,σ2 )
/ (σ , σ )
1
2
(33)
Theory and Applications of Categories, Vol. 5, No. 5
137
is a 2-cell in U-AlgK . Since DK behaves well with change of base, then DK (U Ds, n−1
Ds ) =
).
Thus,
applying
D
,
we
have
that
(DU Ds, s−1
K
Ds
(σ1 DU DU, σ2 DU DU )
(DsDU,τ1 )
(DU Ds,s−1
Ds )
/ (σ1 DDU, σ2 DDU )
Dτ1
(σ1 DU, σ2 DU )
(Ds ,τ2 )
/ (σ , σ )
1
(DDs,τ3 )
2
is a 2-cell in U-AlgK . From this we obtain the equation
/
/
τ1 DU
τ3
/ nn6 QQQ
nnnU Dτ −1 QQQQQ
n
n
n
(/
/ 1
=
/ −1
(34)
τ
2
sDs
/ /6 .
PPP
PPP Dτ1 nnnnn
PP' nnn
We also have that
(U mU,n−1
mU )
(βU DDU, µU DDU )
(s ,σ2 )
γ1
(σ1 , σ2 )
/ (βU DU, µU DU )
(s,σ2 )
/ (σ1 , σ2 )
(mU,γ1 )
is a 2-cell in U-AlgK . Applying DK to it we obtain that the equation
/ PP
PPP
PPP
PPP
P
U
P( Dγ1 PPP(/
PPP τ2
PPP
γ2
PP( / =
/ QQ
QQQ
QQQ
/
OOO τ1 (
OOO
OOO
OOO
OO' OODγ
' 1
/ ,
s−1
mU
(35)
is satisfied.
We are ready to show (31). Start with the second member of the equation, and use
(32). Next make the substitution
/
=
U muDU
−1
−1
/ JU Dσ1
JJ
J
JJ
J
JJ U ms JJJ
JJ
JJ
J$
J$ /
U DDσ1
II
II
II
II
I$
U mDU
/ .
Use now (33) and (34). Make the substitutions
6
mmm O
mmm
m
m
m
−1
mmm
mmm U Dσ1 DU /
/ O
U Dτ1−1
/ ,
=
U Du−1
Ds
/ M
MMM
MMM
MM
U Dσ1 −1 MM&
/ / ,
Theory and Applications of Categories, Vol. 5, No. 5
138
and
/
=
U Dσ1 −1
JJ
JJ
JJ
JJ
J$ U Du−1
mU
U DmU
/
/ IU Dσ1−1
HH
II
HH
HH U Dγ1 IIII
HH
II
H$
/$ ,
using that (Ds, τ1 ) : (σ1 DU, σ2 DU ) → (σ1 , σ2 ) and (mU, γ1 ) : (σ1 , σ2 ) → (σ1 , σ2 ) are
1-cells in U-AlgK . Then the substitution
NNN
NNN U DDσ−1
NNN
1
NN&
−1
NUNDu
NNNDuDU
NNN
U Du−1
Ds
NNN
& NNNs−1
NDuDU
NNN
s−1
NNN
Ds
N& =
/ U DuDDU
/#
/ sDDU
MMM
MMM DU Dσ−1
MMM
1
MM&
/$ .
Finally, use (35).
We show now that we do have a distributive law.
10.2. Theorem. With the above definitions, r : U D → DU and the invertible 2-cells
ω1 , ω2 , ω3 , and ω4 , define a distributive law of U over D.
Proof. We show (coh 3, 6 and 8), leaving the rest to the reader. Start on the left hand
side of (coh 3) and make the following substitutions:
/ J
JJ
JJ
JJ
−1
ηU U
JJ
1$
/ NN
NNN
NN
NNN
NNN σ4 U NNNNN
NN&
NN
/
µU &
NNN
NNN
NNN
dn
NN& / =
/
n
/ NN
NNN
NN
NNN
NNN U duU NNNNN
NN
NN&
U βU &
/
U dn
/
U σ4
σ4
/ =
U dU
µU
/ ,
using (27),
/ NN
NNN
NNN
NNN
'
U Dβ
U
/ =
/ /
/ NN
NNN
NNN
U dn
NNN
/
'
NNN
NNN σ3−1
NNN
NNN
σ4
NNN
NNN
NN'
NN' /
=
$ /
MMM
MMM
MMUMβU
MM& & ,
ww
ww
w
w
{ww
FF µ−1
FFU
FF
FF
F" n−1
dU
σ4
/ H
HH
HH
HH
HH
#
σ2
/ w
w
ww
ww
w
/ , {w
Theory and Applications of Categories, Vol. 5, No. 5
139
using that (dU, σ4 ) : (βU , µU ) → (σ1 , σ2 ) is a 1-cell in U-AlgK ;
/
NNN
NNN −1
NUNηNU
NN& and
/
µ−1
U
=
/ O
O
n−1
u
/ / NN
NNN
NN
NNN
NNN U U du NNNNN
N
NN&
N&
/
NNN n−1
u
NNN
NNN
n−1
dU
NN& / =
/
/ ,
qq8
q
q
−1 qq
ηU
qq
qqq
/ NN
NNN
NNN
NNN
/
'
NNN n−1
MMM
NNN Du
MMM
MMM U du NNN
NN& MM&
/ .
n−1
d
To prove (coh 6), start on the right hand side, and make the following substitutions:
/
pp
p
p
p
U U Du−1
Du
ppp
xppp n−1
/ DuD
p
p
p
pp
U s−1
Du
ppp
p
p
p
x
/ NN
NNN
33
NN
NNN
33
NNNU Du−1 NNNNN
33
NN& U Du
NN&
33
/
U DβU D
p
33σ2 D
pp
p
−1
p
33
sU Du
pp
33
ppp σ3 D−1 p
x
/ 33
pp Dn−1 pppp
p
33
p
Du pp
33 ppppp
ppp
p
xpp
p
x
/
/
=
t
t
t
tt
tt
t
zt
JJ Dσ3−1
JJ
JJ
JJ
J$ σ5 U
/ mn
=
t
tt
tt
t
t
tt
ytt
JJ U Du−1
Dn
JJ
JJ
JJ
JJ
J
−1 % NNN sDn
NNN
NNN
NN& / / NN
NNN
NNN
U Du−1
DuU
NNN
/
&
du−1
NNU
NNN Dn
NNN
s−1
DuU
NNN
/
& NNN s−1
Dn
NNN
DU DβU NN
NNN
&- /
pp
pp n−1
p
p
p
p
p
DDu pp
ppp
ppp
xppp
/ wpp−1
nDuDU ppp
pp
U Du−1
Du
ppp
p
p
/ 3wp
NNN
22
33
NNN
22
3
NNN
33
22
NN'
3
22
DU βU DU p
3
σ
DU
2
3
22
pp
33
22
ppp
p
33
p
22
wppp
33
22 s−1
Du
33
σ3 DU −1qqqq
22
33
qq
22
3 qqqq
q
x
/ ,
U mn
/ K
KKK
KKK
KKK
KK
/
%
using (29)
σ3−1
σ5
/ ,
=
/ NN
NN
U DDβU NNNN
NNN
-'
U DuDU
sDU
,
Theory and Applications of Categories, Vol. 5, No. 5
140
and
/ NN
NNN
NN
NNN
NNN U muU NNNNN
NN&
NN&
/
U mU
=
U DDβU
) U mn
/ NN
NNN
NNN
NNN
'
U DβU
/ ) .
Now use (28). Conclude with the substitution
/ NN
NNN
NN
NNN
NNNU U mu NNNNN
NN&
NN&
/
NNN n−1
DDu
NNN
NNN
n−1
mU
NN& / / NN
NNN
NNN
NNN
/
'
NNN n−1
MMM
Du
NNN
MMM
MMM U mu NNN
NN& MM&
/ .
=
n−1
m
To prove (coh 8), start on the left hand side, and make the substitutions:
/ NN
NNN
NNN
U Du−1
d
NNN
/
&
NNUNDu−1
Du
N
NNN
s−1
NNN
d
N& /
−1
NNN
NNN sDu
NNN
N
NNN DU du NNNNN
NN&
NN& /
=
NNN
NNN
NNN U ηD
NN&
NNN U Ddu
NNN
NNN
NN& =
/ NN
NNN
NN
NNN
NNN U Ddu NNNNN
NN&
NN'
/
NNNU Du−1
u
NNN
−1
NNN
U DudU
NN
−1 & / MMM su
MMM
MMM
s−1
dU
MM& / ,
and
/(
U mu
/ U Du
MMM
MMM
MMMU ηD U
MM&
(/ .
Now, use (30), then the equation obtained from (26). It is easily seen that what remains
equals the identity.
11. (Co-)KZ-doctrines over KZ-doctrines
Distributive laws for KZ-doctrines over KZ-doctrines are formally very similar to distributive laws of co-KZ-doctrines over KZ-doctrines. We start this section by recalling
the definition and some aspects of KZ-doctrines [9] (see [8] as well). Then we define distributive laws of co-KZ-doctrines over KZ-doctrines, and we show that such a distributive
law induces a distributive law between the induced pseudomonads, as defined in section
4. Finally, we show that the composite pseudomonad given by a distributive law of a
KZ-doctrine over a KZ-doctrine is again a KZ-doctrine.
Theory and Applications of Categories, Vol. 5, No. 5
141
11.1. We are still working in a Gray-category A, and K is an object of A.
11.2. Definition. A KZ-doctrine D on K consists of an object D, 1-cells d : 1K → D,
and m : DD → D in A(K, K) and a fully-faithful adjoint string η, * : Dd m; and
α, β : m dD : D → DD such that
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
/
1KDD (dd )−1 ⇓ DD m D.
<
DD
z
D
zz
d D"
zz Dd
d zzz
dDttt
1K
=
DD
(36)
D
The 2-cell δ : Dd → dD is defined as the pasting
IdD
/
t: D JJJ Dd
JJ β −1 ⇓ttt
JJ
J
J
t
$ ⇓ J$
tt m
dD J$
/
DD,
DD
D JJJ
IdDD
and it satisfies the equations δ ◦ d = dd and m ◦ δ = β −1 · η −1 .
In a co-KZ-doctrine U, with u : 1 → U and n : U U → U , the fully-faithful adjoint
string is of the form uU n U u.
Proposition 10.1 of [9] says that D induces a pseudomonad D = (D, d, m, β, η, µ) if µ
is defined as the pasting
IdDDD
/ DDD
@
==
== αD ⇓ =
mD ==
= dDD
DDD
==
DD
m
/ DD
@ ===
== m
==
dm ⇓ dD
=
β ⇓ =
/D
/D
Dm
IdD
11.3. Assume we have a KZ-doctrine
D = (D, d, m, αD , βD , ηD , *D ),
and a co-KZ-doctrine U = (U, u, n, αU , βU , ηU , *U ), with
Id
/ DD,
III αDuuu:
m $
uu dD
DDII
D
DD JJ m
dD uu:
D
uuu
JJ
βD J$
/ D,
D II
Id
III
Dd $
/ D,
:
ηD uuu
uu m
DD
Id
: D JJJDd
JJ
$D $
/ DD,
muuu
DD
uu
Id
and
/ U,
U HH
HHH αU vvv:
vv n
uU #
UU
Id
; U HHHuU
HH
v
v
βU $
/ U U,
UU
Id
n vvv
/ U U,
HHH ηUvvv:
n #
vv U u
U U HH
Id
U
U uuu:
U
uuu
U U
II n
$U III
$
/ U.
Id
Since we are assuming D to be a KZ-doctrine, and U to be a co-KZ-doctrine, then βD
and ηD , and αU , and *U are invertible. Let δ : Dd → dD and υ : uU → U u be the
corresponding induced 2-cells.
Theory and Applications of Categories, Vol. 5, No. 5
142
11.4. Definition. A distributive law of U over D consists of a 1-cell r : U D → DU in
A(K, K), together with an invertible 2-cell
U DFF
{=
FF r
FF
{
F
{
ω
1  F"
{{
/ DU,
D
Du
uD {{{
such that the pasting
DU PP
pp8
DuU
−1PPPPuDU
PPP
−1
PPP
udU
DαU
ω
U
1
'
'
*
pp
/
/
U NNN υ 4 U U
U DU
DU U
rU
U dU nn7
NNN
mm6 m
n
m
n
UNu
−1
−1 nn
mm
NN
D$U
ru mmmmmDU u
U dunnnnU Du
U d NNN&
nn
mm
/ DU
UD
dU ppp
pp
p
pp uU
/* DU
7
Dn
r
is invertible. We denote the inverse by ω2 . We further require the pasting
Id
/ U U D U r / U DU rU / DU U
; HH
9
v: HH
HH Dn
tt
−1
v
ηU D vvv
U ω1
HH
HH
tt ru
t
H
HH
t
v
v
nD HH$
H$
tt DU u D$U
t
vv U uD
U Du
/
/ DU
UD
DU
ω3 = U U DH
H
r
Id
to be invertible, and the pasting
ω4 =
r
Id
/ DU
JJ
: U D II
DdU
v
II
JJDU d
U m vvv JJ
U $ IUIIDd
JJ
vv
r
I
d
Dω2−1
v
D
I
J
v
$
%
v
/ U DD
/ DU D
U DD
rD
Id
Dr
/ DU
v:
v
ηD U vvvv
v
$
vv mU
/ DDU
to be invertible. We also require the following two coherence conditions:
UD
r
/ DU
JJ
JJDuU
J
U uD ks __ uU D uDU ω U −1 JJJ
1
υD
J%
−1
ur
Dn /
U U D U r / U DU rU / DU U
DU
=
/ DU
DuU
IIUIDu
U uD U ω −1 III ru Dυ
II
1
$
DuU
)/ / U DU
UUD
DU U
Dn
/ DU
DdU
D
JdU
JJ
U dD ω D−1 JJdJ−1
r
2
J% dDU δU
*/ /
U DD
DU D
DDU
mU
U D II
r
Ur
rU
/ DU,
and
UD
r
/ DU
JJ
JJDdU
J
__
_ U Dd DU d
U dD ks _
−1 JJ
Dω
JJ
U
δ
2
%
rd
mU /
/
U DD rD DU D Dr / DDU
DU
=
U D JJ
r
rD
Dr
/ DU.
Theory and Applications of Categories, Vol. 5, No. 5
143
Using the equations υ ◦ n = ηU · βU and δ ◦ m = αD · *D , and the coherence conditions,
it is not hard to see that the inverse of ω3 is
r
Id
/ DU
/ DU,
;U D HH
III
:
DuU
v
v
Dα
HH uU D
v
nD vvv HH
U
vv
−1 IIuDU
III
v
v
H
v
HH ur
v
vv
II ω1 U −1
βU D
vv
#
$
#
vv Dn
/
/
/
U U D U r U DU rU DU U
UUD
Id
and the inverse of ω4 is
/ U DD rD / DU D Dr / DDU
II
; 9
v: II
II mU
tt
Uα
d
vv ω2 D
II
II
tt
r
D vv
t
I
II
t
v
β
U
I
t
v
D
Um
U
dD
dDU
I$
I$
t
v
t
v
dU D
/
/ DU.
UD
DU
Id
U DDII
r
Id
Looking at the definition it seems possible to define a distributive law in terms of ω2 ,
defining ω1 with a similar pasting as ω2 was defined above. It can be shown that if we
start with ω1 , define ω2 as above, and then use a similar pasting to define a new ω1 we
have that ω1 = ω1 . And similarly if we start with ω2 .
11.5. Theorem. If D and U are the pseudomonads determined by the KZ-doctrine D
and the co-KZ-doctrine U respectively, then r together with ω1 , ω2 , ω3 and ω4 define a
distributive law of U over D.
Proof. Since U is a co-KZ-doctrine, we have that βU = αU−1 and ηU = *−1
U . With this
in mind, we must show that the coherence conditions (coh 1) to (coh 9) of section 4 are
satisfied. We show that (coh 1), (coh 3) and (coh 4) are satisfied and leave the others to
the reader.
Consider the inverse of the left hand side of (coh 1). To get the inverse of the right
hand side make the following four substitutions
υ ◦ u = u−1
u ,
9 K
ss KKKK
s
s
K%
ss
/ du
/
/
u−1
d
−1
u−1
uu / dU / 9
KKK
−1
KKKU du ssss
s
% s
O
/% O
ω1/ U −1
O
u−1
Du
/
−1
ru
/
=
/ O
=
u−1
Du
* O
Du−1
u
ω1−1
/
/ ,
/
/!,
and
9 KKK
KKDα
sss
s
K% U
s
s
KKK Du−1
9
u
s
s
KKK
s D$U
s
s
% s
=
/ $:
idDu
Theory and Applications of Categories, Vol. 5, No. 5
144
To prove (coh 3) start with the inverse of the right hand side and make the following
substitutions:
υ ◦ n = ηU · βU ,
βU
7 P
ooo PPPPP
o
o
PP(
u−1
ooo
/: ,
6( n
υU
−1
=
U αU
/
/ /
U dn
/
/ rn
9 MM
sss −1 MMMM
s
s
M&
s
/ dn
/
=
/ / J
JJ
JJ
JJ
u−1
Dn
J
/ ω1 U −1 J/ $
/
JJ
JJU α−1
JJ U
JJ
J$ u−1
n
/
u−1
dU
u−1
dU U
A OOO
OO
Du−1 OOOO
OO'
n
8 O
/ O
r
rrrr
rn
rrr −1
rrr ω1 U U /
/ ,
=
/ O
O
=
U d−1
uU
/ / J
JJ
JJ
JJ
Du−1
n
JJ
/ DαU
/$
u−1
Dn
/ ,
JJ
JJ U dn rrr9
JJ rr
$ r
/ −1
ruU
/
r8 O
rrr
r
r
rr
rrr DαU U/
=
/ O
/ ,
nn6
n
n
nnn
nnn
n
n
/ n
DU α−1
U
/ O
Dµ−1
U
/ ,
and
/
NNN
−1
NNDU
NNNαU
NNN
& /
=
Dµ−1
U
/ : I
uu II
uuD$ U III
u
u
U
/$
Dn
/ .
Comparing the pasting obtained with the left hand side of (coh 3), we see that we must
show that
/:
/9 K
/9
JJ
= Dn ◦ rU ◦ U ω2−1 .
KK
η ttU d−1 ss r −1 ss
J
u s
u s
KKK
JJ Utt
J$ tt / sss / sss D$U
/ 9%
4
−1
s
nd jjjjj
ω
s
3
jj
sss
jjjjj
/ s
/
Now, pass every 2-cell of the left hand side to the right hand side, except for ηU . It is not
hard to see, with the help of (coh 1), that the resulting equation is satisfied.
Start with the inverse of the left hand side of (coh 4), and make the following sequence
of substitutions:
/ O
/=
/
/
HH
O
=
β D {
H
{{
{{
{
{
/ {
µU D−1
U
HβHU U D
HH
HH
H$ u−1
n
/ ,
Theory and Applications of Categories, Vol. 5, No. 5
/ O
/:
U βU D ttt
t
u−1
tt
n
tt
/ t
O
=
/
{{
{
{
u−1
r
{{
}{{
/ CCuuU D
CC
−1
CC
CC U ur
!
/ =
/
/
JJ
JJβU U D
−1
JJ
uU r
JJ
J$ / / J
JJ
JJ
uuDU
JJ
−1JJ
/ U ω1 U
/$
=
=
/
/
AA
AAβU D
AA uuU D
AA
/ ,
/ E
EE
EE
EE
u−1
r
E"
/ u
uDU z
z
zz
u−1
Ur
zz
z
/ , }z
/ O
/ ,
O
βU DU tt9
t
t
nr
tt
tt
t
/
A OOO
OO
uDuU OOOO
OO'
8 O
/ O
r
rrrr
u−1
rU
rrr −1
rrr ω1 U /
/ ,
/ O
rr8 O
r
r
r
u−1
Dn
rrr DαU
rrr
/ ,
/
/ J
JJ
JJ
uDuU
JJ
J
/ U DαU J/ $
/ J
JJ
JJ
−1
JJ
uDn
J
/ ω1 U −1 J/ $
=
=
A OOO
OO
Du−1 OOOO
OO'
n
/ O
rr8 O
rrr
rn
rrr −1
rrr ω1 U U /
/ ,
/ J
JJ
JJ
JJ
JJ
/ DαU
/$
=
8 O
rrr
r
r
rrr
rrr DαU U/
and
145
Du−1
n
/ O
Dµ−1
U
/ ,
11.6. Assume now that D is a KZ-doctrine as before, but that U is also a KZ-doctrine.
The data for D is unchanged, and the data for U is now:
U U HH
n
Id
HHH #
U
/ U U,
αU vv:
vvvuU
U U HH n
uU vv;
U
vvv
Id
βU
HHH
$
/ U,
/ U,
U HH
HHH ηU vvv:
vv n
Uu #
UU
Id
U Uu
n uu: IIII
uuu $U I$
/ U U.
UU
Id
We have υ : U u → uU .
Copying almost exactly the definition of a distributive law of a co-KZ-doctrine over a
KZ-doctrine, we obtain the concept of a distributive law of the KZ-doctrine U over the
KZ-doctrine D. If U and D are the pseudomonads induced by U and D respectively, we
obtain, in a very similar way, a distributive law between the pseudomonads U and D. Let
V be the composite pseudomonad obtained from this last distributive law, as in section
5.
Theory and Applications of Categories, Vol. 5, No. 5
146
11.7. Theorem. The composite pseudomonad obtained from a distributive law between
KZ-doctrines is again a KZ-doctrine.
Proof. We follow the notation introduced before the statement of the theorem, and the
notation for V is the same as in section 5. According to theorem 11.1 of [9], it suffices to
show that p vV with counit βV . Define ϑ : V v → vV as the pasting
2 DU
= UFF
FF
FF
FF
FFDU dU
DuU
FF
DU
FF
DdU
FF
DudU FF
δU
FF
!
"
dDU
,
/ DU DU.
DDU
5
DuDU lll
ll
duDU lll
ldU
l
uDU
l
DU
lll
,
DU u
----Dυ
U DU
It is not hard to see that ϑ ◦ v = pp and that p ◦ ϑ = βV−1 · ηV−1 . Using these two equations,
an easy calculation shows that p vV with unit
Id
V vV
,
V βV−1 ⇓
&
/VV
V V OO ϑV ⇓ 2 V V V
V p oo7
OOvV
o
OO V
vp ⇓ oooo
p OOOO
o vV
o
OOO
o
ooo
'
V
and counit βV .
References
[1] M. Barr and C. Wells. Toposes, triples and theories. Springer-Verlag, 1985.
[2] J. Beck. Distributive laws. Lecture Notes in Mathematics 80, Springer-Verlag, 1969. pp. 119-140.
[3] B. Day and Ross Street. Monoidal bicategories and Hopf algebroids. Advances in Math. Vol. 129,
Number 1, 1997, pp. 99-157.
[4] S. Eilenberg and G. M. Kelly. Closed categories. In: Proceedings of the Conference on Categorical
Algebra, La Jolla 1965, ed. S. Eilenberg et. al., Springer Verlag 1966; pp. 421-562.
[5] J. W. Gray, Formal category theory: Adjointness for 2-categories. Lecture Notes in Mathematics
391, Springer-Verlag, 1974.
[6] R. Gordon, A. J. Power and Ross Street. Coherence for tricategories. Memoirs of the American
Mathematical Society 558, 1995.
[7] G. M. Kelly. Coherence theorems for lax algebras and for distributive laws. Lecture Notes in
Mathematics 420, Springer-Verlag, 1974.
Theory and Applications of Categories, Vol. 5, No. 5
147
[8] A. Kock. Monads for which structures are adjoint to units. Journal of Pure and Applied Algebra
104(1), 1995, pp. 41-59.
[9] F. Marmolejo. Doctrines whose structure forms a fully faithful adjoint string. Theory and applications of categories, Vol. 3, No. 2, pp. 24-44. 1997.
http://tac.mta.ca/tac/volumes/1997/n3/n3.dvi
[10] R. Rosebrugh and R. J. Wood. Distributive adjoint strings. Theory and applications of categories,
Vol. 1, No. 6, 1995, pp. 119-145. http://tac.mta.ca/tac/volumes/1995/n6/v1n6.dvi
[11] Ross Street. The formal theory of monads. Journal of Pure and Applied Algebra 2, 1972, pp. 149168.
Instituto de Matemáticas
Universidad Nacional Autónoma de México
Area de la Investigación Cientı́fica
México D. F. 04510
México
e-mail: quico@matem.unam.mx
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