Theory and Applications of Categories, Vol. 26, No. 1, 2012, pp. 1–29.
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
F. MARMOLEJO AND R.J. WOOD
Abstract. We show that colax idempotent pseudomonads and their algebras can be
presented in terms of right Kan extensions. Dually, lax idempotent pseudomonads and
their algebras can be presented in terms of left Kan extensions. We also show that a
distributive law of a colax idempotent pseudomonad over a lax idempotent pseudomonad
has a presentation in terms of Kan extensions.
1. Introduction
This paper follows [Marmolejo and Wood, 2010] and builds on the idea in [Manes, 1976],
which was actually preceded by [Walters, 1970], that a monad can be presented without
iterating the underlying endofunctor. [Marmolejo and Wood, 2010] extended Manes’
notion of an extension operator to handle algebras but we note now that algebras were
treated in a somewhat similar manner in [Walters, 1970] too. Our treatment of algebras
also enabled “no iteration” descriptions of distributive laws and wreaths. Because the
values of the endofunctor of a monad are term objects, the no iteration description in
effect removes the need to mention terms of terms and (terms of terms of terms). This is
particularly helpful in the descriptions of distributive laws and wreaths where the intent
is to rewrite M-terms of A-terms as A-terms of M-terms.
When we turn to higher dimensional monads the no iteration idea is even more helpful.
For then the terms tend to be n-sorted, with n ≥ 2. For example, in completion monads
with respect to classes of limits, the terms are categorical diagrams comprised of both
objects and arrows. It is in fact completion monads, precisely colax idempotent pseudomonads, about which we have most to say. Such a pseudomonad (D, d, m, · · ·) is what
is also called a “coKZ doctrine”, and characterized by adjunctions dD ⊣ m ⊣ Dd. We
caution the reader that in [Marmolejo, 1997], our main reference for these pseudomonads,
the subject matter is presented in terms of lax idempotent pseudomonads “KZ doctrines”,
for which the adjunctions are reversed to give Dd ⊣ m ⊣ dD.
The extension operator in [Manes, 1976] and those in [Marmolejo and Wood, 2010]
satisfy equations. It will come as no surprise that if pseudomonads (on 2-categories say)
are described in similar terms then the equalities of those papers must be replaced with
invertible 2-cells — which must themselves satisfy equations. However, colax idempotent
The first author gratefully acknowledges financial support from PAPIIT UNAM, project IN110111-2.
The second author gratefully acknowledges financial support from the Canadian NSERC.
Received by the editors 2011-09-30 and, in revised form, 2012-01-09.
Transmitted by R. Street. Published on 2011-12-31.
2000 Mathematics Subject Classification: 18B35, 06D10, 06B23.
Key words and phrases: (co-) lax idempotent pseudomonads, KZ-doctrines, pseudo-distributive laws.
c F. Marmolejo and R.J. Wood, 2012. Permission to copy for private use granted.
⃝
1
2
F. MARMOLEJO AND R.J. WOOD
pseudomonads have all but one of their 2-cell equations given by adjunction equations.
Thus it might be hoped that if colax (or lax) idempotent pseudomonads are described by
extension operators then their 2-cell equations might also mediate universal properties.
This is the case. The extensions which appear in describing colax [lax] idempotent pseudomonads are right [left] Kan extensions! The precise definition (Definition 3.1) in terms
of Kan extensions is somewhat similar to the conditions given in [Bunge, 1974] in what is
called a coherently closed family of U -extensions (U is a 2-functor), furthermore, the way
we extend the function of objects to a pseudofunctor from the data given in Definition
3.1 is similar to the construction of a lax adjoint to U given in [Bunge, 1974].
The algebras for a colax (or lax) idempotent pseudomonad are also defined in terms
of Kan extensions and proven to be essentially the same as the usual algebras.
In Section 2 we begin by recalling the characterization of a colax idempotent pseudomonad D = (D, d, · · ·) and its algebras, in terms of adjunctions, as given in [Marmolejo,1997]. Important equations involving the derived modification δ : dD → Dd are
also recalled. In Section 3 we define right Kan pseudomonads and algebras for these. Section 4 provides a construction of a right Kan pseudomonad D′ from a colax idempotent
pseudomonad D and a construction of a colax idempotent pseudomonad D′ from a right
Kan pseudomonad D. In Section 5 we show that starting with either notion as D, the
2-category of algebras for D is 2-equivalent to the 2-category of algebras for D′ .
We recall in Section 6 that morphisms between pseudomonads on 2-categories can
be described in terms of 2-functors between their underlying 2-categories, together with
liftings to their 2-categories of algebras. Moreover, these can also be described, see [Marmolejo and Wood, 2008] in terms of transitions which are a pseudo version of Street’s
morphisms of monads [Street, 1972]. In Section 6 we use the work of the previous sections and these observations to give a description of transitions between colax idempotent
pseudomonads in terms of extensions. Since distributive laws can be elegantly described
in several ways in terms of extensions and one of their duals we are able in Section 7
to give a description of distributive laws between certain pseudomonads in terms of extensions. We note that the distributive law described in [Marmolejo, Rosebrugh, Wood,
2002], whose algebras are constructively completely distributive lattices, was produced
this way, as a Kan extension. Another example is the distributive law of the small limit
completion pseudomonad over the small colimit completion, whose algebras are the completely distributive categories [Marmolejo, Rosebrugh, Wood, to appear]; we also have
the lextensive categories as algebras for the pseudomonad obtained from a distributive
law of the finite completion pseudomonad over the finite sum completion pseudomonad;
or regular categories as algebras for the finite limit completion pseudomonad over the
regular factorizations pseudomonad with base catker as defined in [Centazzo and Wood,
2002], and many more. To illustrate how these distributive laws work in the setting of
Kan extensions we examine, in Section 8, the distributive law of coFam over Fam.
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
3
2. Preliminaries
For the convenience of the reader, we recall in this section the definition of co-lax idempotent pseudomonad (also known as co-KZ pseudomonad). They first appeared in the
papers of Kock [Kock, 1973] and Zöberlein [Zöberlein,1976]. In this section we largely
follow (the dual of) the development given in [Marmolejo, 1997].
Let K be a 2-category. A co-lax idempotent pseudomonad D = (D, d, m, α, β, η, ε) on
K consists of a pseudofunctor D : K → K, together with strong transformations d : 1K → D
and m : D2 → D, and modifications
D?
1D
??
??
dD ?
/D
?

α ≃ 

 m
D2
1
D2
/ D2
2
D2 ?
? D ??
?
? D ??
?


? dD
??m
?? η 
m 
Dd 
 β ???
 Dd
m ??
 ε ≃ ??


/ D2
/ D,
D
D
D2
1
1
D2
(1)
D
with α and ε invertible, that render dD ⊣ m ⊣ Dd, and such that the coherence condition
2
: D JJ m
JJ
JJ
t
ε
tt
$
/D
/D
JJ 1D
:
t
JJ
tt
JJ α ttm
t
dD $
1K
d
: D JJ
JJDd
JJ
$
dtttt
Ddttt
ttt
1K JJ
=
JJJ
J
d J$
D2
dd
D
D
ttt:
t
tt dD
2
m
/D
(2)
is satisfied. It is shown in [Marmolejo, 1997] that any such structure induces a pseudomonad, whose structure is given by (D, d, m, α−1 , ε−1 , µ), where µ is the pasting
1D3
Dm
/ D3
/ D2
>
>}
AA
666
6
}
AA ηD
}
}
AAm
6666 dD }}
}
AA
}
AA
6
}} 2 dm 6
}} α−1 A
mD AA
}
dD
}
}
}
/ D,
/
2
D
D
D3 AA
m
(3)
1D
and furthermore, that for a pseudomonad (D, d, m, α−1 , ε−1 , µ) to be co-lax idempotent
it suffices that there exists a modification β such that α, β : dD ⊣ m is an adjunction;
equivalently, that there exists a modification η such that η, ε : m ⊣ Dd is an adjunction.
Recall as well that we can then produce a 2-cell δ : dD → Dd as the pasting
1D2
/ D2 ,
?
?

??m η 
dD 
?


−1
?  Dd
 α /D
D
2
?D ?
1D
that this pasting is equally the pasting of ε−1 and β at m, that δ · d = d−1
d , that m · δ =
−1 −1
ε α , and that δ · m is the pasting of β and η at 1D2 .
4
F. MARMOLEJO AND R.J. WOOD
The 2-category D-Alg of D-algebras is defined as follows. Its objects are adjunctions
b
ζ, ζ : dB ⊣ B,
1B
B EE
EE
EE
dB E"
ζ
/B
<
y
≃ yyy
yyy B
B EE
y<
yy b EEEdB
y
EE
y
ζ
yy
"
/ DB
DB
B
DB
(4)
1DB
with invertible unit. The invertibility of ζ is automatic if d is fully faithful. Recall as well
that ζb is completely determined by ζ as the pasting
3B
B
dB
dDB
DB
δB
d−1
B
*
2
4D B
Dζ −1
DdB
%
/ DB,
7
DB
1DB
and that all we have to do to verify that a ζ as above determines an object in D-Alg is to
show that the equation
1B
3B
B
dB
d−1
B
dDB
DB
δB
*
2
4D B
Dζ −1
DdB
ζ
/
w; B
w
w
ww
wwwB
ww
%
/ DB,
7
DB
= 1B
(5)
1DB
is satisfied. (Note that replacing B by D, B by m, and ζ by α in the definition of ζb gives
us β = α
b.)
A 1-cell from (B, B, ζ) to (A, A, ξ) is a 1-cell H : B → A such that the pasting
H
B EE
y<
EEdB
B yy
EE
b
yy
ζ
E"
y
y
/
DB
DB
1DB
d−1
H
1A
/A
/A
EE
y<
EE
y
EE ξ yyy
dA E"
yy A
/ DA
(6)
DH
is invertible. Given H, K : (B, B, ζ) → (A, A, ξ), a 2-cell in D-Alg is simply a 2-cell τ : H →
K in K. Provisionally write D′ for the pseudomonad (D, d, m, α−1 , ε−1 , µ) described above.
It is shown in [Marmolejo, 1997] that D-Alg is 2-isomorphic to D′ -Alg, the usual category
of algebras for a pseudomonad, since the associativity constraint needed to complete a
D-algebra (B, B, ζ) to a D′ -algebra is given uniquely by the pasting
1D2 B
DB
/ D2 B
/ DB
AA
>
6666
}>
AA
}
AA B
6666 dB }}}
AA ηB
}}
AA
}
}}
A
} ζ −1
AA
dB 66
}
mB AA
dDB
}}
}}
/ B,
/
DB
B
D2 BA
B
1B
(7)
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
5
while for a 1-cell H : (B, B, ζ) → (A, A, ξ), the pasting (6) uniquely completes H to a
1-cell of D′ -algebras.
3. Right Kan pseudomonads and their algebras
We define co-lax pseudomonads in terms of right Kan extensions. Later on we shall
show that they are the usual co-lax pseudomonads as in the previous section, but for the
moment (and just to be able to distinguish one from the other in this paper) we will call
them right Kan pseudomonads.
3.1. Definition. A right Kan pseudomonad D on K is given as follows:
i) A function D : Ob(K) → Ob(K).
ii) For every A ∈ K, a 1-cell dA : A → DA.
iii) For every 1-cell F : B → DA, a right Kan extension of F along dB
dB /
DB
EE DF EE EE~ FD
F EE
E" B EE
(8)
DA
with DF invertible (the latter being automatic if the 1-cell dB is fully faithful).
Subject to the axioms
a) For every A in K,
/ DA
EE
EE EE
1DA
dA EEE
" dA
A EE
DA
exhibits 1DA as a right Kan extension of dA along dA.
b) For every G : C → DB and F : B → DA the 2-cell
dC /
DC
EE DG EE EE~ GD
G EE
E" C EE
DB
FD
DA
exhibits F D GD as a right Kan extension of F D G along dC.
(9)
6
F. MARMOLEJO AND R.J. WOOD
3.2. Remark. Observe that we can also define an effect ( )D on 2-cells: given φ : F →
G : B → DA in K, we define φD : F D → GD as the unique 2-cell such that
B
/ DB
dB
DG } G
GD
ks
φD
dB
B
F
=
FD

{ 
φ
G
( / DB
DF 

{ 
D
F
+ DA.
DA
We clearly obtain a functor ( )D : K(B, DA) → K(DB, DA).
We now define the 2-category of algebras for a a right Kan pseudomonad D in terms
of right Kan extensions. We denote it by D-Alg and we define it as follows. An object B
in D-Alg consists of an object B in K together with an assignment, to every F : C → B,
of a right Kan extension F B : DC → B of F along dC
/ DC
II BF 
II 
II {  F B
F III
$ C II
dC
(10)
B
with BF invertible (automatic if dC fully faithful), in such a way that for every G : X →
DC in K, the diagram
/ DX
EE DG EE EE~ GD
G EE
E" X EE
dX
(11)
DC
FB
B
exhibits F B · GD as a right Kan extension of F B · G along dX.
A 1-cell H : B → A in D-Alg is a 1-cell H : B → A in K such that for every F : C → B,
the diagram
/ DC
EE BF EE B
EE~ EE F
F
E" C EE
dC
(12)
B
H
A
exhibits F B · H as a right Kan extension of F · H along dC. A 2-cell τ : H → K : B → A
is simply a 2-cell τ : H → K in K. Composition is as in K. It is not hard to show that
composition of 1-cells in D-Alg results in a 1-cell in D-Alg.
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
7
3.3. Remark. As in Remark 3.2 we can, for any B in D-Alg, induce an effect ( )B on
2-cells: given φ : F → G : C → B, we define φB : F B → GB as the unique 2-cell such that
/ DC
dC
C
BG } GB
G
φB
ks
dC
C
F
FB
=

{ 
φ
G
) / DC
BF 

{ 
B
F
+ B,
B
thus inducing a functor ( )B : K(C, B) → K(DC, B).
4. Right Kan pseudomonads versus co-lax idempotent pseudomonads 1
In this section we construct a colax idempotent pseudomonad from a right Kan pseudomonad, and vice versa. The constructions are given in the following two theorems.
4.1. Theorem. Every right Kan pseudomonad on K induces a co-lax idempotent pseudomonad on K.
Proof. Assume we have a right Kan pseudomonad D on K. We first extend D to a
pseudofunctor D : K → K. Given φ : F → F ′ : B → A in K, define DF = (dA · F )D , and
define Dφ : DF → DF ′ as (dA · φ)D , that is, Dφ is the unique 2-cell such that
dB
B
DdA·F ′
F′
A
z
zzzz
x zzz
DF ′
dA
/ DB
Dφ
ks
dB
B
DF
=
F′
/ DA
φ
ks
A
F
DdA·F zzz
zz
x zzz
dA
/ DB
DF
/ DA
(using the fact that the left most square exhibits DF ′ as a right Kan extension). It is then
immediate that D(1F ) = 1DF and that for ψ : F ′ → F ′′ we have D(ψφ) = (Dψ)(Dφ). If
G : C → B, define DG,F : DF · DG → D(F · G) as the unique (invertible) 2-cell such that
/ DC
EE
EEDG
EE
G
E"
D(F ·G)
G,F
D
ks
B
DB =
DdA·F ·G zzzz
yy
y
z
F
zz
yy
y zzz
|yy DF
/ DA
A
C
dC
dA
C
DdB·G
G
B
F
/ DC
z
zz
zzzzz DG
z
z
y
/ DB
dB
z
DdA·F zzzzz DF
z
y zzz
/ DA
dC
A
dA
8
F. MARMOLEJO AND R.J. WOOD
Observe that the inverse of the 2-cell DG,F is the unique 2-cell ρ : D(F · G) → DF · DG
such that
C
G
B
dC
/ DC
zzz
zzzz DG y zzz
/ DB ρ :: dB
::~ :
DF : DdB·G
G
D(F ·G)
=
B
dB





/ DB
D(F ·G)
PPP { 
P
−1
P
PPP
DdA·F dA PP'
~ / DA.
DdA·F ·G
F
DB
DA
/ DC
dC
C
DF
(using (9)). It is not hard to see that for any γ : G → G′ and φ : F → F ′
′
′
D(F · γ)DG,F = DG ,F (DF · Dγ) and D(φ · G)DG,F = DG,F (Dφ · DG).
Since both 1DA and D(1A ) are right Kan extensions of dA along dA, there is a unique
isomorphism DA : 1DA → D(dA) such that
A
1A
A
dA
/ DA
D(1A )
DdA·1A
DA
tttt
v~ tttt
dA
ks
1DA
= 1dA .
/ DA
It is not hard to see that
DF,1A (DA · DF ) = 1DF = D1B ,F (DF · DB ),
as well as
DG·H,F (DF · DH,G ) = DH,F ·G (DG,F · DH),
therefore D : K → K is a pseudofunctor.
Then we extend d to a strong transformation d : 1K → D by defining dF = DdA·F
for F : B → A (all the relevant equations necessary to show that d is indeed a strong
transformation appear above).
Next we define m : D2 → D such that for every A,
mA = 1DA D ,
and, using (9), define for F : B → A, mF : DF · mB → mA · D2 F as the unique 2-cell
such that
/ D2 B U
2
D
UUU1UDB D
= BCC
{
UUUU
CC 1 D
{
U*
dDB {{{

CC DB
DdDA·DF  D2 F
DF
=
CC
{
DB

{

D

CC
1
{

DB {
mF wwwww
{
!
{ 
w
w
w
/ DB
/
DF
DB
DA dDA D2 A UUUU w wwww
1DB
UUUU
D 1DA U* 1DA D
5 DA
DB
dDB
1DA
(13)
DF
/ DA.
9
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
The inverse of mF is the unique 2-cell θ such that
/ D2 B
EED 1DB ==== 2
EE ==D F
EE ~ ==
1DB EE
mB
=
E" DB= θ D2 A
==ks
==
=
mA
DF ===
DBEE
dDB
dDB
DB

 D2 F



{ 
dDA /
DA KK
D2 A .
KKD 1
KK DA KK mA
~ K
1DA KK
K% DdDA·DF
DF
=
DA
/ D2 B
DA
It is not hard to see that m : D2 → D is a strong transformation.
Now define αA = D1DA −1 , then (13) tells us that α : 1DA → m · dD is a modification.
Define εA : mA · DdA → 1DA as the unique 2-cell such that
A
/ DA
EE
EE
EEDdA
EE
EE
"
εA D 2 A
1DA  dA
=
DA
/ DA

 DdA

{
dDA /
DA KK
D2 A .
KKD 1
KK DA KK mA
~ K
1DA KK
K% dA
mA
&
dA
A
DdDA·dA
DA
The inverse of εA is the unique 2-cell ρ such that
A
dA
DA
/ DA
dA
DdDA·dA ~ DdA
1DA
/ D 2 A ρ = 1dA .
:: dDA
:
~
:
D 1DA mA :: ~ 2 DA
(14)
1DA
It is not hard to show that ε : m · Dd → 1D is a modification by pasting the relevant
equation with dF . Define βA : dDA · mA → 1D2 A as the unique 2-cell such that
DA
dDA
/ D2 A
/ DA
βA 
dDA =

{ 
1DA
, 2
mA
D A
2
D
< AEE
EE
yy
y
dDAyy
EEmA
EE
y D 1DA
y
EE
y
yy
"
/ DA
DA
1DA
dDA /
D2 A.
Finally define ηA : 1D2 A → DdA · mA as the unique 2-cell such that
dDA /
/ D2 A
2
DA
D
9
O
K
9 A.
?
?
s
KKK
s
?
?
KK αA−1
s
βA
s
???
s
KK
K1
1DA sss44
KK mA KKDKK2 A
s 4444 mA ? # ssss
s
=
s
K
s

~

K
4
s 1 2
1DA KK
−1 4
ss
K% { ηA KK%
sss D A
ss εA
/
/
DA DdA D2 A
DA DdA D2 A
DA KK
dDA
10
F. MARMOLEJO AND R.J. WOOD
By Section 2, the 2-cell above is δA : dDA → DdA. It is not hard to see that β and η are
modifications and that they determine, together with α and ε, adjunctions dD ⊣ m ⊣ Dd.
Furthermore, the coherence condition (2) is given by (14).
4.2. Theorem. Every co-lax idempotent pseudomonad D on K induces a right Kan
pseudomonad on K.
Proof. Let D be a co-lax idempotent pseudomonads with structure (1). We then take D
and d on objects for items i) and ii) of Definition 3.1. For item iii) we define F D = mA·DF
and show that
dB /
B
DB





{ 
dF
F
DF
/ D2 A
−1
EαA
EE EE ~ mA
E
1DA EE
E" DAEE
dDA
(15)
DA
exhibits F D as a right Kan extension of F along dB. So take H : DB → DA and ψ : H ·
dB → F . We show that the 2-cell
5 DA NN
1DA
NNN
NNdDA
NNN
NNN
dDB
d−1
αA
N&
H
,
/
2
2
δB D
A
2D B
DH
6
Dψ
DdB
H
DB
mA
'
/ DA
(16)
DF
is the unique 2-cell H → F D that produces ψ when pasted with (15). So paste the above
−1
−1
2-cell with (15), substitute δB · dB by d−1
dB , then substitute the pasting of dH , ddB , Dψ
and dF by dDA · ψ, and cancel αA with its inverse, thus obtaining ψ. Assume now that
we have a 2-cell θ : H → F D such that pasting it with (15) equals ψ. Substitute Dψ in
(16) by D of the pasting of θ with (15). We show that the resulting 2-cell equals θ. For
this replace the pasting of δB and DdF by the pasting of dDF and δDA. Now replace
−1
the pasting of d−1
H , Dθ and dDF by the pasting of θ and dmA . Paste µA and its inverse
at the composite mA · DmA (where µ : m · Dm → m · mD is the pasting (3)). Replace
−1
the pasting of αA, d−1
and DαA−1 by
mA and µA by mDA · αDA, and the pasting of µ
mA · εDA. The pasting of αDA, δDA and εDA is the identity, leaving just θ.
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
11
The proof of a) is similar, given κ : K · dA → dA, the relevant 2-cell to consider is
5 DA NN
1DA
NNN
NNdDA
NNN
NNN
dDA
d−1
αA
N&
K
,
/
2
2
δA D A
2D A
DK
6
DdA
Dκ
εA
K
DA
mA
'
/ DA.
6
DdA
1DA
And the proof of b) is also similar, for a 2-cell ψ : L · dC → mA · DF · G, the relevant
2-cell is
3 DA N
1DA
N
NNN
NNdDA
NNN
NNN
−1
dDC
dL
αA
N&
'
,
/
/ DA.
2
2
δC
DC NN
D
A
2D C
<
<
DL
mA
NNN
yy
yy
y
y
NNN DdC
Dψ
yy
yy
NN
µA
yyyDmA
yymA
DG NNN
y
y
y
N&
y
y
/ D2 A
D2 BKK D2 F / D3 A
mDA pp8
KK
pp
KK
−1
KK
ppp
mF
p
p
K
mB KK
pp DF
K%
ppp
L
DB
We compare these constructions in Section 6 below.
5. D-Alg versus D-Alg
5.1. Theorem. Let D be a right Kan pseudomonad on K, and produce the colax idempotent pseudomonad (also called D) as in Theorem 4.1. There is a 2-equivalence Φ : DAlg → D-Alg such that the diagram
D-AlgE
EE
EE
EE
E"
Φ
K
/ D-Alg
y
yy
yy
y
y
y| y
commutes, where the un-labeled arrows are forgetful 2-functors.
Proof. We define Φ : D-Alg → D-Alg as follows. For τ : H → K : B → A in D-Alg, define
−1
−1
Φ of it as τ : H → K : B−1
1B → A1A . Φ will be a 2-functor if we can show that B1B and H
are in D-Alg. To show the first of these we must show that equation (5) is satisfied, in
12
F. MARMOLEJO AND R.J. WOOD
this case the equation is
1B
δB
d−1 B
*
1B
2
4D B
D(D1B )
DdB
/
;B
ww
w
ww
www B
ww 1B
D−1
1
dB
dDB
DB
1B
3B
B
%
B
/ DB,
7
D1B B
= 11B B ,
1DB
but this follows from the fact that (10), with F = 1B , is a right Kan extension. Again,
since for F = 1B , (12) is a right Kan extension, we obtain the inverse of
H
< B EE
EEdB
1B B yyy
EE
−1
yy Bd
E"
y
1B
y
/
DB
DB
d−1
H
1DB
1A
/A
/A
EE
y<
EE A−1
y
EE 1A yyy
dA E"
yy 1A A
/ DA
DH
as the unique 2-cell γ such that
/ DC
==
EE BF ==
EE ==DH
EE~ B ==
1B EE
1
B
=
E" C EE
dC
B ks
H
γ
B
=
DA
/ DB

 DH
{ 
dA
A EE A / DA
EE 1A 
EE { 
EE
1A A
dC EEE
" H
1 A
A
dB
A
dH
DA
In the opposite direction define Ψ : D-Alg → D-Alg as follows. For an algebra ζ as
in (4), we define Ψ(ζ) such that for every H : X → B, its extension is B · DH, and the
corresponding 2-cell is
/ DX
dH 
H
DH

{ 
dB
B EE −1 / DB
EE ζ EE E ~ B
1B EEE
E" X
dX
(17)
B
To see that Ψ(ζ) is well defined, we must show that (17) exhibits B · DH as a right Kan
extension of H along dX. Given K : DX → B and κ : K · dX → H, the unique 2-cell
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
13
K → B · DH that pasted with (17) equals κ is
DX
5 B NN
1B
NNN
NNNdB
K
NNN
NNN
dDX
d−1
N&
K
,
/ DB
2
δX 2D X
DK
7
DdX
Dκ
ζ
B
&
/ B.
DH
Furthermore, we must show that for any G : Y → DX and H : X → B, the 2-cell
/ DY
dG 
G
DG

{ 
dDX /
DXEE
D2 X
EαX
EE −1
EE ~ mX
E
1DX EE
E" Y
dY
DX
(18)
DH
/ DB
B
/B
exhibits B · DH · mX · DG as a right Kan extension of B · DH · G along dY. Given
N : DY → B and ν : N · dY → B · DH · G, the 2-cell
3 B NN
1B
NNN
NNNdB
N
NNN
NNN
ζ
dDY
d−1
N&
N
,
&
/
/ B,
2
DY NN δY DB
2D Y
<
<
DN
B
y
y
NNN
yy
yy
NNN DdY
yy
Dν
yy
NNN
ζ
y
2
y
yy DB
NNN
DG
yy B
&
yy
yy
/ DB
D2 XKK D2 H / D2 B
8
mB
p
p
KK
pp
KK
p
−1
p
KK
mH
pp
mX KKK
ppppp DH
K%
p
DX
where ζ2 is the 2-cell given by (7), is the unique 2-cell that pasted with (18) equals ν.
We thus conclude that Ψ(ζ) is an object of D-Alg.
Given a 1-cell L : ζ → ξ (with ξ : idC → C · dC) in D-Alg, we want to show that
L : Ψ(ζ) → Ψ(ξ) is a 1-cell in D-Alg. Thus we must show that
/ DX

dH 
H
DH

{ 
dB
B EE −1 / DB
EE ζ EE E ~ B
1B EEE
E" X
dX
B
(19)
L
/C
14
F. MARMOLEJO AND R.J. WOOD
exhibits L · B · DH as right Kan extension of L · H along dX for any H : X → B. Given
N : DX → C and ν : N · dX → L · H, the unique 2-cell N → L · B · DH that pasted with
(19) equals ν is
3 C NN
1C
NNN
NNNdC
NNN
NNN
dDX
d−1
N&
N
,
/ DC
2
δX
DX NN
2D X
DN
jjj5
j
NNN
j
j
j
NNN DdX
Dν
jjjj
N
jjjjjjDL
DH NNNN
j
N&
jjjj
N
DB
B
ξ
&
/ C,
<
C
y
yy
y
y
χ
y
yyy L
yy
/B
where χ is the inverse of the 2-cell induced by L that corresponds to (6), given by the
fact that L is a 1-cell of algebras. Thus we define Ψ(L) = L.
For a 2-cell λ : L → L′ : ζ → ξ in D-Alg, define Ψ(λ) = λ.
It is routine to verify that Φ ◦ Ψ and Ψ ◦ Φ are isomorphic to the corresponding
identities.
Similar arguments produce the following
5.2. Theorem. Given colax idempotent pseudomonad D, produce its associated right Kan
pseudomonad (also called D), as in Theorem 4.2. Then D-Alg and D-Alg are equivalent.
6. Right Kan pseudomonads versus co-lax idempotent pseudomonads 2
If U and D are pseudomonads on the 2-categories L and K respectively then, following
[Marmolejo, 1999] we can describe morphisms from (L, U) to (K, D) in terms of liftings
of 2-functors F : L → K to 2-functors Fb : U-Alg → D-Alg (that commute with the forgetful 2-functors). In Theorem 3.5 of [Marmolejo and Wood, 2008] we showed that such
liftings are essentially the same as transitions from U to D along F . The latter are a
pseudo version of the morphisms of monads found in [Street, 1972] (where they are called
monad functors) and consist of strong transformations r : DF → F U together with two
invertible modifications (corresponding to the two equalities of [Street, 1972]) subject to
two equations. We refer the reader to [Marmolejo and Wood, 2008] for the definitions of
these and of coherent isomorphisms between them.
We consider the particular case of D = (D, d, m, αD , βD , ηD , εD ) and U = (U, u, n, αU , βU ,
ηU , εU ) colax idempotent pseudomonads (with D as in (1), but with subindex D on the
2-cells that conform D, and we use the same letters for the corresponding 2-cells that
conform U, but with subindex U; thus the structure for U is
U?
?
uU
1U
?? αU
??
/U
?

≃ 

 n
U2
1
U2
/ U2
2
U2?
? U ??
?
? U ??
?


??uU
?? ηU 
??n
n 
U u 
??

?

 U u
n ??
 βU  εU ≃ ?
/ U2
/ U.
U
U
U2
1
1
U2
U
(20)
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
15
We follow the same pattern with δ, thus δD : dD → Dd, and δU : uU → U u). It follows from
the previous Section that morphisms of monads between them and hence also transitions,
can be described in terms of algebras for the corresponding right Kan pseudomonads.
6.1. Theorem. Let U and D be colax idempotent pseudomonads on 2-categories L and
K respectively. A transition from U to D along a 2-functor F : L → K can be given
by the following data: for every A in L, a D-algebra (F U A, ( )λ ), such that for every
L : B → U A in L,
F (LU ) : (F U B, ( )λ ) → (F U A, ( )λ )
is a morphism of D-algebras. Every transition from U to D along F is coherently isomorphic to one that arises in this way.
Proof. For every A in L define rA = (F uA)λ and ω1 A = λF uA :
/ DF A
ω1 A= λF uA (F uA)λ = rA
} dF A
FA
F uA
*
F UA
To make r a strong transformation observe that, for every G : B → A, rA · dF G =
(F uA)λ · DdF A·F G exhibits rA · DF G as a right Kan extension of rA · dF A · F G along
dF B. Thus we define rG as the unique 2-cell such that
/ DF B
BB
BB
BB rB
BB
FG
DF
G
BB
dF G BB
!
/
F AHH dF A DF A r
FUB
HH
ks G |||
HH ω1 A ||
HH ~ ||
H
|
H
rA
F uA
| F UG
HH
H# }|||
FB
dF B
dF B /
DF B
HH
HH ω1 B HH rB
H F uB H~ H
HH
# F B HH
FG
=
F AHH
DF A
HH F uG HH HH ~ H
F uA HH
HH
# FUA
F UG
F U A.
The inverse of rG is the unique 2-cell θ (given by the fact that F U G · ω1 B = F ((uA ·
G)U ) · λF uB exhibits F U G · rB as a right Kan extension of F U G · F uB along dB) such
that
/ DF B
BB
HH ω B 
BB
HH 1 
BBDF G
HH { 
BB
rB
H
BB
F uB HH
BB
HH
!
# F B HH
dF B
FUB
F UG
DF A
||
||
|
||
|| rA
|
}|
|
ks
FUA
θ
F BB
=
BB
BB
BB
B
F G BBB
B!
F uB
FUB
/ DF B
dF B
DF G
} F AHH dF A / DF A
HH ω A HH 1 HH ~ F u−1
rA
G H
F uA HH
HH
~
# / F U A.
dF G
F UG
16
F. MARMOLEJO AND R.J. WOOD
It is routine to verify that r : DF → F U is a strong transformation and the equation
defining rG above tells us that ω1 : r · dF → F u is a modification.
To define ω2 A we observe that rA · αF A−1 = (F uA)λ · D1DF A exhibits rA · mF A as
a right Kan extension of rA · mF A along dDF A, thus we can define ω2 A as the unique
2-cell such that
dDF A /
D2 F A
::
::
::
:: α F A−1

:: D
:: 

mF A
1DF A ::: { 
::
::
::
::
DF A
:
DrA
/ DF U A
/ D2 F A
drA zzzz
rA
DrA
zzz
y zzz
dF U A /
F U A QQ
DF U A
QQQ ω1 U Aqq
QQQ qqq
rU A
Q t|
F uU A QQQ(
F αU A−1vv F U 2 A
vvvv
v~ vvvv
1F U A
F nA
,
rU A
=
F U 2A
} ω2 A
DF A
rA
dDF A
DF A
F nA
/ FUA
FUA
(This is the equation in Theorem 2.3 of [Marmolejo and Wood, 2008].) To induce the
inverse of ω2 A we observe first that rU A · DrA ∼
= (F uU A · rA)λ : in one direction take
the unique 2-cell χ such that
dDF A
DF A
rA
dF G
/ D2 F A
/ D2 F A
λF uU A·rA (F uU A·rA)λ
rA
F U A F uU A / F U 2 A,
dDF A
DF A
DrA
/ DF U A
A
χ
HHdFωU1 U
HH A ks
HH { 
H A
F uU A HHrU
HH
# |
(F uU A·rA)λ
F U AHH
=
F U 2A
while in the other take the unique 2-cell π such that
DF A
/ D2 F A
::
::
::DrA
::
::
λ
(F uU A·rA)
dDF A
rA
λF uU A·rA } F UA
F uU A
DF U A
rU A
ks
π
/ F U 2A
DF A
rA
=
dDF A
/ D2 F A
DrA
zzz
zzzz
y zzz
/ DF U A
dF G
F U AHH
HHdF U A
HHω1 U A 
HH 
H
F uU A HH{ H
H# rU A
F U 2 A.
The isomorphism χ just exhibited and the equality F nA = F ((1U A )U ) show that the
−1
pasting of drA , ω1 U A and F αA
exhibits F nA · rU A · DrA as a right Kan extension of
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
17
rA along dDF A. Thus, the inverse of ω2 A is the unique 2-cell θ such that
/ D2 F A
zzz
rA
zzzzz DrA
z
z
y
dF U A /
F U A QQ
DF U A
QQQ ω1 U Aqq
QQQ qqq
rU A
Q t|
ks
F uU A QQQ(
−1
2
F αU A vv F U A
vvvv
v~ vvvv
1F U A
F nAu
,
DF A
dDF A
/ D2 F A
NNN αD F A−1vvv
NNN vvvvv
N v~ vv
mF A
1DF A NNNN
N& DF ANN
mF A
drA
DF A
θ
=
dDF A
DF A
rA
rA
F U A.
FUA
We need to verify that ω2 : F n · rU · Dr → r · mF is a modification. Given any G : B → A,
one shows that the pasting of dDF G and αD F A−1 followed by rA exhibits rA·mF A·D2 F G
as a right Kan extension of rA · DF G along dF B. Then one has to prove that the two
pastings that need to be equal to show that ω2 is a modification, are equal when preceded
by dDF B and pasted with dDF G and αD F A−1 .
The coherence conditions in Definition 2.1 of [Marmolejo and Wood, 2008] remain to
be shown. The first ends in r, so it suffices to show that both pastings are equal when
preceded by dF and pasted with ω1 . The following commutative diagram shows this:
ω2 ·DdF ·dF
F n·rU ·Dr·DdF ·dF
W
WWWWW
WFWWn·rU
dF
WWW·Dr·d
WW+
/
F n·rU ·dr
F n·rU ·DF u·dF
−1
F n·ru
·dF
F n·rU ·dF u
F n·F U u·r·dF
F n·F U u·ω1
/
ω2 ·dDF ·dF
F n·rU ·dF U ·r·dF
F n·rU ·dF U ·ω1
OOO
OOO
OOOr·εD F ·dF
O
/ r·mF ·dDF ·dF
WWWWW OOOOO
WWWWW OOO
WWWWW OO
r·αD F −1 ·dF
WW+ '
/ r·dF
/ F n·F uU ·r·dF
r·mF ·ddF
F n·rU ·Dr·dDF ·dF
F n·rU ·Dω1 ·dF
r·mF ·DdFO·dF
F n·ω1 U ·r·dF
F n·F uU ·ω1
α−1
U ·r·dF
/ F n·F uU ·F u
ω1
WWWWWα−1 ·F u
3
g
ggg
UW
g
W
g
W
g
g
WWWWW
gg
WWWW+ ggggg F n·F uu
/ F n·F U u·F u
/ F u.
F n·rU ·dF U ·F u
F n·ω1 U ·F u
F εU ·F u
For the other condition, observe first that, for every A, rA · mF A · αD DF A−1 exhibits
rA·mF A·mDF A as a right Kan extension of rA·mF A along dD2 F A. It then suffices to
show that both pastings are the same when preceded by dD2 F and pasted with αD DF −1 .
18
F. MARMOLEJO AND R.J. WOOD
The following commutative diagram shows that these are equal:
r·µ F ·dD 2 F
D
/ r·mF ·mDF ·dD2 F
\\\\\\\\\\\\
$$
\\\\\\\\\r·mF
·dmF
\\\\\\\\\\\\
\\\\\\\\\\\\
r·mF ·αD DF −1 $$
\\\\.
$
ω2 ·DmF ·dD 2 F
·mF $$
c1 r·mF ·dDF
c
c
c
,
c
c
c
c
c
c
ccω ·dDF ·mF
,,
$$
F n·rU ·Dr·dmF
aaaa0 F n·rU ·Dr·dDF
,,
LLL·mF 2
$$
aaaaaaaaaaaaaa
a
a
LLL
2F
r·αD F −1 ·mF ,,
$$
F n·rU ·Dr·DmF
·dD
LLL
O
,,
$$
LLFLn·rU ·dr ·mF
,,
L%
$
F n·rU ·DF n·DrU
·dDF
U
·Dr
QQQ
G
,, $$
QFQQn·rU ·DF n·drU ·Dr
F n·rU8 ·dF UA·r·mF
,
p
(
AA
p
,, $$
F n·rU ·dF U ·ω2 pp
AA
p
F n·rU ·DF n·dF UOO2 ·rU ·Dr
p
AAF n·ω1 U ·r·mF ,, $$
OOFOn·rU ·dF n ·rU ·Dr pppp
AA
,, $$
p
O
p
OOO
F n·rU ·
,, $$
'
ppp
F n·F <uU ·r·mF
P
−1
,$
P
2
2
F
n·rU
·dF
U
·F
n·rU
·Dr
Dω2 ·dD F
PPP
F n·rn ·dF U ·rU ·Dr
MMM
yy
F n·F uU ·ω
PPP ,, $$
MMMF n·ω U ·F n·rU ·Dr y2yy
F n·rU ·DF n·DrU ·dDr
P , $
F α−1 ·r·mF P'
MMM 1
yy
U
y
M
y
r·mF
MM&
:
yy
uu H
F n·F U n·rU 2 ·dFFU 2 ·rU ·Dr
u
F
n·F
uU
·F
n·rU
·Dr
u
F
u
9
=
F
== F α−1 ·F n·rU ·Dr uu
FF F n·F U n·ω U 2 ·rU ·Dr rrr
u
FF
1
r
== U
u
F n·rU ·DF n·DrU ·D2 r·dD2 F
r
uω
r
FF
==
uu 2
rr
−1 ·DrU ·D 2 r·dD 2 F
FF
F n·rn
==
uu
rrF n·F un ·rU ·Dr
F#
r
u
r
r·mF
u ·αD DF −1 F µU ·rU 2 ·dF U 2 ·rU ·Dr
F n·rU ·Dr
F n·F U n·F uUJ2 ·rU ·Dr
JFJ n·F α U −1 ·rU ·Dr }}> P ] r·mF ·mDF ·dD2 F
JJ U
P!!
F n·F U n·rU 2 ·DrU ·D 2 r·dD2 F
}}
JJ
JJ
!!
}}
}
F µU ·rU 2 ·DrU ·D 2 r·dD 2 F
2
J
F µU ·F uU ·rU ·Dr J
}
J%
!!
}}
/ F n·F nU ·F uU 2 ·rU ·Dr
2
!!
F n·F nU ·rU 2 ·dF
·Dr
O U ·rU
F n·F nU ·ω1 U 2 ·rU ·Dr
!!
2
2
2
F n·F nU ·rU ·DrU ·DV r·dD F
VVVV
F n·rU ·αD F U −1 ·Dr
!! ω2 ·mDF ·
F n·F nU ·rU 2 ·drU ·Dr
VVVV
VVVV
!!
2
F n·F nU ·rU ·DrU ·dDr V*
!! dD2 F
/
2 ·DrU ·dDF U ·Dr
F
n·rU
·mF
U
·dDF
U
·Dr
F
n·F
nU
·rU
F n·ω2 U ·
1
d
F n·ω2 U ·dDF U ·Dr
!!
d
dddd
ddddddd
d
d
F n·rU ·Dr·αD DF −1 !
d
d
2
d
d
Dr·dD F
!!
ddd
dddddFdn·rU
·mF U ·dDr
ddddddd
!
d
d
d
d
d
d
d
/
2
2
F n·rU ·mF U ·D r·dD F
F n·rU ·Dr·mDF ·dD2 F.
r·mF ·DmF
·dD2 F
O
2
F n·rU ·m−1
r ·dD F
Assume now that we have a transition (r, ω1 , ω2 ) from U to D along F . Consider the
composite
U-Alg
Fb
/ D-Alg
Ψ
/ D-Alg,
where Fb : U-Alg → D-Alg is the lifting of F determined by the transition (r, ω1 , ω2 ) as in
Proposition 2.2 of [Marmolejo and Wood, 2008], and Ψ : D-Alg → D-Alg was defined in
the proof of Theorem 5.1. If we apply this composite to the free U-algebra αU A, A in
L, we obtain the following U-algebra structure on F U A: for H : X → F U A, H λ is the
composite
DX
DH
/ DF U A
rU A
/ F U 2A
DnA
/ FUA
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
19
and λH is the pasting
/ DX


H
DH
{ 
dF U A /
F U AMM
DF U A
MMMω1 U A 
MMM { 
rU A
F uU A MMM&
X
dX
dH
F αU A−1
(21)
F U 2A

 F nA
{ 
* F U A.
1F U A
Furthermore, since for every L : B → U A we have that LU : αU B → αU A is a morphism of U-algebras, the same composite of functors tells us that F (LU ) : (F U B, ( )λ ) →
(F U A, ( )λ ) is a D-algebra morphism. According to the first part of this proof, the ω
c1 of
the induced transition from U to D along F is
/ DF A
dF uA 

F uA
DF uA
{ 
dF U A /
F U AMM
DF U A
MMMω1 U A 
MMM { 
rU A
F uU A MMM&
FA
dF A
F αU A−1
F U 2A

 F nA
{ 
* F U A,
1F U A
with its corresponding ω
c2 , and the invertible modification that makes this and (r, ω1 , ω2 )
coherently isomorphic is given by
1F U
ll5 F U RRRRR
RFRRU u
−1
RRR
ru
RR) F εU / FU2
/ DF U
r lllll
DF
l
lll
lll
DF u
rU
(
Fn
/ F U.
This completes the proof.
7. Distributive laws
In this section we deal with distributive laws. We treat the particular case of a distributive
law of a colax idempotent pseudomonad over a lax idempotent pseudomonad, but observe
that the other cases are similar. We point out that the composite pseudomonad resulting
from a distributive law of a colax idempotent pseudomonad over another colax idempotent
pseudomonad turns out to be colax idempotent (see Theorem 11.7 in [Marmolejo, 1999]).
We begin with the following
20
F. MARMOLEJO AND R.J. WOOD
7.1. Lemma. Let D be a pseudomonad on K and let U be a colax idempotent monad (as
in (20)) on K. If there is a distributive law of U over D, then
(i) For every A, d−1
uA exhibits dU A as a right Kan extension of DuA · dA along uA.
(ii) For every L : B → U A in K,
uB /
UB
EE UL EE EE LU
L EE
E" B EE
UA
dU A
DU A
exhibits dU A · LU as a right Kan extension of dU A · L along uB.
Proof. Let (r, ω1 , ω2 , ω3 , ω4 ) be a distributive law of U over D as in Proposition 5.1 in
[Marmolejo and Wood, 2008]. Actually, the only part of the structure for a distributive
law that we need to prove this is r, w1 , w2 and the coherence condition (10) of that article.
For (i) let H : U A → DU A and θ : H · uA → DuA · dA, then the unique 2-cell H → dU A
that pasted with d−1
uA is θ is given by the pasting
UA
1DU A
3 DU A
EE
E
H
EE
EE
E ω U A−1 DuU A
u−1
H
DαU A
uU A
uDU A EEE1"
 (
+
/
/
2
2
δU A U
DU
A
DU
A
3U A
U H ll5
rU A pp8
lll
U uA
l
Uθ
−1
ppp DεU A l
l
ruA
lll U DuA
ppp
U dA
p
p
,
p DU uA
U DA [[[[[[[[[[[ppp
1DU A
rA
DU A
ω2 A
4
DnA
'
/ DU A.
6
dU A
For (ii) let M : U B → DU A and λ : M · uB → dU A · L, then the unique 2-cell M →
dU A · LU that pasted with dU A · UL is λ is given by the pasting
UB
3 DU A
EE
EE
M
EE
EE ω U A−1 −1
uM
uU B
uDU A EEE1"
 +
/ U DU A
δU B 3 U 2B
U M lll5
l ω2 U A
U uB
Uλ
lll
llllU
dU A
UL
l
2A
, 2
dU
U A VVVVVV
VVVV
VV+
σ
nA
1 UA
U
L
1DU A
DuU A
Dα A
rU A
U
(
/ DU 2 A
5
dnA
DnA
'
/ DU A,
8
dU A
where σ : nA · U L → LU is the unique 2-cell that pasted with UL equals the pasting of uL
and αU A.
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
21
For the next theorem we take U a colax idempotent pseudomonad with structure as
in (20), but D is now a lax idempotent pseudomonad. We take the data for D as follows:
D?
1D
?? ηD
??
?
Dd
/D
?


≃

 m
D2
1
D2
/ D2
2
D2 ?
? D ??
?
? D ??
?


?

α
?? D 
??m
m
dD 
??Dd

?

 dD
?
m ??
 εD  βD ≃ ?
/ D2
/D
D
D
D2
1
1
D2
D
so that Dd ⊣ m ⊣ dD.
7.2. Theorem. Assume that U is a colax idempotent monad on K, and that D is a lax
idempotent monad on K such that the conditions (i) and (ii) of Lemma 7.1 are satisfied.
Then a distributive law of U over D can be given by the following data:
(iii) For every A in K, a U-algebra structure (DU A, ( )λ ),
such that the following two conditions are satisfied:
(iv) For every L : B → U A, D(LU ) : (DU B, ( )λ ) → (DU A, ( )λ ) is 1-cell of U-algebras.
(v) For every H : C → DU A, (H λ )D : (DU C, ( )λ ) → (DU A, ( )λ ) is an algebra morphism.
Proof. According to Theorem 6.1 we get a transition from U to U along D if we define,
for every A in K, rA = (DuA)λ , define ω1 A as the 2-cell λDuA :
/ U DA
II λDuA 
II
II {  (DuA)λ
DuA III
$ DAII
uA
DU A,
define rG , for G : B → A, as the unique 2-cell such that
/ U DB
BB
BB
BB rB
BB
DG
BB
uDG U DG
BB
!
DAHH uDA / U DA r
DU B
HH
ks G |||
HH ω1 A ||
HH ~ ||
H
|
F
DuA HH rA
|| U G
HH
# }||
DB
uDB
DU A
/ U DB
HH
HH ω1 B HH rB
H DuB H~ H
HH
# DBHH
DG
=
DAHH
uDB
U DA
HH DuG HH
HH ~ H
DuA HHH
H# DU G
DU A,
22
F. MARMOLEJO AND R.J. WOOD
and define ω3 A as the unique 2-cell such that
uU DA /
U 2 DA
::
::
::
:: α DA−1

::U
:: 

nDA

:
1U DA :: { 
::
::
::
::
U DA
:
U DA
U rA
/ U DU A
/ U 2 DA
zzz
rA
zzzzz U rA
z
z
y
uDU A /
DU A QQ
U DU A
QQQ ω1 U Aqq
QQQ qqq
rU A
Q t|
DuU A QQQ(
DαU A−1vv DU 2 A
vvvv
v~ vvvv
1DU A
DnA
,
uU DA
U DA
urA
rU A
=
DU 2 A
} ω3 A
rA
DnA
/ DU A
DU A
We define ω2 A as the unique 2-cell such that such that
uA
A
/ UA
BB
BB
BB U dA
BB
BB
BB
dU A
!
A
DA
zzz
zzzz U dA
y zzz
uDA /
DAHH
U DA
HH ω1 A vvvv
HH vvvvv
HHv~ v
rA
H
DuA HH
HH
# udA
dA
U DA
dA
sk ω2 A |||

d−1

||

uA 

||

|

rA

|
{ 
|
}||
/ DU A
/ UA
uA
=
DU A.
DuA
Then ω2 is an invertible modification. Now we define ω4 A as the unique 2-cell such that
U D2 APP
n7
PPrDA
PPP
n
n
n
PP'
nn
2
U mA
DU DA
D A
umA zz
ω
A
z
4
z
DrA
zzz
~ y zzz
mA
U DAPP
D2 U A
PPPrA
nn7
n
n
PPP
nnn
PP' mU A
ω1 A
nnn uDA
/ DU A
DA
uD2 Annn
U D2 APP
n7
PPrDA
PPP
PP'
ω1 DA
/ DU DA
DuDA
Dω1 A DrA
~ .
2
D uA
D2 U A

m−1
uA 

mU A

{ 
/ DU A.
uD2 Annn
D2 A
nn
nnn
=
DuA
mA
DA
DuA
One induces the inverse of ω4 A using the fact that
U D2 APP
n7
PPrDA
PPP
ω1 DA
PP'
/ DU DA
uD2 Annn
D2 A
nn
nnn
DuDA
DrA
/ D2 U A
mU A
/ DU A
exhibits mU A · DrA · rDA as a right Kan extension of mU A · DrA · DuDA along uD2 A;
the proof that it is indeed a right Kan extension follows from the fact that mU A · DrA ≃
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
23
(rA)D = ((DuA)λ )D : (DU DA, ( )λ ) → (DU A, ( )λ ) is a 1-cell of U-algebras. To show
that ω4 is a modification, one shows that for every G : B → A,
U D2 BOO
7
uD2 A
ooo
2
ooo
D B OO
OOO
O
D2 G '
U
OOD2 G
uD2 G
OO'
U D2 AOO
o7
OUOOmA
o
o
o 2
OO'
oouD
A
2
U
D A OO umA
7 DAOOOrA
uDA
OOO
OOO
ooo
o
OO'
o
O'
ω1 A
oo
mA
/ DU A
DA
DuA
exhibits rA · U mA · U D2 G as a right Kan extension of DuA · mA · D2 G, since rA · U mA ·
U D2 G ≃ (DuA · mA · D2 G)λ .
Next we show that (r, ω2 , ω4 ) is an op-transition from D to D along U . Coherence
condition (7) of [Marmolejo and Wood, 2008] follows from the fact that ω1 A exhibits rA
as a right Kan extension of DuA along uDA, using the defining equation of ω2 A. And
coherence condition (8) of [Marmolejo and Wood, 2008] follows from the fact that
U D3 AOO
7
uD3 A
ooo
ooo
D3 A OO
OOO
O
DmA '
U
OODmA
OO'
U D2 AOO
o7
OUOOmA
o
o
o 2
OO'
oouD
A
2
U DAOO
D A OO umA
o7
OOrA
uDA
OOO
OOO
ooo
OO'
o
o
ω1 A
'
o
mA
/ DU A
DA
uDmA
DuA
exhibits rA · U mA · U DmA as a right Kan extension of DuA · mA · DmA along uD3 A,
this because rA · U mA · U DmA ≃ (DuA · mA · DmA)λ .
Thus we have a transition (r, ω1 , ω3 ) from U to U along D and an op-transition
(r, ω2 , ω4 ) from D to D along U . We are left with the verification that the coherence
conditions of Proposition 5.1 of [Marmolejo and Wood, 2008] are satisfied. Condition
(10) of that paper is the defining equation of ω2 , while (11) of the same paper follows
from the fact that dU A · αU−1 exhibits dU A · nA as a right Kan extension of dU A along
uU A. And (12) of that paper is the defining equation for ω4 , leaving us only with coherence condition (13) of the same paper. This coherence condition follows from the fact
that rA · U mA · αU DA exhibits rA · U mA · nD2 A as a right Kan extension of rA · U mA
along uU D2 A (since rA · U mA · nD2 A ≃ (rA · U mA)λ ).
We must now show that every distributive law of U over D, with U colax idempotent
and D lax idempotent, arises essentially in this way. Let (r, ω1 , ω2 , ω3 , ω4 ) be a distributive
law of U over D. Then we have that conditions (i) and (ii) of Lema 7.1 are satisfied, and we
must obtain conditions (iii), (iv) and (v) of Theorem 7.2, and show that the distributive
law obtained from Theorem 7.2 is essentially the distributive law (r, ω1 , ω2 , ω3 , ω4 ).
24
F. MARMOLEJO AND R.J. WOOD
Observe that (D, ω1 , ω3 ) is a transition from U to U along D. Then Theorem 6.1 gives
us the U-algebra structure on DU A corresponding to (21), which in this case assigns to
an H : X → DU A the right Kan extension
/ UX
uH 

H
UH
{ 
uDU A/
DU AMM
U DU A
MMMω1 U A 
MMM { 
rU A
DuU A MMM&
X
uX
DαU A−1
DU 2 A

 DnA
{ 
* DU A,
1DU A
and for every L : B → U A, D(LU ) is a 1-cell of U-algebras. This gives us conditions (iii)
and (iv) of Theorem 7.2.
We are left with showing that, for any H : C → DU A, (H λ )D : (DU C, (−)λ ) →
(DU A, (−)λ ) is a U-algebra morphism. To do this we observe that (DU C, (−)λ ) and
(DU A, (−)λ ) are the images under the 2-functor Ψ : U-Alg→ U-Alg of the U-algebras
given by
1DU C
/ DU C
y<
>>
UUUU
y
>>
UUUU DαU C yy y
yy DnC
>> DuU C UUU*
>>
2
>
−1
DU
C
<
uDU C >>ω1 U C
>>
yyyy
>>
y
yy rU C
DU C
> UUUU
and
U DU C
1DU A
/ DU A
y<
>>
UUUU
y
>>
UUUU DαU A yy y
yy DnA
>> DuU A UUU*
>>
2
>
−1
DU
A
<
uDU A >>ω1 U A
>>
yyyy
>>
y
yy rU A
DU A
> UUUU
U DU A
respectively (these in turn are the images of the free algebras αU C and αU A under the
lifting U-Alg → U-Alg induced by the transition (D, ω1 , ω3 )), thus it suffices to show that
(H λ )D = DU C
DU H
/ DU DU A
DrU A
/ D2 U 2 A
D2 nA
/ D2 U A
mU A
/ DU A
is a 1-cell between these latter U-algebras. According to (6), we must show that
DnA · rU A · U mU A · U D2 nA · U DrU A · U DU H · U DnC · U rU C · δU DU C
is invertible; and one uses the available isomorphisms to produce nDU C just after δU DU C
to conclude that the above 2-cell is indeed invertible. One then applies the construction
given in Theorem 7.2 to produce a new distributive law (s, π1 , π2 , π3 , π4 ). The claim is
that the original distributive law (r, ω1 , ω2 , ω3 , ω4 ) is coherently isomorphic to this new
one in the following sense:
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
25
7.3. Definition. Let U and D be pseudomonads on K, and let (r, ω1 , ω2 , ω3 , ω4 ) and
(s, π1 , π2 , π3 , π4 ) be distributive laws of U over D. We say that the distributive laws
are coherently isomorphic if there is an invertible α : r → s that makes the transitions
(r, ω1 , ω3 ) and (s, π1 , π3 ) coherently isomorphic, and makes the op-transitions (r, ω2 , ω4 )
and (s, π2 , π4 ) coherently isomorphic.
7.4. Theorem. Let U be a colax idempotent monad, D a lax idempotent monad on K,
and (r, ω1 , ω2 , ω3 , ω4 ) a distributive law of U over D. If (s, π1 , π2 , π3 , π4 ) is the distributive
law produced just before Definition 7.3, then (r, ω1 , ω2 , ω3 , ω4 ) and (s, π1 , π2 , π3 , π4 ) are
coherently isomorphic distributive laws.
Proof. Theorem 6.1 already gives us (r, ω1 , ω3 ) and (s, π1 , π3 ) coherently isomorphic by
the 2-cell
1DU
ll5 DU RRRRR
RDU
rlllll
u
RRR
ru
l
(22)
RRR Dε−1
U lll
UD
ll
U Du
/ U DU
R)
/ DU 2
rU
Dn
(
/ DU.
We must show that it also makes (r, ω2 , ω4 ) and (s, π2 , π4 ) coherently isomorphic. We
have that sA = DnA · rU A · U DuA and, π1 is the pasting
/ U DA


DuA
U DuA
{ 
uDU A/
DU AMM
U DU A
MMMω1 U A 
MMM { 
rU A
DuU A MMM&
DA
uDA
uDuA
DαU A−1
1DU A
DU 2 A

 DnA
{ 
* DU A.
To show that (22) at A pasted with π2 equals ω2 we use the fact that d−1
uA exhibits dU A
as a right Kan extension of DuA · dA along uA and the defining equation of π2 , namely
A
/ UA
BB
BB
BB U dA
BB
BB
BB
dU A
!
uA
U DA
dA
sk π2 A |||


||


||

|

DnA·rU
A·U DuA

{ 
||
}||
/ DU A
d−1
uA
DA
A
/ UA
zzz
zzzz U dA
y zzz
uDA /
DAHH
U DA
HH π1 A vvvv
HH vvvvv
HHv~ v
DnA·rU A·U DuA
H
DuA HH
HH
# dA
=
uA
DuA
The case for ω3 and π3 is similar to the one just shown.
udA
DU A.
26
F. MARMOLEJO AND R.J. WOOD
7.5. Remark. Of course we still have not shown that Definition 7.3 is good, in the sense
that the structures induced (liftings, composite pseudomonads, coherent structures) are
essentially the same for two coherently isomorphic distributive laws. However, this would
take us too far from the objectives of the present paper. We defer the treatment of this
issue to a paper that will deal with the “no-iteration” version of the algebras for a general
pseudomonad, and the corresponding version of a distributive law.
8. Example
Let U be coFam on Cat. That is, U : Ob(Cat) → Ob(Cat) is given as follows. For a
category A, the objects of U A are finite families ⟨Ai ⟩i∈I of objects of A. A morphism
⟨Ai ⟩i∈I → ⟨Bj ⟩j∈J in U A consists of a function φ : J → I together with a family of
morphisms ⟨fj : Aφ(j) → Bj ⟩j∈J in A. The identity on ⟨Ai ⟩i∈I is (1I , ⟨1Ai ⟩i∈I ), whereas
composition of (φ, ⟨fj ⟩j∈J ) : ⟨Ai ⟩i∈I → ⟨Bj ⟩j∈J and (ψ, ⟨gk ⟩k∈K ) : ⟨Bj ⟩j∈J → ⟨Ck ⟩k∈K is
(φψ, ⟨gk · fψ(k) ⟩k∈K ).
The functor uA : A → U A sends an object A to the family with exactly one element
⟨A⟩{∗} , and f : A → B to (1{∗} , ⟨f ⟩{∗} ).
We observe that U A has finite products. Given a finite set I, and for every i ∈ I an
element ⟨Aij ⟩j∈Ji in U A, then
∏
⟨Aij ⟩j∈Ji = ⟨Aij ⟩(i,j)∈⨿i∈I Ji ,
i∈I
with the i-th projection given by
⨿
(σi : Ji → i∈I Ji , ⟨1Aij ⟩j∈Ji ) : ⟨Aij ⟩(i,j)∈⨿i∈I Ji → ⟨Aij ⟩j∈Ji .
∏
Given a functor F : B → U A, F U : U B → U A is such that F (⟨Bj ⟩j∈J ) = j∈J F Bj ,
and given a morphism (γ, ⟨gj ⟩j∈J ) : ⟨Ck ⟩k∈K → ⟨Bj ⟩j∈J , define F U on it such that the
diagram
F U (⟨Ck ⟩k∈K )
F U ((γ,⟨gj ⟩j∈J ))
F U (⟨Bj ⟩j∈J )
πγ(j)
πj
commutes for every j ∈ J. Then the diagram
uB /
UB
II
II
I
FU
F III
$ B II
UA
/ F Cγ(j)
gj
/ F Bj
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
27
commutes (provided we make the convention that a unary product is simply the object
involved). And it exhibits F U as a right Kan extension of F along uB. Indeed, given
uB /
UB
II θ II II  H
F II$ B II
U A,
then the unique natural transformation θb : H → F U that preceded by uA is θ, is given,
at ⟨Bj ⟩j∈J , by the morphism that makes the diagram
H(⟨Bj ⟩j∈J )
b j ⟩j∈J
θ⟨B
H((pjq,⟨1Bj ⟩{∗} ))
F U (⟨Bj ⟩j∈J )
/ H(⟨Bj ⟩{∗} )
F U ((pjq,⟨1Bj ⟩{∗} ))
θBj
/ F U (⟨B ⟩
j j∈J )
commute for all j ∈ J.
U
It is a routine
∏exercise to verify that F : U B → U A preserves finite products, and
that ⟨Bj ⟩j∈J = j∈J ⟨Bj ⟩{∗} in U B. Then we can verify condition b) of Theorem 3.1.
Indeed, given G : C → U B, H : U C → U A and θ : H · uC → F U · G, then the unique
natural transformation θb : H → F U · GU that preceded by uC is θ is given, at ⟨Ci ⟩i∈I in
U C, by the unique arrow that makes the diagram
H(⟨Ci ⟩i∈I )
b i ⟩i∈I
θ⟨C
H(πi )
F U GU (⟨Ci ⟩i∈I )
/ H(⟨Ci ⟩{∗} )
F U GU (πi )
θCi
/ F U GU (⟨Ci ⟩{∗} )
commute for all i ∈ I.
We have shown, using the techniques of this paper, that U is a colax idempotent
monad. It is well known that the algebras for U are categories with finite products and
functors that preserve finite products.
Dually, as D we take Fam. Thus DA = (U (Aop ))op , and the rest of the structure can
be read from this from the description of U. Of course, D is a lax idempotent monad.
It is well known that there is a distributive law of U over D; the main ingredient being
the fact that if A has finite products then FamA also has finite products. Here we verify
the conditions of this paper.
We observe first that DU A has finite products. Indeed, given a finite set I, and for
every i ∈ I an element ⟨⟨Aijk ⟩k∈Kij ⟩j∈Ji in DU A, then the product of the family is given
by the object
⟨⟨Ait(i)k ⟩k∈⨿i∈I Ki,t(i) ⟩t∈∏i∈I Ji ,
28
F. MARMOLEJO AND R.J. WOOD
∏
with∏the i-th projection given by the projection πi : i∈I Ji → Ji together with, for every
t ∈ i∈I Ji , the morphism
⨿
(σi : Kit(i) → i∈I Kit(i) , ⟨1Ait(i)k ⟩k∈Kit(i) ) : ⟨Ait(i)k ⟩k∈⨿i∈I Ki,t(i) → ⟨Aijk ⟩k∈Kij .
It is not hard to verify that the conditions of Lemma 7.1 are satisfied. Indeed, to see
that d−1
uA exhibits dU A as a right Kan extension of DuA · dA along uA, take θ : H · uA →
DuA · dA, then the unique 2-cell θb : H → dU A that pasted with d−1
uA produces θ is
constructed as follows. Given ⟨Ai ⟩i∈I in U A, we observe that
∏
⟨⟨Ai ⟩{∗} ⟩{∗} = ⟨⟨Ai ⟩i∈I ⟩{∗}
i∈I
b i ⟩i∈I : H(⟨Ai ⟩i∈I ) → dU A(⟨Ai ⟩i∈I ) = ⟨⟨Ai ⟩i∈I ⟩{∗} is the unique arrow
in DU A. Thus θ⟨A
such that the diagram
H(⟨Ai ⟩i∈I )
b i ⟩i∈I
θ⟨A
H((piq,⟨Ai ⟩{∗} ))
⟨⟨Ai ⟩i∈I ⟩{∗}
/ H(⟨Ai ⟩{∗} )
πi
θAi
/ ⟨⟨Ai ⟩{∗} ⟩{∗}
commutes.
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Instituto de Matemáticas, Universidad Nacional Autónoma de México
Area de la Investigación Cientı́fica, Circuito Exterior, Ciudad Universitaria
Coyoacán 04510, México, D.F. México
Department of Mathematics and Statistics, Dalhousie University
Chase Building, Halifax, Nova Scotia, Canada B3H 3J5
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rjwood@mathstat.dal.ca
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