DISTRIBUTIVE LAWS FOR PSEUDOMONADS
F. MARMOLEJO Transmitted by Ross Street
ABSTRACT. We define distributive laws between pseudomonads in aGray-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 pseu-domonads. We will see that the structure obtained from a distributive law between pseudomonads is a pseudomonad. We define a distributive law between pseudomonads as wesaid above, that is to say, asking for commutativity up to isomorphism of thetwo triangles and thetwo pentagons. Weproposeninecoherenceconditions for theseisomor-phisms. See section 4 below. We observe that the coherence conditions of [7] and the ones proposed in this paper coincideif in our setting weask for commutativity on thenoseof 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
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 distributivelaw is thesamething as a monad in the2-category MND(C), whoseobjects aremonads inC. 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 provethat, for monads,
a distributivelaw, a lifting of onemonad structureto thealgebras of theother, 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 whosestructureis compatiblewith two given pseudomonads is.
Weshow how to obtain a compositepseudomonad with compatiblestructurefrom 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.
Weseethen, that theninecoherenceconditions can beshown to hold if wedefinea 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 KZ-doctrineover 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, thedistributivelaw 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 thesamepaper. Wethus work in theframework ofGray-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 changeof base2-functors, changeof basestrong transformations and thechangeof base modifications that we will need in later sections. Change of base turns out to be a Gray-natural transformation.
In section 4 we define distributive laws for pseudomonads by replacing commutativity on thenoseby commutativity up to isomorphism. Wegiveheretheninecoherence conditions that these isomorphisms should satisfy.
from a distributivelaw. This is what wedo 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.
WeintroducetheGray-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 pseu-domonads with compatiblestructure. In thefollowing section, wego from a pseudomonad in PSM(A) to a distributivelaw.
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. Comparewith [10], whereit is shown that oneof thetriangles 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] wewill work with a Gray-category A, whe re Gray is thesymmetric monoidal closed category whose underlying category is 2-Catwith tensor product as in [6]. A Gray-category is a Gray-category enriched in the Gray-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 e ve ry pair of obje cts 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). Wewill denoteM by juxtaposition, M(G, F) = GF for F ∈
A(A,B) and G∈ A(B,C). Given f :F → F′ inA(A,B) and g :G→ G′ in A(B,C) we
will denotetheinvertible2-cellMg,f by
GF gF //
Gf
gf
G′F
G′f
GF′
gF′ //G
′F′.
What the definition of being cubical means for M is thefollowing: 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), wehavethat ( )F : A(B,C) → A(A,C) and G( ) : A(A,B) → A(A,C) are2-functors, ( )f : ( )F → ( )F′ and g( ) : G( ) → G′( ) arestrong transformations,
equations are satisfied
GF gF
**
g′F 44
γF
Gf
Gf′
_ _ _ _
ks
Gϕ
g′ f
G′F
G′f
GF′
g′F′ //G
′F′
= GF gF //
Gf′
gf′
G′F
G′f
G′f′
_ _ _ _
ks
G′ϕ
GF′
gF′
++
g′F′ 33
γF′G′F′,
GF gF //
Gf
gf
G′F
G′f
GF′ gF′ //
Gf′′
gf′′
G′F′
G′f′′
=
GF′′ gF′′//G
′F′′
GF gF //
G(f′′◦f)
gf′′ ◦f
G′F
G′(f′′◦f)
GF′′
gF′′//G
′F′′,
and
GF gF //
Gf
gf
G′F g
′′F
//
G′f
g′′ f
G′′F
G′′f
=
GF′
gF′ //G′F′ g′′F′//G′′F′
GF (g
′′◦g )F
//
Gf
(g ′′◦g
)f
G′′F
G′′f
GF′ (g′′◦g )F′//G
′′F′,
and if eitherf org is an identity, thengf is an identity 2-cell. Now, for every objectAof
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 objectD inA, andκ:k →k′ :K →K′ inA(C,D) thefollowing 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
3.1. Definition. A pseudomonad D on an object K of a Gray-category A is a pseu-domonoid in the Gray monoidA(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
D dD //
IdD
!!
D D D D D D D D
~ β DD
m
D Dd
oo
IdD
}}
zzzz zzzz :: ::
Ya
η
D
DDD Dm //
mD
µ DD
m
DD m //D,
such that the following two equations are satisfied:
DDDD DDD
DDD DD
DDD
DD D
Dµ
⇐=
µD
⇐= µ
⇐=
DDm
//
DmD
""
E E E E E E E
mDD
Dm
""
E E E E E E E
Dm //
mD
m
mDEEE""
E E E
m //
=
DDDD DDD
DD
DD DDD
DD D,
m−m1
⇐= µ
⇐=
µ
⇐=
DDm
//
mDD
Dm
$$
J J J J J J J J
mD
m
$$
J J J J J J J
J m
Dm //
mDLLLL%%
L L L L L
m //
(1)
DD DDD
DD
D
DD µ ⇓ DdD
//
Dm v::
v v v
mDHH$$
H H
m
##
F F F F F
m
;;
x x x x x
= DD
DDD
DDD
DD D.
Dβ⇓
ηD⇓ DdDtt::
t t t
DdDJJJ$$
J
J //
Dm
$$
J J J J J
mD
::
t t t t t
m
// (2)
It is shown in [9] that the following three equations hold for any pseudomonad D:
1K D D
DD
DD β ⇓
η⇓ d //
dDttt:: t t t
DdJJJ$$
J J J
IdD
//
m
$$
J J J J J J
m
::
t t t t t t
= 1K
D
D
DD D,
d−1
d ⇓
d zz<<
z z z z
dDDD""
D D D
dD
""
D D D D D
Dd
<<
z z z z z
m //
(3)
DD DDD DD
DD D
⇐
βD
⇐=µ
dDD
//
IdDD
""
E E E E E E E E E E
Dm
//
mD
m // m
=
DD DDD
D DD
D,
⇐=
dm
⇐β
dDD
//
m
Dm
dD //
IdD
""
E E E E E E E E E E E E
m
DD
DDD DD
DD D
⇐
Dη
⇐=µ
DDd
IdDD
""
E E E E E E E E E E E
Dm //
mD
m // m
=
D.
DD D
DDD DD
⇐η ⇐m=d
IdD
""
E E E E E E E E E E E
m // m
//
DDd
mD // Dd
(5)
We recall as well the 2-categories of algebras for a pseudomonadD. Le t X beanother
object in A. An object in the 2-category D-AlgX consists of an object X in A(X,K),
together with a 1-cell x:DX →X and invertible 2-cells
X dX //
IdX DDD""
D D D D
D
~ ψ DX
x
X
DDX Dx //
mX
χ DX
x
DX x //X,
such that the following two equations are satisfied
DDDX DDX
DDX DX
DDX
DX X
Dχ
⇐=
µX
⇐= χ
⇐=
DDx
//
DmX
""
E E E E E E E
mDX
Dx
""
E E E E E E E
Dx //
mX
x
mXEEE""
E E E
x //
=
DDDX DDX
DX
DX DDX
DX X,
m−x1
⇐= χ
⇐=
χ
⇐=
DDx
//
mDX
Dx
$$
J J J J J J J J
mX
x
$$
J J J J J J J
J x
Dx //
mXLLLLL%% L L L L
x //
(6)
DX DDX
DX
X
DX χ ⇓ DdX
//
Dx v::
v v v
mXHH$$
H H
x
##
F F F F F
x
;;
x x x x x
= DX
DDX
DDX
DX X.
Dψ⇓
ηX⇓ DdXtt::
t t t
DdXJJJ$$
J
J //
Dx
$$
J J J J J
mX
::
t t t t t
x
// (7)
It is shown in [9] that for every object (ψ, χ) in D-AlgX, thefollowing equality holds:
DX dDX//
Id
$$
I I I I I I I I I
βX
DDX Dx //
mX
χ DX
x
DX x //X
= DX dDX//
x
dx
DDX
Dx
X dX //
Id
$$
I I I I I I I I I I
ψ DX
x
X.
A 1-ce ll (h, ρ) : (ψ, χ) → (ψ′, χ′) in D-Alg
X consists of a 1-cell h : X → X′ in A(X,K),
together with an invertible 2-cell
DX Dh //
x
ρ DX′
x′
X h //X′,
that satisfies the following two equations:
X dX //
Id
""
D D D D D D D D
~ ψ
DX Dh //
x
ρ DX′
x′
X h //X
= X dX //
h
dh
DX
Dh
X′ dX′//
IdEEEE""
E E E E
~ ψ′
DX′
x′
X′,
(9)
DDX DDX′
DX DX′
DX
X X′
Dρ
⇐=
χ
⇐= ρ
⇐=
DDh
//
Dx
""
E E E E E E E
mX
Dx′
""
E E E E E E E
Dh //
x
x′
xEEEE"" E E
h //
=
DDX DDX′
DX′
DX′ DX
X X′.
m−h1
⇐= χ′
⇐=
ρ
⇐=
DDh
//
mX
Dx′
$$
J J J J J J J J
mX′
x′
$$
J J J J J J J
J x′
Dh //
x
%%
L L L L L L L L L
h //
(10)
A 2-ce llξ : (h, ρ)→(h′, ρ′) : (ψ, χ)→(ψ′, χ′) is a 2-ce llξ:h→h′ such that thefollowing
condition is satisfied:
DX Dh
**
Dh′ 44
Dξ
x
ρ′
DX′
x′
X h′ //X
′
= DX Dh //
x
ρ DX′
x
X h
))
h′ 55
ξ X′.
(11)
Given another object Z of A, and K ∈ A(Z,X), wecan definea changeof base 2-functor K : D-AlgX → D-AlgZ. If ξ : (h, ρ)→ (h′, ρ′) : (ψ, χ) → (ψ′, χ′) is in D-AlgX,
then its image under K is ξK : (hK, ρK) → (h′K, ρ′K) : (ψK, χK) → (ψ′K, χ′K). If
k : K → K′ then we define the strong transformation k : K → K′ such that k
(ψ,χ) =
(Xk, x−k1) and k(h,ρ)=h−k1. If κ:k →k
′ :K →K′ in A(Z,X), thenκ
(ψ,χ) =Xκ defines
a modification κ:k→k′. We have actually defined a Gray-functor D-Alg :Aop→Gray.
For every object Z, wehavean obvious forgetful 2-functorD-AlgZ →A(Z,K). These
4. Distributivelaws
Let D= (D, d, m, βD, ηD, µD) and U = (U, u, n, βU, ηU, µU) bepseudomonads on thesame
object K of the Gray-category A. A distributive law of U over D consists of a 1-cell r:U D →DU inA(K,K), together with invertible 2-cells 444
ω4
DDU mU
OO
subject to the following coherence conditions:
(coh 1)
&&
L
""
(coh 4)
nDU
DnU
DuD
((
DDu
%%
DDu
nDD
U U D
nD
U r
""
D
""
D
nDD
}}
{{{{
{{{{ U rD
""
D
""
D
DnD
}}
{{{{ {{{{
DU r
""
D
rDU
||
DrU
||
zzzz
zzzz ~ω4U
DDU U
DDn
(coh 7)
U D U dD //
Id
&&
dU D
((
Q Q Q Q Q Q Q Q Q Q Q Q Q
r
U β−
1
D
~ ω2D
U DD U m //
rD
U DU
r
~ dr
DU D
Dr
~ ω4
DU dDU //
Id
88
βDU
DDU mU //DU
= idr.
(coh 8)
U D U Dd //
Id
%%
r
U ηD
~ rd−1
U DD U m //
rD
U D
r
DU DU d //
DdU
((
Q Q Q Q Q Q Q Q Q Q Q Q Q
Id
99
~ Dω2
DU D
Dr
} ω4
ηDU−1
DDU mU //DU
= idr.
(coh 9)
U DDD U Dm //
rDD
U mDPPPPP((
P P P P P P
P U DD
| U µD U m
''
N N N N N N N N N N N
DU DD
DrD
| ω4D
U DD U m //
rD
U D
r
DDU D
DDr
mU D PPPP((
P P P P P P P P
DDDU
mDU
((
P P P P P P P P P P P P
| mr
DU D
Dr
yzzzzω4
DDU mU //DU
= U DDD U Dm //
rDD
{ rm−1
U DD
rD
U m
&&
N N N N N N N N N N N
DU DD DU m //
DrD
DU D
Dr
U D
r
DDU D
DDr
{ Dω4 {{
{ {
y ω4
{ { { {
y µDU
DDDU DmU //
mDU
''
P P P P P P P P P P P
P DDU
mU
&&
N N N N N N N N N N N
DDU mU //DU.
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
5. Thecompositepseudomonad given by a distributivelaw
Assumewehavea distributivelaw of U overD as in section 4. The first question is how
to producea compositepseudomonad from U, D, and thedistributivelaw. This is what
wedo 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 thecomposite
DU DU DrU //DDU U DDn //DDU mU //DU;
βV is defined to be the pasting
DU DU DrU
&&
M M M M M M M M M M
U DU
dU DUrrrrr99 r r r r
r
DDU U
DDn
%%
L L L L L L L L L L
DDU DuDU
Dω1U
OO
DDuU
88
q q q q q q q q q q
//
DDβU
_ ___d+3
−1
uDU
DDU mU
$$
I I I I I I I I I
DU uDU
>>
} } } } } } } } } } } } } } } } } }
} dDU
33
h h h h h h h h h h h h h h h h h h h h h h h h h
Id //
βDU
DU;
ηV as thepasting
DU Id //
DU u
##
H H H H H H H H
H
DηU
DU Id //
DdU
**
V V V V V V V V V V V V V V V V V V V V V V
ηDU
DU;
DU U
Dn
55
l l l l l l l l l l l l l l l
DdU U
++
X X X X X X X X X X X X X X X X X X X X X X X X X X
DU dU
""
F F F F F F F F F F F F F F F F F F F F
F Dd−n1 DDU
mU
::
v v v v v v v v v
Dω2U−1 DDU U
DDn
99
r r r r r r r r r r
DU DU DrU
88
and µV as thepasting
DU DU DU DU DrU //
DrU DU
Dr−rU1
DU DDU U DU DDn //
DrDU U
Dr−Dn1
DU DDU DU mU //
DrDU
DU DU
DrU
DDU U DU DDU rU //
DDnDU
DDU DU U DDU Dn //
DDrU U
DDr
−1
n
DDU DU
DDrU
Dω4U
DDω3U
DDDU U U DDDU n //
DDDnU
DDDµU
DDDU U DmU U //
DDDn
Dmn
DDU U
DDn
DDU DU DDrU //
mU DU
m−rU1
DDDU U DDDn //
mDU U
m−Dn1
DDDU DmU //
mDU
µDU
DDU
mU
DU DU DrU //DDU U
DDn //DDU mU //DU.
5.1. Theorem. V = (V , v, p, βV, ηV, µV), as defined above, is a pseudomonad on the
object K.
Proof. Observethat thepasting of µV and ηVV−1 is
//
""
E E E E E E E E E E E E E E E E E
E //
))
R R R R R R R R R R R R R R R
DηUDU−1
//
Dω2U DU
Dr−1
rU
//
Dr−1
Dn
//
//
DdnDU
//
DDr −1
n
Dω4U
))
R R R R R R R R R R R R R R R
++
DDω3U
//
DDDµU
//
Dmn
//
ηDU DU−1
m−rU1
//
m−Dn1
//
µDU
// // //.
Wemust show that this pasting equals p◦V βV. Substitutethepasting of DdnDU and
DηUDU−1 by thepasting ofDd−U uDU1 and DDηUDU. Then use (coh 2). With the help of
(2) show that
DDr−uU1
//
DDr−n1
//
//
DDDµU
DDDηUU−1
//
//
=
))
R R R R R R R R R R R R R R R
//
55
l l l l l l l l l l l l l l l
DDU DβU
DDrU
DDDn
and makethesubstitution. Makethesubstitution
followed by the substitutions
&&
N
&&
N
&&
N
and
//
and
DdDn
DDn
mU
.
Use(coh 7), and finish with thesubstitution
//
&&
In regards to the other condition, observe that the pasting of V µV, µVV and µV is:
DDDµUDU
DDDµU
Dmn
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 wedo thefollowing. First
makethesubstitution
//
&&
N
&&
N
&&
N
&&
N
DrU
Then make the substitutions
//
DDDµU
DDDDµU
and
//
DrDnDU−1
&&
N
&&
N
&&
N
nDU
&&
N
&&
N
DU DDDµU
&&
N
&&
N
DDDU DµU
''
Now use (coh 4). Proceed with the following substitutions
//
DDDrnU−1
DDDU DµU
&&
M
&&
M
&&
M
DDDDµUU
&&
M
DDDDµU
DDDDµU
//
DDDDµU
&&
L
&&
M
followed by
and then
&&
N
&&
N
&&
N
&&
N
µDDU
//
&&
M
&&
M
&&
M
followed by the substitution
//
m−1
DnDU
&&
N
&&
N
&&
N
&&
N
&&
N
DDDDµU
&&
N
DDDµU
''
Use(coh 9) to show that
and makethesubstitution. Next thesubstitution
&&
N
&&
N
&&
N
&&
N
&&
N
&&
L
followed by the substitution
//
&&
M
&&
L
&&
L
&&
M
&&
N
&&
N
&&
N
&&
N
&&
N
&&
M
&&
N DDDDµU equals the pasting of DDDµU, DDmnU and DDmn. Use(coh 6). To finish
theproof, makethesubstitution
//
&&
M
&&
L
&&
L
&&
M
&&
N
&&
N
&&
N
&&
N
&&
N
&&
M
&&
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 bepseudomonads on thesameobject K of the Gray-category A. Give n
another pseudomonad V = (V , v, p, βV, ηV, µV) on thesameobject K, wesay that V is
compatible with thepseudomonads D and Uif V =DU and there are invertible 2-cells:
U dU θ1
< < < < < < <
| | | |
z
1K v //
u
BB
DU,
DU DU p
θ2
""
F F F F F F F F
DU U Dn //
DU dUwwww;; w w w w w
DU,
DU DU p
θ3
%%
J J J J J J J J J
DDU
DuDUsssss99 s s s s s
mU //DU,
subject to coherence conditions. To describe the coherence conditions introduce the fol-lowing pastings:
θ4 = DU DU U DU Dn //
DU DU dUTTTTTTTT))
T T
pU
DU θ−
1 2
DU DU
p
p−dU1
DU DU DU pDU
DU p
55
k k k k k k k k k
µV
DU DU
p
))
S S S S S S S S S S
DU U
DU dU
55
k k k k k k k k k k
Dn //
θ2
DU,
and
θ5 = DDU DU
mU DU
//
DuDU DUTTTTTTTTTT))
Dp
θ3DU−1
DU DU
p
Dup
DU DU DU DU p
pDU
55
k k k k k k k k k
µ−V1 DU DU
p
))
S S S S S S S S S S
DDU DuDU
55
j j j j j j j j j j
mU //
θ3
DU.
The coherence conditions are:
DU
DU u
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
DU v
))
T T T T T T T T T T T T T T T T T T
Id
//
ηV
DU,
DU θ−1
1 DU DU
p mmmmmm66 m m m m m m m m 9999 θ2
= DηU,
DU U
DU dU
99
s s s s s s s s s
s Dn
DU U DuU
DD
DdU
//
Id
77
DDU mU //
DuDU
99
DDU
DuDU
%%
uDU
;;
dDU
DD
DDn
DDU DuDU
55
mDU
Dθ−
1 3
DDU
mU
DDU DuDU
44
6.1. Proposition. Assume we have invertible 2-cells θ1, θ2, and θ3, satisfying the
co-herence conditions. Then
(Dn, θ4) : (βVU, µVU)→(βV, µV) (18)
is a 1-cell in V-AlgK, and the following are 2-cells in V-AlgK:
(βV, µV)
Id
//
(DU u,p−1
u ) NNNN''
N N N N N N
N
DηU
(βV, µV)
(βVU, µVU)
(Dn,θ4)
77
p p p p p p p p p p p
(19)
(βVU, µVU)(Dn,θ
4)
**
V V V V V V
(βVU U, µVU U)
(DU n,p−n1g)33
g g g g g
(DnU,θ4UWW)++
W W W W
DµU (βV, µV),
(βVU, µVU)(Dn,θ
4)
44
h h h h h h
(20)
We also have that
(p, θ5−1) : (βDU DU, µDU DU)→(βDU, µDU) (21)
is a 1-cell in D-AlgK, and the following are 2-cells in D-AlgK:
(βDU DU, µDU DU)(p,θ−1 5 )
,,
X X X X X X X
(βDU DU DU, µDU DU DU)
(DU p,m−U p1)
22
d d d d d d d d
(pDU,θ5DU−1Z),,
Z Z Z Z Z Z Z
µV (βDU, µDU),
(βDU DU, µDU DU)(p,θ
−1 5 )
22
f f f f f f f
(22)
(βDU DU, µDU DU)
(p,θ−51)
θ2
))
S S S S S S S S S S S S S S
(βDU U, µDU U)
(DU dU,m−U dU1kk)kkkk55 k k k k k k k k k
(Dn,m−n1)
//(βDU, µDU),
(23)
(βDU DU, µDU DU)(p,θ−1 5 )
,,
X X X X X X X
(βDU DU U, µDU DU U)
(DU Dn,m−U Dn1 )22
d d d d d d d
(pU,θ5U−1)ZZ,,
Z Z Z Z Z Z Z
θ4 (βDU, µDU),
(βDU U, µDU U) (Dn,m
−1
n )
22
f f f f f f f
(24)
Furthermore, if we define σ1 as,
U DU
dU DU θ1DU QQ((
Q Q Q Q Q
DU DU p
βV OOOOO''
O O O
DU uDU
@@
vDU
33
g g g g g g g g g g g g g g g g g g
σ2 as
U U DU U dU DU //
nDU
dU U DU
d−U dU DU1
((
P P P P P P P P P P P P
U DU DU U p //
dU DU DU
d−U p1
U DU
dU DU
dnDU
DU U DU DU dU DU//
DnDU
((
R R R R R R R R R R R R
R
θ2DU
DU DU DU DU p //
pDU
µV
DU DU
p
U DU dU DU //DU DU p //DU,
γ as
U U
n
U dU
//
dU U
&&
L L L L L L L L L L
d−U dU1 U DU
dU DU
DU U DU dU//
Dn dn
θ2
''
O O O O O O O O O O O
DU DU p
U dU //DU.
then(σ1, σ2)is an object inU-AlgK, and (dU, γ) : (βU, µU)→(σ1, σ2)is a 1-cell inU-AlgK. Proof. Weshow first that (Dn, θ4) is a 1-cell in V-AlgK. To show that (Dn, θ4) satisfies
(9), start on theleft hand sideand makethesubstitutions:
//
βVU
$$
J J J J J J J J J
//
p−dU1
//
= // // ,
//
OO
vDU dU
OO
βVDU ttt99 t t t t t t
//
βVDU
$$
J J J J J J J J J
//
µV
//
= // // ,
//
OO
vp
OO
βV ttt99 t t t t t t
and
//
θ2
$$
DU θ2−1
vv
vDU dU
//
vp
//
= vDn.
To show that (Dn, θ4) satisfies (10), start on the left hand side, cancel DU θ2 and its
inverse, make the substitution
//
&&
N N N N N N N N N N N
DU p
−1
dU NNNN&&
N N N N N N N
&&
N N N N N N N N N N
N µVU //
p−dU1
//
=
//
OOOO''
O O O O O O O O
p−DU dU1
//
&&
M M M M M M M M M M M
p−dU1
''
N N N N N N N N N N
N µVDU
Then use(1), and concludewith
DU DU θ−21
$$
//
//
p−DU dU1
p−p1
// //
=
//
p−1
Dn
//
A A A A A A A A DU θ−1
2
>>
} } } } } } } }
Next we show thatDηUin (19) is a 2-cell in V-AlgK. To show that satisfies (11), start
on theleft hand sideof (11). Use(12) to substitutethepasting of DU DηU and DU θ−21
for thepasting of DU βV and DU DU θ−11. Then make the substitution
DU DU θ1−1
$$
//
//
p−u1
p−dU1
// //
=
//
p−v1
//
@ @ @ @ @ @ @
@ DU θ−1 . 1
>>
| | | | | | | |
Now use(5), and then use(12) onceagain.
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 substitutethepasting ofθ2and DµU, by thepasting ofDU d−n1,DU θ4
and θ2U. Makethesubstitution
//
OOOOO''
O O O O O O O
p−n1
//
&&
M M M M M M M M M M M
DU d−n1
&&
N N N N N N N N N N
N p−1
dU
// ,
=
//
''
O O O O O O O O O O O O
DU DU d
−1
n
''
O O O O O O O O O O O O
''
O O O O O O O O O O O
O p−1 //
dU U
p−Dn1
// .
Sincewealready know that (Dn, θ4) is a V-algebra morphism, we can substitute the
pasting of µV, p−Dn1, and θ4, by thepasting of DU θ4, θ4 and µV. By thedefinition of θ4,
wecan substitutethepasting of µVU, p−dU U1 and θ2U by thepasting of DU θ2U and θ4U.
Finally, use(16).
That (p, θ−51) is a morphism of D-algebras, and that µV, θ2, and θ4 are2-cells of
D-algebras are similar and left to the reader.
Now weshow that (σ1, σ2) is an object inU-AlgK. Paste ηUDU−1 onto theleft hand
sideof (7), and makea substitution on it to obtain
//
d−U uDU1
''
O O O O O O O O O O O
O //
d−U dU DU1
''
O O O O O O O O O O O
O //
d−U p1
//
ηUDU
11
//
θ2DU
''
O O O O O O O O O O O
O //
µV
// .
Also paste ηUDU−1 on what turns out to betheright hand sideof (7), and makethe
following two substitutions:
##
ηVDU
µV
// .
Comparethepasting wearriveat with (25), and useequation (12). Now, the left hand side of (6) in this case, is the following pasting:
//
µUDU
Make the following sequence of substitutions:
//
µUDU
//
&&
N
&&
N
&&
M
&&
N
&&
N
&&
N
&&
N
&&
N
and
//
&&
N
&&
N
&&
N
&&
N
&&
N
&&
N
On theother hand, theright hand sideof (6) in this caseis thepasting
//
&&
N
&&
M
On this pasting makethefollowing substitutions:
//
&&
N
nDU
// //
=
//
&&
N
&&
M
&&
N
pppppp ppppp
// //
=
//
&&
N
pppppp ppppp
//
p−dU DU1
xx
pppppp ppppp
//
p−p1
xx
pppppp ppppp
and
//
p−dU DU1
//
θ2DU
..
&&
N N N N N N N N N N
N =
//
((
P P P P P P P P P P P P P
A A A A A A A A A A A A A A A A
A DU θ2DU
PPPP((
P P P P P P P P P
θ4DU
((
P P P P P P P P P P P P P
µVDU−1
//
(definition of θ4).
Compare both results, and use equation (16). Theclaim about (dU, γ) is left to the reader.
Next weprovethat thecompositeof two pseudomonads via a distributivelaw as defined in section 5, is compatible.
6.2. Theorem. Assume we have a distributive law as in section 4 and let V the
com-posite pseudomonad defined in section 5. If we define θ1 =iddU◦u, and θ2 as the pasting
DU DU DrU
&&
M M M M M M M M M M
DU U
DU dUrrrrr99 r r r r r
DdU U //
Dn
Dω2U
Ddn
DDU U
DDn
DU DdU //
Id
**
U U U U U U U U U U U U U U U U U U U U U U U
ηDU−1
DDU
mU
DU,
and θ3 as the pasting
DU DU DrU
&&
M M M M M M M M M M
DDU
DuDUrrrrr99 r r r r r
DDuU //
Dω1U
Id //
DDβU DDU U
DDn
%%
K K K K K K K K K K
DDU mU //DU,
Proof. Conditions (12), (13), and (14) are fairly easy and left to the reader. The
remain-ing ones arealso easy oncewehaveshown that
θ4 = DU DU UDU Dn//
DrU
DDU U UDDU n//
DDnU
DDµU DDU U
DDn
DDU
mU
DU U Dn //DU,
and θ5 = DDU DUmU DU//
DDrU
mrU
DU DU
DrU
DDDU UmDU U//
DDDn
mDn
DDU U
DDn
DDDU mDU //
DmU
µDU DDU
mU
DDU mU //DU.
To show the first equality above, start with the definition of θ4 and makethefollowing
substitutions:
DU Dd−1
n
&&
N
&&
N
&&
N
&&
N
and
&&
N
&&
N
makethesubstitutions:
//
''
O O O O O O O O O O O O
DDd
−1
U n
''
O O O O O O O O O O O O
''
O O O O O O O O O O O O
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? DDd //
nU
DDDµU
//
DDdn
77
o o o o o o o o o o o o
= //
//
DDµU
DDdn
// // ,
and
// //
DDdn
Dmn
//
//
m−dU1
µDU
//
ηDU−1 ;;
//
= mU ◦DDn◦DηDU U−1.
Finally, use(coh 8).
Theproof for θ5 is very similar, except that we need the equation:
U D uU D//
Id
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
n n n n
s{
βUD−1
U U D U r //
nD
U DU
rU
h h h h
px ω
3
DU U
Dn
U D r //DU
= U D uU D //
r
ur
U U D
U r
DU uDU //
DuU
&&
L L L L L L L L L L
Id
))
ω1U
U DU
rU
DβU DU U
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 byDn◦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 theGray-category PSM(A) of pseudomonads on a Gray-category
A.
Theobjects of PSM(A) arepseudomonads in A.
Given pseudomonads D on K, and U on L, wedenotethe2-category PSM(A)(D,U)
that
U-Alg G // Φ
D-Alg
Φ
A( ,L)
G0( )
//A( ,K)
commutes, where Φ is the forgetful Gray-natural transformation defined in section 3. We can considerU-Alg andD-Alg as trihomomorphisms andGas a tritransformation. A 1-cell
(G, G0)→(H, H0) in [D,U] is a pair (g, g0), whereg :G→H is a strict trimodification,
and g0 :G0 →H0 is in A(L,K), such that
U-Alg G
++
H 33
g D-Alg Φ
A( ,K)
= U-Alg
Φ
A( ,L)
G0( )
,,
H0( )
22
g0( ) A( ,K).
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), whereγ :g →his a perturbation, andγ0 :g0 →h0is inA(L,K),
such that
U-AlgX D-AlgX
A(X,K)
gX ⇓ ⇓g′X
++
33
ΦX
γX
//
=
U-AlgX
A(Z,L) g0( )⇓ ⇓h0( )A(X,K).
ΦZ
++
33
γ0( )
//
Vertical and horizontal compositions are the obvious ones. The rest of the operations are defined as follows:
(H, H0)(G, G0) = (HG, H0G0)
(H, H0)(g, g0) = (Hg, H0g0) (h, h0)(G, G0) = (hG, h0G0)
(H, H0)(σ, σ0) = (Hσ, H0σ0) (τ, τ0)(G, G0) = (τ G, τ0G0)
8. Liftings
8.1. Definition. A lifting is a pseudomonad in the Gray-category PSM(A).
In more detail, observe that a pseudomonad D in PSM(A) has to havedomain a
pseudomonad U = (U, u, n, βU, ηU, µU) in A, with domain someobject K of A. Observe
that D is of theform
Taking the second entries, we obtain a pseudomonad D = (D, d, m, βD, ηD, µD) in A on
thesameobject K. Furthermore, given an object X of A, wecan defineDX =DX, and
dX =dX etcetera, to obtain a pseudomonad DX = (DX, dX, mX, βX, ηX, µX) in Gray, on
the2-category U-AlgX. D and thefamily DXX behave well with forgetful 2-functors
and changeof base.
9. From a pseudomonad with compatible structure to a pseudomonad in
PSM(
A
)
Assumewehavea pseudomonad Vthat is compatiblewith pseudomonads Dand U as in
section 6. We want to define a pseudomonadD on theobjectUof PSM(A). LetX bean
object of A. We begin by defining a 2-functor DX :U-AlgX → U-AlgX. Give n an obje ct
(ψ, χ) in U-AlgX, with
U X x
!!
D D D D D D D D
~ ψ X
uX {{{==
{ { { { {
Id //X,
U U X
nX
U x
//
χ U X
x
U X x //X,
Definethefirst entry of DX(ψ, χ) as
U DX
U DuX
&&
N N N N N N N N N N N
dU DX
θ1DX
d−U DuX1
_ _ _ _
ks
U DU X
dU DU X
''
O O O O O O O O O O O
DU DX DU DuX //
vDuX
DU DU X pX
&&
N N N N N N N N N N N
DX uDX
FF
DuX //
Id
..
vDX
99
s s s s s s s s s s
DU X
vDU X
77
o o o o o o o o o o o
//
βVX
DU X Dx Dψ
$$
I I I I I I I I I
and the second entry of DX(ψ, χ) as
U U DXU U DuX//
nDX
U U DU X U dU DU X //
dU U DU X
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
nDU X
U DU DU X U pX //
dU DU DU X
d−U dU DU X1
U DU X U Dx //
U DuU X
&&
M M M M M M M M M M
Id
_ _ _ _
ks
U DβUX
U Du−x1
U DX
U DuX
_ _ _ _
ks
dU DnX
U DU U X
U DnX
xx
qqqq qqqqqq
U DU x // dU DU U X
d−U DU x1
U DU X
dU DU X
d−1
U pX
U DU X
dU DU X
_ _ _ _
ks
θ4X−1
DU DU U X
DU DnX
xx
qqqq qqqqqq
DU DU x// pU X
p−x1
DU DU X
p
n−DuX1 dnDUX
DU U DU X
DnDU X θ2DU X
''
N N N N N N N N N N N
DU dU DU X
//
DU DU DU X
pDU X
DU pX//
µVX DU DU X
pX
DU U X
DnX
xx
rrrrrr rrrrr
DU x //
Dχ
DU X
Dx
U DU X
U DuX//U DU X dU DU X //DU DU X pX //DU X Dx //DX.
If (h, ρ) : (ψ, χ) →(ψ′, χ′) in U-Alg
X, then define the first entry of DX(h, ρ) as Dh, and
thesecond as thepasting
U DX U Dh //
U DuX
U Du −1
h
U DX′
U DuX′
U DU X U DU h //
dU DU X
d
−1
U DU h
U DU X′
dU DU X′
DU DU XDU DU h//
pX
p
−1
h
DU DU X′
pX′
DU X DU h //
Dx
Dρ
DU X′
Dx′
DX Dh //DX′
Given ξ: (h, ρ)→(h′, ρ′) in U-Alg
X, define DX(ξ) = Dξ.
9.1. Proposition. The above definitions make DX : U-AlgX → U-AlgX a 2-functor.
Furthermore, if we define D :U-Alg→U-Alg as DX at every object X of A, then D is a
Gray-transformation, and (D, D )∈[U,U].
Proof. Thehardest part of theproof is to show that DX(ψ, χ) is inde e d an obje ct in
U-AlgX. This is what we do, and we leave the rest to the reader. We must show that
(6) and (7) aresatisfied. To show (7), pass ηUDX to the left hand side, and perform the
following sequence of substitutions:
//
++
//
ηUDX−1
n−DuX1
//
dnDU X
// //
= // //
DηUDU X−1
&&
N N N N N N N N N N N
U uDuX
88
r r r r r r r r r r
r // d
−1
U uDU X
88
r r r r r r r r r r
r //
88
r r r r r r r r r r
//
DηUDU X−1
22
θ2DU X
&&
N
&&
N
>>
}
&&
N
&&
N
pppppp ppppp
= //
&&
N
&&
N
&&
N
&&
N
&&
N
&&
N
&&
N
&&
N
and
To get from the left hand side of (6) to the right hand side, make the following sequence of substitutions:
//
&&
N
&&
N
&&
N
pppppp
pppppp θ4DU X
xx
pppppp ppppp
&&
N
&&
N
&&
//
&&
N
&&
N
&&
//
DnX
&&
N
pppppp
ppppppU DµUX
xx
//
&&
N
pppppp
ppppp d−U DU U x1
pppppp ppppp
and
//
DuX
We are now ready to define dX : 1→DX, and mX :DXDX →DX. Thefirst entry of
(dX)(ψ,χ) is dX, whilethesecond is thepasting
U X U dX //
U uX
++
V V V V V V V V V V V V V V V V V V V V V V V V
dU X
; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;
x
U duX
U DX
U DuX
d−U uX1 U U X
dU U X
U dU X //
d−U dU X1 U DU X
dU DU X
DU X DU uX//
Id //
DηUX−1
DU U X DU dU X//
DnX
((
Q Q Q Q Q Q Q Q Q Q Q
Q
θ2X
DU DU X
pX
dx
DU X
Dx
X dX //DX,
and (dX)(h,ρ)=dh.
Definethefirst entry of (mX)(ψ,χ) as mX, and thesecond as thepasting
U DDX U mX //
U DDuX
))
S S S S S S S S S S S S S S S
U DuDX
U muX
U DU
U DuX
U DU DX
dU DU DX
U DU DuXSSSSSSSS))
S S S S S S
S
U Du−DuX1 U DDU X
U DuDU X
dU DDU X
))
S S S S S S S S S S S S S S
S U mU X //
_ _ _ _
ks
dU DuDU X
d−U mU X1
U DU X
dU DU X
DU DU DX
pDX
DU DU DuXSSSSSS))
S S S S S S S S
S
d−U DU DuX1
_ _ _ _
ks
p−DuX1
U DU DU X
dU DU DU X
DU DDU X
DU DuDU X
uu
kkkkkkkkk
kkkkkk DU mU X
))
R R R R R R R R R R R R R
DU DX
DU DuX
DU DU DU X
pDU X
uu
kkkkkkkkk kkkkkk
DU pX //
DU θ3X
DU DU X
pX
DU DU X
DdU DU X
Id
,,
Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y
ηDU DU X µVX
DDU DU X mU DU X //
DpX
θ5X
DU DU X
pX
))
R R R R R R R R R R R R R
DDU X
DDx
mU X //
mx
DU X
Dx
DDX mX //DX.
Define (mX)(h,ρ) =mh.
9.2. Proposition. With the above definitions, dX : 1 → DX and mX : DXDX → DX
X of A, we have that, (d, d ) : 1 → (D, D ) and (m, m ) : (D, D )(D, D ) → (D, D ) are 1-cells in the 2-category [U,U].
Proof. Thesizeof thediagrams obtained is what makes it hard to provethat (mX)(ψ,χ)
satisfies (10). To get from theleft hand sideof (10) to theright hand sidemakethe following three substitutions:
// jjjjjj
DU θ3X TTTTT))
jjjjjjjjj jjj
DdU DnX TTT))
jjjjjj
)) jjjjjj
))
hhhhhhhh hhhhhh
dU DnX
hhhhhhhh
//
&&
N
&&
N
&&
N
&&
N
pppppp ppppp
&&
N
&&
N
&&
N
&&
N
&&
N
&&
N
&&
N
&&
N
&&
N
&&
N
and
=
//
n−mX1
((
P P P P P P P P P P P P P
++
X X X X X X X X X X X X X X X X X X X X X X X X X X
((
P P P P P P P P P P P P P
n−DuX1
PPPPP((
P P P P P P P
P //
U muX
((
P P P P P P P P P P P P P
((
Q Q Q Q Q Q Q Q Q Q Q Q Q
dnDU X
θ2DU X
!!
d−U dU DU X1
PPPPP((
P P P P P P P
P U Du−DuX1
PPPPP((
P P P P P P P
P //
d−U mU X1
((
P P P P P P P P P P P P P P P P P P P P P P P P P P P
''
P P P P P P P P P P P P
P d−U DU DuX1
dU DuDU X
ww
nnnnnn nnnnnn
n
DU θ3X
++
X X X X X X X X X X X X X X X X X X X X X X X X X
// .
The fact that the three equalities above are indeed satisfied and the rest of the proof are left to the reader.
It is easily seen that, defining η, β, and µ at every X as (ηX)(ψ,χ) =ηDX, (βX)(ψ,χ) =
βDX, and (µX)(ψ,χ) =µDX, weobtain 2-cells (η, η), (β, β ), and (µ, µ ) in [U,U]. Wethus
have:
9.3. Theorem. D = ((D, D ),(d, d ),(m, m ),(β, β ),(η, η ),(µ, µ )) is a pseudomonad in
PSM(A), on the object U.
10. From a pseudomonad in PSM(
A
) to a distributivelaw
Assumethat wehavea pseudomonad in PSM(A) as in section 8. We produce a distribu-tivelaw as follows. Consider DK:U-AlgK→U-AlgK. Sincethediagram
U-AlgK DK
//
ΦK
U-AlgK
ΦK
A(K,K)
D( )//A(K,K)
commutes, wehavethat DK(βU, µU) is of theform (σ1, σ2) with
U DU
@ @ @ @ @ @ @
s
| σ1
DU
uDU
????
Id //DU,
U U DU U s //
nDU
σ2
U DU
s
U DU s //DU.
Furthermore, since the diagram
U-AlgK DK //
U
U-AlgK
U
U-AlgK DK
commutes, where U is the change of base 2-functor defined in section 3, then DK(n, µU) :
DK(βUU, µUU) → DK(βU, µU) is of theform (Dn, σ3) : (σ1U, σ2U) → (σ1, σ2) with σ3 of
theform
U DU U U Dn //
sU
σ3
U DU
s
DU U Dn //DU.
Similarly, it can beshown that (dK)(βU,µU) is of theform (dU, σ4), with
U U U dU//
n
σ4
U DU
s
U dU //DU.
If weapply DK to (σ1, σ2) weobtain anotherU-algebra (σ′1, σ2′), with
U DDU
F F F F F F F F
s′
####
σ′
1
DDU
x x x x x x x x x
uDDU
<<<<
////DDU,
U U DDUU s
′
//
nDDU
σ′
2
U DDU
s′
U DDU s′ //DU.
Applying DK to (s, σ2) : (βUDU, µUDU)→(σ1, σ2), weobtain a morphism (Ds, τ1) with
U DU DU U Ds //
sDU
τ1
U DDU
s′
DU DU Ds //DDU.
Wealso havethat (mK)(βU,µU) is of theform (mU, γ1), with
U DDUU mU //
s′
γ1
U DU
s
DDU mU //DU.
Defineσ5 as thepasting
U DDU U mU //
U Dσ−11
Id
U DuDU
U DU
s
U DU DU
U Ds
''
O O O O O O O O O O O
sDU
τ1
DU DU
Ds
''
O O O O O O O O O O
O U DDU
s′
γ1
It is with thehelp of σ1 to σ5 that wedefinethedistributivelaw as follows:
Define r=s◦U Du, ω1 as thepasting
U D U Du
''
N N N N N N N
D uDss99
s s s s
DuJJJ$$
J J J
J uDu U DU
s σ1 NNN&&
N N N N
DU uDU Id //
88
p p p p p p p
DU,
ω2 as thepasting
U D U Du
''
N N N N N N N
U U dsss99
s s s s
U u
$$
J J J J J J
J U du
Id 11
U DU
sNNNN&& N N N
U U n
ηU−1 NN&&
N N N N N N
U dU
88
p p p p p p
p
σ4
DU,
U dU
88
p p p p p p p p
ω3 as thepasting
U U D U U Du//
nD
U U DU U s //
nDU
U DU U DuU//
Id
%%
L L L L L L L L L
L
U DU U sU //
U Dn U DβU
DU U
Dn
n−Du1 σ2
U DU
s
%%
K K K K K K K K K
K σ−1 3
U D U Du //U DU s //DU,
and ω4 as thepasting
U D U Du //U DU s //DU
////
U mu
U DDU U mU
OO
U DuDU//U DU DU sDU
((
Q Q Q Q Q Q Q Q Q Q Q Q
////
σ5
U DD U DuD//
U DDu
77
o o o o o o o o o o o
U m
OO
U Du −1
Du
U DU D sD //
U DU Du
66
n n n n n n n n n n n n
s −1
Du
DU D DU Du //DU DU
Ds //DDU. mU
OO
Theproof that weobtain a distributivelaw in this way is based on thebehavior of the 2-cellsσ1-σ5. Asidefrom theobvious conditions that comefrom thefacts that (σ1, σ2) is a
U-algebra, (Dn, σ3) and (dU, σ4) arehomomorphisms ofU-algebras, we have the following
conditions.
10.1. Proposition. The following are 2-cells in U-AlgK:
DµU: (Dn, σ3)◦(DU n, s−n1)→(Dn, σ3)◦(DnU, σ3U).
dn : (Dn, σ3)◦(dU U, σ4U)→(dU, σ4)◦(n, µU). (27)
Furthermore, the following equations are satisfied:
DDU uDDU //
DuDU
%%
sDU
&&
N
&&
N
&&
N
&&
N
eeeeeeeee
''
nDDU
n
−1
mU
U U DU
nDU
U s
&&
L
&&
L
sDU
U DDU UU mU U//
DsU
DU DnOOOO''
sDU
DDU U
DDn
''
DsU
σ
5U
DDU U
DDn
&&
N
sDU
DU
DdU
&&
M
DdU ηDU
DU,
DDU mU
&&
M
nDU
σ
4DU
U DU DU
sDU
DDU mU βDU
&&
M
nDU
_ _ _ _
ks
U Du−mU1
U DDU
U DuDU
U mU
//U DU
s
U DDDU
U DmUnnnnn66
n n n n n n n
U DuDDU
_ _ _ _
ks
s−mU1
U DU DU
sDU
U DU DDU
U DU mU
66
n n n n n n n n n n n n
sDDU
DU DU
Ds
σ5
DU DDU
DU mU
66
n n n n n n n n n n n n
DU DuDU
DU DU DU
DsDU
Dσ
5
DDU DU
DDs
µDU
DDU mU //DU
DDDU
DmU
66
n n n n n n n n n n n n
mDU //DDU mU
88
q q q q q q q q q q
=
U µDU
U DDU U mU //U DU
s
U DDDU
U DmUmmmmmm66 m m m m m m m
U mDU // U DDuDU
U DuDDU
vv
llllll llllll
l
U muDU
_ _ _ _
ks
U Du−DuDU1
U DDU U mU
88
q q q q q q q q q q
U DuDU
U DU DDU
U DU DuDU
((
R R R R R R R R R R R R R
sDDU
_ _ _ _
ks
s−DuDU1
U DDU DUU mU DU//
U DuDU DU
U DU DU
sDU
DU DDU
DU DuDU
((
R R R R R R R R R R R R
R U DU DU DU
sDU DU
σ
5DU
DU DU DU
DsDU
σ5
DDU DU mU DU //
DDs
ms
DU DU
Ds
DU.
DDDU mDU //DDU
mU
88
q q q q q q q q q q
(31)
