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Theory and Applications of Categories, Vol. 1, No. 6, 1995, pp. 119{145.

DISTRIBUTIVE ADJOINT STRINGS
ROBERT ROSEBRUGH AND R. J. WOOD
Transmitted by Michael Barr

ABSTRACT. For an adjoint string V a W a X a Y : B ;! C, with Y fully
faithful, it is frequently, but not always, the case that the composite V Y underlies an
idempotent monad. When it does, we call the string distributive. We also study shorter
and longer `distributive' adjoint strings and how to generate them. These provide a new
construction of the simplicial 2-category, .

1. Introduction

Consider a string of adjoint functors, V a W a X a Y : B ;! C, with Y fully faithful.
The composite V Y is a well-pointed endofunctor so that it is natural to ask whether it
underlies an idempotent monad on B. Somewhat surprisingly, in light of the examples
that come readily to mind, this is an additional property for a string of adjoint functors.
If the string above has also Y a Z then it is equivalent to ask whether the composite
ZW underlies an idempotent comonad. Since the question makes sense in any bicategory
and any functor Y has a right adjoint in the larger bicategory of profunctors, it follows that

the question can be asked for a shorter string of adjoint functors, W a X a Y : B ;! C,
with Y fully faithful, the situation that Lawvere 8] refers to as a unity and identity of
adjoint opposites and abbreviates by UIAO.
In fact, these observations allow us to ask our question for a UIAO in a 2-category
with proarrow equipment.
We begin with a section of examples and a counterexample. After a brief section on
comonads and distributive laws we settle the original question and prove some related
exactness results. Here the point of view is that certain adjoint strings, which we call
distributive, admit a calculus of what might be called cosimplicial kernels. We speak here
of constructing \shorter" adjoint strings.
It transpires that the same set of conditions also permit the construction of cosimplicial cokernels. We speak of constructing \longer" adjoint strings. The shortening and
lengthening constructions are related, as we note. It becomes clear that our distributivity
conditions
nd their paradigm in , the simplicial 2-category and we close with a section
that addresses this point.
Research partially supported by grants from NSERC Canada. Diagrams typeset using M. Barr's
diagram macros.
Received by the editors 26 June 1995.
Published on 8 September 1995
1991 Mathematics Subject Classication : 18A40, 18C15 .

Key words and phrases: adjoint functor, distributivity, simplicial 2-category.
c Robert Rosebrugh and R. J. Wood 1995. Permission to copy for private use granted.

119

Theory and Applications of Categories,

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120

Throughout this paper we work in a 2-category K equipped with proarrows (;) :
K ;! M which satisfy the axioms in 21], restated below without comment. In Section
4 we introduce Axiom 5, a weakening of the Axiom (S) which appeared in 16]. Thus, our
results apply not only to functors but also, for example, to geometric morphisms between
toposes. However, we will refer to the arrows of K as functors and to the arrows of M as
profunctors, whenever possible, so that familiarity with 21] is a prerequisite only for the
fullest extent of the results.
Axiom 1 (;) : K ;! M is a homomorphism of bicategories which is the identity on
objects and locally fully faithful.

Axiom 2 For every arrow Y : B ;! C in K, there is an adjunction
 : Y a Y in
M.
Y

Y

Axiom 3 M has
nite sums with injections in K. Universality restricts to K and the
right adjoints of injections provide also product projections in M.
Axiom 4 M has Kleisli objects for monads with injections in K. Universality restricts
to K and the right adjoints of injections provide also Eilenberg-Moore projections in M.
Arrows of M that are not assumed to be arrows of K are denoted by slashed arrows
of the form C - B.

2. Examples and a Counterexample

1) For B any category, take C to be B2 and V = codomain : C ;! B. Then V a (W =
identity ) a (X = domain ). If B has a terminal object then we have also X a Y where
Y B = (B ;! 1). Here V Y B = 1, for all B , so that V Y underlies an idempotent monad.
2) Consider V = connectedcomponents a discrete a objects a indiscrete = Y : set ;!

cat. Now V Y S = 1 for S 6=  and V Y  =  so that V Y is idempotent.
3) In Example 2) replace cat by ord. The same conclusion holds.
4) In Example 2) replace cat by top and rename objects as points . However, if top is to be
understood as the category of all topological spaces then we do not have a functor V left
adjoint to W = discrete . We have merely a UIAO as in the second paragraph of the Introduction. Here the profunctor Z , right adjoint to Y = indiscrete in the bicategory of categories and profunctors, has, for a set S and a topological space T , Z (S
T ) = top(Y S
T ).
The composite ZW , for sets S1
S2, is given by ZW (S1
S2) = top(Y S1
W S2). Write
 1 2 : top(Y S1
W S2) ;! set(S1
S2 ) for the inclusion. This de
nes the components of
a transformation  : ZW ;! 1set . (Recall that the identity profunctor is the hom functor.) An element of ZW ZW (S1
S2) is an equivalence class of pairs (Y S1 ;! W S
Y S ;!
S

S

Theory and Applications of Categories,

Vol. 1, No. 6

121

) (equivalence being de

ned by the usual -condition). It is a generality, to be established shortly, that
=
. The eect of this transformation on an equivalence
class is to provide the composite 1 ;!
;! ;! 2, where
;! is
the canonical continuous function. Thus, idempotence of (
) amounts to the assertion that any 1 ;! 2 admits a factorization as above, unique up to the equivalence
in question. This is easily verifed after noting that any 1 ;! 2, for non-empty 1
and 2, is given by a constant.
5) Let be a constructively completely distributive lattice as in 15]. Write : ;! D
for the down-segment embedding of into its lattice of down-closed subsets. Then is the
supremum function and , the de
ning adjoint for constructive complete distributivity,
classi
es the totally below relation, which is an order ideal : - . In this example
the ambient 2-category is ord and the relevant proarrows are order ideals. Moreover, the
composite
is . Its idempotence expresses the fact that  implies there exists
such that   .

6) In Example 5) replace by an ordered set and D by I , the ordered set of downclosed and up-directed subsets of . With as before the adjoint string now prescribes
that is a continuous ordered set and  is known as the way below relation. The theorem
which asserts that  is idempotent is often known as the Interpolation Lemma.
7) Entirely analogous to Example 6) is the idempotence of the \wavy hom" for a continuous category as in 4].
8) Also related to Example 5) is the string a a a a : set ;! setsetop , with
the Yoneda embedding, which was shown in 13] to characterize set among categories
with set-valued homs. Here
has constant value 1 and
has constant value .
9) In 17] co
brations were studied in the context of proarrow equipment. It was observed
there that the de
ning adjoint strings for both left co
brations and right co
brations have
the property in question. For the particular case of toposes, geometric morphisms and
left exact functors this example was
rst pointed out in 14].
10) In the simplicial 2-category, , any UIAO of the form n ;! n + 1 satis
es the

idempotence condition. This example provides the paradigm for Sections 4, 5 and 6 of
this paper. In Example 1) the string a a is obtained from a string in by
exponentiation.
11) We display below the counterexample promised earlier. In the following, B is the
ordered set of natural numbers and C is the `long fork' above . The eects (from the
left in the diagram) of
and are indicated by the tailed arrows. Note that
( ) = + 2 which shows that
is not idempotent.
W S2

ZW 

Z W

Y S

WS

Y S

WS

WS

YS

Z W


YS

WS

YS

WS

S

S

L

Y

L

L

L

ZW
a

a

c

b

b

L

L

L

L

Y

L

U

V

W

X

Y

Y

VY

XU

V

W

X

h

V
W
X

VY

n

n

Y

V Y

L

X

W

c

L

Theory and Applications of Categories,

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Vol. 1, No. 6

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This counterexample exhibits other aspects of our problem and will be referenced later.

3. Idempotents and Distributive Laws

We start with a functor Y : B ;! C. There may or may not be a functor right adjoint
to Y but in any event we have an adjunction
 : Y a Z with Z a profunctor. (Note that
we cannot assume that Z has a right adjoint.) We refer to Y as an adjoint string of length
1 from B to C. A functor X and an adjunction 
 : X a Y provide an adjoint string of
length 2 from B to C. A further functor W and an adjunction 
 : W a X produces a
string of length 3 and so on. Note that our somewhat informal de
nition is not to convey
any notion of maximality: a string of length n starting from Y might very well underly
a string of length n + 1 starting from Y . Obviously, a very systematic, integer-labelled
de
nition could be provided but it would transcend our present needs.
The functor Y : B ;! C is always assumed to be fully faithful. In the generality
of proarrow equipment this means that the unit, , for the adjunction Y a Z is an
isomorphism. (For strings of length greater than 1 this de
nition agrees with that given
in terms of representability.) In fact, fully faithfulness is really a property of an adjoint
string. For if we have X a Y then the counit,  , is an isomorphism if and only if the unit
for Y a Z , , is an isomorphism. This follows by dualizing the following folk-lemma. We
have used it in a variety of earlier papers. Some history of it and a detailed proof can be
found in 5].

Theory and Applications of Categories,

Vol. 1, No. 6

123

3.1. Lemma. If a a : B ;! C then the counit for a , , is an isomorphism
if and only if the unit for a , , is an isomorphism. When this is the case there is a
transformation : ;! , unique with the property
W

X

Y

W



W

X

X

Y





Y





=

X



;1 :

Note that the characterizing equation for can be solved explicitly to give






=

Y



=

Y

and similarly

W

;1

;1 :W:

It follows that for longer strings,    a a a a , the functors   
:
B ;! C are all fully faithful and there are canonical transformations    ;! ;!
as above. The latter give rise, by adjointness, to transformations    ; ;
U

V

W

X

Y

U
W
Y

U

V

W

Y

X

Z

satisfying characterizing equations which will be introduced as required.
For a suciently long string, write = , = , =
, =
and so on,
C, of arrows (note that G is
giving rise to an adjoint string,    a a a : C
typically merely a profunctor) which underly idempotent comonads and monads. Indeed,
with the nomenclature above, the counit for is , the unit for is and the counit
for is . Recall that a pointed endoarrow, : 1C ;! , is said to be well-pointed
if
= . Idempotent comonads and monads are much simpler than their general
counterparts. The following lemma will serve to summarize.
3.2. Lemma. For a monad (
), :
;! is an isomorphism if and only if
( ) is a well-pointed endoarrow. A well-pointed endoarrow ( ) underlies a monad if
and only if = : ;!
is an isomorphism.
Of course, a similar lemma holds for comonads and we will not always comment on
obvious dualizations in the sequel. Idempotence also greatly simpli
es the equations
required of distributive laws.
3.3. Lemma. For idempotent comonads, ( ) and ( ), a transformation :
;!
is a distributive law if it satises either of the following equations.
G

S

H

YZ

T

T

H

H

WX

S

WV

G

G

T

YX





T





T

T

T







TT

T

T


T


T

T

T

TT

G


H


HG

G

;; @@
;
@@
;
;
-@R
@
;
@@
;
;
@@R ;;

G

G

GH



H

HG

H

H



GH

Theory and Applications of Categories,

Vol. 1, No. 6

124

Proof. Write  for the comultiplication of (G
) and
for the comultiplication of (H
 ).

So = ;1 = and = ;1 = . Recall the equations for a distributive law and
label them ` ',` ', ` ',` ' according to the single structural transformation that appears in
each. Thus ` ' and ` ' are the displayed triangles and ` ' and ` ' are pentagons. From
idempotence of it is easy to show that ` ' implies ` ' and similarly ` ' implies ` '. Now
given ` ' construct the ` ' diagram and adjoin the diagrams ` ' and ` ' . Adjoin to
the resultant diagram via evident arrows from each instance of
and . Join
to via
;! ;! . Now ` ' follows from a few naturality observations. A
similar diagram chase produces ` ' from ` '.
In fact, idempotence can be characterized in terms of distributivity.
3.4. Lemma. For a comonad (
), is idempotent if and only if 1GG :
;!
is a distributive law.
For an idempotent comonad and a general comonad , existence of a distributive
law :
;! is a property, rather than extra structure. Semantically, this is clear
in CAT. We give a syntactic proof.
3.5. Lemma. For a comonad (
) and an idempotent comonad ( ), there is at
most one distributive law :
;! .
Proof. First observe that for any such ,
is an isomorphism, for any -coalgebra
C. For in this case is also an -coalgebra and the inverse to
is
:X
;
1
 ( ) . In particular this consideration applies to the -coalgebra : C
C
so that in the following naturality square both the top and left sides are isomorphisms.
G



G





H







H







G













G 

 G

GH

G

GH G

GG

G

G

GG

H

GG

G

HG

G





GH

H


HG



GC

C

C

GH G



G



GH

G

HG









C

H

H

GC

G C

H

GH H

H

-

H

H GH

GH 

H G

?

GH



- ?
HG

Thus is explicitly given by
 ( );1 
 ( ) ;1 =
 ( );1.
In an adjoint string of comonads and monads, mates of distributive laws are distributive laws.
3.6. Lemma. For an adjoint string,    a a a : C - C of comonads
(  
) and monads (   ), the bijections


H G

GH

GH

S

H
G

H

G H

T

H G

GH

G

S
T

GH

;!

H G = SG

;!

GS = T S

;!

ST =






mediated by the adjunctions, restrict to distributive laws (those involving both a monad
and a comonad being what have been called \mixed" distributive laws).

Theory and Applications of Categories,

Vol. 1, No. 6

125

We note that mixed distributive laws of the form (comonad)(monad) ;! (monad)(comonad), which do appear in the sequence suggested above, appear to be rare in
the literature. They play the same role with respect to Kleisli objects as do the more familiar form, (monad)(comonad) ;! (comonad)(monad), with respect to Eilenberg-Moore
objects. An excellent reference for the latter is 11].
If any one of    S
H
T
G is idempotent then they are all idempotent. (While this
is obvious in any event, it is interesting to note that it follows from Lemma 3.4 and an
evident variant of Lemma 3.6.) In light of Lemma 3.5, the sense of distributivity given
by Lemma 3.6 is a property of an adjoint string of idempotent comonads and monads.
Bearing in mind also Lemma 3.4, we make the following de
nitions.
3.7. Definition. An adjoint string of length 1, Y : B ;! C, is said to be distributive
if Y is fully faithful. An adjoint string H a T a G, where G underlies a comonad, is said
to be distributive if 1GG : GG ;! GG is a distributive law and there exists a distributive
law GH ;! HG. A fully faithful adjoint string of length 3, in other words a UIAO,
W a X a Y : B ;! C, is said to be distributive if the corresponding string of comonads
and monads, WX a Y X a Y Z , is distributive.
Note that our terminology is also suggested by Examples 5) through 8).

4. Shorter Adjoint Strings

Given a UIAO, W a X a Y : B ;! C, recall the transformation  : W ;! Y
introduced in Lemma 3.1. We de
ne  : Z ;! X as the transformation corresponding to
 by adjointness but it is also the unique solution of

  Y =

;1

The explicit solutions

 = X   ;1Z
 = ;1X  Z
follow from the characterizing equation. The characterizing equations for  and  also
give X   =  ;1 = Y  .
4.1. Lemma. The following diagram commutes.
ZW
Z

?

ZY

W - XW
 ;1

;1

- 1B?

Theory and Applications of Categories,

Vol. 1, No. 6

126

Proof. Insert X Y in the centre of the diagram and join Z Y to X Y by  Y , X W to X Y
by X  and X Y to 1B by  . The resulting quadrilateral commutes by naturality and each

of the triangles expresses a characterizing equation.
De
ne :
;! 1B to be the composite transformation above and observe, from
the proof, that it is given symmetrically by =  .
4.2. Lemma. The arrow
is well-augmented by in the sense that
=
.
Proof. Consider the following diagram.


ZW







ZW



-

W

WZ

ZW 

Z

Z W

WX



?

YZ

- 1C?



Insert
in the centre of the diagram and join
to
by
,
to
;
1
by
and
to
by
. The resulting quadrilateral commutes by
naturality and each of the triangles commutes from the explicit descriptions of and .
;1 the topNow apply (;) to the diagram displayed above. Since
=
followed-by-right composite yields
using the top-followed-by-right description of
in Lemma 4.1. Similarly, the other composite is seen to be
.
Of course, by duality, Lemma 4.2 establishes our earlier assertion that
is a wellpointed endofunctor, for adjoint strings of length 4. From either point of view we can
now state and prove a Theorem which answers our opening question.
4.3. Theorem. For a UIAO, a a : B ;! C, (
) underlies an idempotent
comonad if and only if the UIAO is distributive.
Proof. It suces to show that invertibility of
=
is equivalent to the existence of a distributive law
;!
. From invertibility of :
;! 1B
and adjointness we have bijections
W XY Z

WX

W XY Z

W X

WZ

W XY Z

W

YZ

Y Z

W XY Z

Z



Z

W

W



W

ZW 



Z W

VY

W

X

Y

Z W


ZW 

Y ZW X

Y ZW X

;!

Z W

W XY Z

W XY Z = Y ZW X

;!



;!
;!

W Z = ZW

XY

ZW ZW

and a diagram chase shows that if a transformation
satis
es either
one of the equations for a distributive law then its counterpart
;!
provides
a section for
=
and conversely. However, such a section is necessarily an
isomorphism. This follows from naturality and the equation
=
.
It is now possible to explain the generation of Counterexample 11) and rationalize the
names of the objects of the ordered set C displayed there. For if Theorem 4.3 is stated for
adjoint strings of length 4 then by Lemma 3.6 the relevant distributive law is ;! .
Thus, in an ordered set counterexample there must not be  but all composites of
Y ZW X

W XY Z

ZW

ZW 

ZW ZW

Z W

ZW 

Z W

TS

TS

ST

ST

Theory and Applications of Categories,

Vol. 1, No. 6

127

and either or reduce by adjunction inequalities and idempotence. The remaining
ideas are somewhat similar to those found in 18].
For the moment, let B be any object and let = ( ) be an idempotent comonad
in M on B. In our working terminology, is a profunctor. Then a -coalgebra in M
with domain Y is an arrow : Y - B, in M, together with a -coalgebra structure
transformation : ;! . By idempotence, the usual requirements for such reduce
to
= 1B and this implies, by naturality and = , that = ( );1. Thus, being
a -coalgebra is a property of . If both and 0 are -coalgebras then any transformation ;! 0 is a coalgebra homomorphism in the usual sense. Write M(Y B)C for the
category of -coalgebras in M with domain Y. It is just the full subcategory of M(Y B)
determined by the which invert . If : Y - B is a -coalgebra then composition
with de
nes, for every object X in M, a functor M(X Y) ;! M(X B)C . Recall
that an Eilenberg-Moore object for is a -coalgebra, : BC - B, such that, for all
X, composing with provides an equivalence of categories, M(X BC ) ;! M(X B)C .
It is clear from the discussion that if is Eilenberg-Moore for then it also provides an
inverter for the transformation : ;! 1B .
Recall the proarrow equipment for toposes and geometric morphisms extensively studied in 14], namely the transformational dual of toposes and left exact functors. It does
admit Eilenberg-Moore objects for comonads in M. However, it was shown in 12] that
the paradigm for proarrows, namely categories and profunctors in the usual sense, does
not. The paradigm does admit a weaker notion which we now describe.
For = ( ) an idempotent comonad on B in M, suppose that : Y ;! B is a
-coalgebra with in K. In this event, composing with , M(X Y) ;! M(X B)C ,
has a right adjoint which is given by composing with , the right adjoint of in M.
(This follows from the fact that M(X B)C is a full subcategory of M(X B).) Write
K(X B)C for the full subcategory of M(X B)C determined by the -coalgebras in K.
Henceforth we assume the following.
Axiom 5 For every idempotent comonad (B ) in M there is a -coalgebra : B( ) ;!
B in K such that, for each X, the adjunction given by composing with , M(X B( )) ;!
M(X B)C restricts to an equivalence K(X B( )) ;! K(X B)C .
One could say that the Axiom provides, for each idempotent comonad in M, an
Eilenberg-Moore object as seen by K. With an obvious extension of such terminology, it
is clear that : B( ) ;! B provides an inverter as seen by K for . For categories
and profunctors, : B( ) ;! B was
rst described in 20]. In that context, a variety
of descriptions of B( ) were given in 12]. Note that the Axiom ensures that if the
idempotent comonad is in K then : B( ) ;! B in K is a true Eilenberg-Moore
object in K and may be written : BC ;! B. In this case, regarding as an idempotent
comonad in M, : BC ;! B is also an Eilenberg-Moore object in M. (The limit in
question is preserved by all homomorphisms of bicategories in particular, it is preserved
by (;) : K ;! M.) It may well be the case that for in M, not necessarily in K,
that : B( ) ;! B provides an Eilenberg-Moore object in M. In any event, writing
for the right adjoint of we have
;! corresponding by adjointness to the
H

S

T

C

C


C

C

C

B

b

B

CB

b

B :b

C

C

B

B

B

B

C

b

B

C

B




C




B



C

B

B




C

C




I

I




I



C




C

C

C


C

B

B

B







B

B













C


C

C

I

I







I

C




C



I

C

C

C

I

C

I

C

I

C

I

Q

C

I

IQ

C

C




C

Theory and Applications of Categories,

Vol. 1, No. 6

128

coalgebra structure I ;! CI . In 12] the following observation was made in the case of
(;) : CAT ;! PRO.
4.4. Lemma. A functor I : B(C ) ;! B as provided by Axiom 5, with right adjoint Q in
M, is an Eilenberg-Moore object for C in M if and only if the canonical transformation
IQ ;! C is an isomorphism.
Eilenberg-Moore coalgebras for idempotent comonads in a bicategory are representably
fully faithful. For the weaker notion of Axiom 5 we have:
4.5. Lemma. The functors I : B(C ) ;! B provided by Axiom 5 are fully faithful and
the unit for I ; a Q; : M(X
B)C ;! M(X
B(C )) is an isomorphism.
Proof. For X = B(C ), the 1B(C ) component of the unit of the adjunction given in
Axiom 5 is 1B(C) ;! QI , the unit for the adjunction I a Q in M and, by Axiom 5,
it is an isomorphism because 1B(C) is in K. Thus I is fully faithful and the rest of the
statement of the Lemma follows from this.
Thus, by Theorem 4.3, a distributive UIAO, that is a distributive adjoint string of
length 3, W a X a Y : B ;! C, gives rise to I : A = B(ZW ) ;! B, a distributive
adjoint string of length 1.
Lawvere has taken the point of view that a UIAO, W a X a Y : B ;! C, provides C
with the structure of an oriented cylinder. Both the top and bottom are copies of B. The
former is provided by W , the latter by Y and the orientation by  : W ;! Y . He further
points out in 9] that the top and bottom are not necessarily disjoint, in the sense that
part of the top may be isomorphic to part of the bottom in C. The following theorem
shows that this overlap is provided precisely by I : A = B(ZW ) ;! B.
4.6. Theorem. If W a X a Y : B ;! C is a distributive UIAO then I : A =
B(ZW ) ;! B is the inverter in K of  : W ;! Y : B ;! C.
Proof. We have already remarked that I : B(ZW ) ;! B is the inverter as seen by K
of  : ZW ;! 1B : B - B. It suces to show, for a functor B : X ;! B, that B
is an isomorphism if and only if B is an isomorphism. Since B = ;1B:ZB the \if"
part is clear. On the other hand, if B is an isomorphism with inverse b : B ;! ZW B
then the transformation W B:Y b : Y B ;! W B can be shown, with the help of Lemmas
4.1 and 4.2, to be the inverse of B : W B ;! Y B .
There remains the question of whether or not I : B(ZW ) ;! B is actually EilenbergMoore in M for the comonad ZW . (After all, by construction, Y : B ;! C is EilenbergMoore for G and W : B ;! C is Eilenberg-Moore for H .) Again writing Q for the right
adjoint of I in M, Lemma 4.4 shows that this determination rests on the invertibility of
the canonical transformation IQ ;! ZW . We will show that IQ ;! ZW can fail to be
an isomorphism.
To explain, it is convenient to generalize, temporarily, the situation with which we are
preoccupied. So let G and H be idempotent comonads in M on C, without our usual
adjointness assumptioms, for which there exists a distributive law, GH ;! HG. Assume

Theory and Applications of Categories,

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129

that admits an Eilenberg-Moore object, : CH ;! C. From the general theory of
comonads, the comonad restricts to a comonad j on CH . That is we have j
=
with j =
, where is right adjoint to . Invoking our Axiom 5 we have
H

W

G

G

X GW

G

X

GW

WG

W

CH ( j)
I

?
CH

- C( )

J

G

G

'
;

W

Y

- C?

in K. (The functor is the \
ll-in" that results from
being a -coalgebra in K. It
is necessarily fully faithful because the composite
is fully faithful.) Each functor has
a right adjoint in M, say in the case of and in the case of . The isomorphism
;! gives, by adjointness, a transformation, ;! . In 1] invertibility of
;! , a Beck condition, was called distributivity for the adjoint square and the
condition is satis
ed when and are Eilenberg-Moore coalgebras.
Returning to our case of interest, we have CH = B = C( ) with also Eilenberg =
and we can take = . However, the
Moore. Here j =
=
resulting adjoint square,
-B
B( )
J

WI

G

WI

Q

YJ

WI

JQ

ZW

I

Z

Y

JQ

I

ZW

Y

G

G

X GW

ZW

XY ZW

J

?
B

I

I

ZW

I

Y

'
;

W

Y

- C?,

may fail to be distributive, even in the paradigm (;) : CAT ;! PRO.
Counterexample: Let B be the rationals with the usual order. Let the objects of C
be pairs ( ), with a rational and in f0 1g, ordered by ( )  ( 0 0) if and only if
0 and = 1 and 0 = 0. De
ning
 0 and  0 or
= ( 0), ( ) = and
= ( 1) produces a UIAO in ord and hence in CAT. Direct calculation shows that
the profunctor
: B - B is the order ideal : B - B, which is an idempotent.
However, the inverter of  is : 0 ;! B so that the composite is 0 : B - B.
b
i

b

Yb

b

i

b

i

i

b < b




i

b
i

i

Wb

b
i

b


X b
i

b

b


ZW

<

W

Y

I

IQ

The reader who is familiar with 3] may
nd the following to be more natural.
Counterexample: Let B be the closed unit interval and : B ;! C the downsegment embedding into the lattice of down-closed subsets of B. This is a special case
of 5) in Section 2. It follows from remarks in 3] that
: B - B is the order ideal
: B - B but, again, the inverter of  is empty so that the composite is 0.
Y

ZW

<

W

Y

IQ

Theory and Applications of Categories,

Vol. 1, No. 6

130

However, even in the general situation that we described in the second to last diagram, either diagonal composite CH (Gj) ;! C is Eilenberg-Moore as seen by K for the
composite comonad GH . In short, we always have CH (Gj) ' C(GH ), for idempotent
comonads in M, when H admits an Eilenberg-Moore object and there exists a distributive
law GH ;! H G.
We would like now to investigate the generation of distributive UIAOs from longer adjoint strings by several instances of the \shortening" procedure that we have just discussed.
We will see that the subtlety of constructed functors actually providing Eilenberg-Moore
objects in M disappears but that the longer starting string must satisfy higher order
distributivity conditions. At
rst these conditions appear somewhat strange but they are
satis
ed in naturally occuring examples. Moreover, at this stage a pattern emerges which
enables us in Section 6 to deal with strings of arbitrary length. It is convenient to begin
with a lemma that admits Theorem 4.3 as a corollary.
4.7. Lemma. If Y : B ;! C and Y : B ;! C are fully faithful arrows in K with right
adjoints Z and Z respectively, possibly in M, then a transformation Y ZY Z ;! Y Z Y Z
is a distributive law if and only if the transformation ZY ;! ZY Z Y ZY , corresponding
by adjointness, is the inverse of Z Y : ZY Z Y ZY ;! ZY , where  and  are the
respective counits.
Proof. A very direct calculation suces.
Lemma 4.7 admits a simple interpretation in CAT. Considering B and B to be
subcategories of C, the distributive law in question asserts that every arrow of the form
c : Y B ;! Y B in C admits a factorization,
0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

c YB
c Y B ;!
s Y B ;!
;!
1
0
with unique tensor product (see 12]), c1 B sB c0. The UIAO situation, where Y = W a
X a Y , simpli
es this condition by the requirement that s = B0 : W B0 ;! Y B0. The
YB

0

0

1

0

0

0

0

0

case when GH = Y ZW X admits an Eilenberg-Moore object is the further specialization
to invertible B0.
Suppose now that U a V a W a X a Y : B ;! C is a distributive adjoint string
of length 5. This gives rise to a string of idempotents, L a S a H a T a G, where we
have extended our earlier terminology with S = W V and L = U V . The distributive laws
in Lemma 3.6 now continue to include explicitly LT ;! T L and HL ;! LH . Thus we
can apply Theorems 4.3 and 4.6 to the distributive UIAOs W a X a Y and U a V a W
to produce I : B(Z W ) ;! B the inverter of W ;! Y and J : BXU ;! B the inverter
of U ;! W . (Of course X U is a comonad in K and we have oserved that in this case
the requisite Eilenberg-Moore object exists.) But we have also the idempotent monad
V Y with XU a V Y a Z W and it is convenient to revise, perhaps extend, a well-known
result of 2]. We refer to the theorem which states that if a monad M is left adjoint to a
comonad C (in CAT) then the category of M -algebras is isomorphic to the category of
C -coalgebras via an isomorphism that identi
es the forgetful functors. This theorem can
be generalized in many ways. Here we collect just what we need, without proof.

Theory and Applications of Categories,

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131

4.8. Lemma. If : B ;! B is an idempotent monad in K with right adjoint (possibly
not in K) then B( ) ;! B provides both an Eilenberg-Moore object in K for and an
Eilenberg-Moore object in M for .
4.9. Lemma. If a monad in K has an Eilenberg-Moore object : BM ;! B then
has a left adjoint, , and if is idempotent then : B ;! BM provides a Kleisli object
for .
Axiom 4 for proarrow equipment (;) : K ;! M ensures that every monad in K
has a Kleisli object : B ;! BM in K.
4.10. Lemma. If a monad in K has a left adjoint then : B ;! BM provides a
Kleisli object for the comonad .
Finally, let us explicitly state a dual of Lemma 4.9.
4.11. Lemma. If a comonad in K has a Kleisli object : B ;! BD then has a left
adjoint, , and if is idempotent then : BD ;! B provides an Eilenberg-Moore object
for .
It follows, from Lemmas 4.8 through 4.11, that if we start with an adjoint string of
length 5, a a a a : B ;! C, where a a , or equivalently a a ,
is a distributive UIAO, then our construction generates a fully faithful adjoint string of
length 3, a a : A ;! B, where we can take A to be B( ). Note, for future
reference, that is Eilenberg-Moore for
and that is Eilenberg-Moore for both
and
.
Considering just the composable UIAOs a a and a a and the fact that
inverts ;! we have an instance of Lawvere's interpretation of Hegel's \Aufhebung"
as described in 9].
Note that if, as before, we write for the right adjoint of in M then we do in this
case have

=
(by Lemmas 4.8 and 4.4). In fact, a distributive adjoint string of length 4 ensures this
conclusion. We have also
=
and

=
To see that such compatible composable adjoints strings do not arise in the absence of
distributivity, even if the construction of inverters is available generally, it is instructive
to return to the Counterexample in 11) of Section 2. Inspection shows that the functor
there has a further left adjoint, , given by 0 = , 1 = 1, 2 = , 3 = and so on
up the right hand side of the long fork. (In fact this has itself a left adjoint which does
not have a further left adjoint.) The inverter of ;! is 0 : 1 ;! B but the inverter
of ;! is 0 ;! B.
Our constructed string, a a : A ;! B, cannot be shown to be a distributive
UIAO without further conditions on the given data, a a a a : B ;! C.
M

C

C

M

C

M

P

I

M

I

P

M

M

P

M

D

P

D

D

J

P

D

P

J

D

U

J

V

W

P

X

Y

U

V

W

W

I

X

Y

ZW

J

XU

I

VY

ZW

J

U

P

I

U

V

W

J

W

Q

I

IQ

ZW

IP

VY

JP

X U:

V

U

U

h

U

U

t

W

X

U

ts

U

U

W

W

Y

J

P

I

U

V

Y

Theory and Applications of Categories,

Vol. 1, No. 6

132

4.12. Lemma. Let be either an idempotent monad or an idempotent comonad on C.
Assume the same of and the existence of a distributive law
;! . If the
distributive law is invertible then its inverse is also a distributive law and, conversely, if
there is a distributive law
;!
then it is the inverse of the original distributive
law.
4.13. Definition. A string of idempotent comonads and monads, a a a a ,
is said to be distributive if there are distributive laws
E

E

0

EE

0

E E

EE

0

0

E E

0

L

!
!
!

GH
GS
GL

S

H

T

G

HG
SG
LG:

A fully faithful adjoint string of length 5, U a V a W a X a Y : B ;! C, is said to
be distributive if the corresponding string of idempotents is distributive.
The law S G ;! GS is an equivalent of the law GH ;! H G, by Lemma 3.6. Therefore, by Lemma 4.12, the law GS ;! S G is an isomorphism. Also T L ;! LT is an
equivalent of GS ;! S G and we will
nd it convenient to use this formulation. Since
LT ;! T L is another equivalent of GH ;! H G, T L ;! LT is an isomorphism too. Still
another equivalent of GS ;! S G is LG ;! GL so that the last two distributivities in
the de
nition above could be combined as a single isomorphic distributivity GL ;! LG.
In the proof of the following theorem we content ourselves with exhibiting the existence
of the requisite arrows and isomorphisms. It should be clear to the reader by now that this,
not coherence, is the central problem. In fact, a full coherence theorem for distributive
adjoint strings will appear elsewhere.
4.14. Theorem. If U a V a W a X a Y : B ;! C is a distributive adjoint string of
length 5 then J a P a I : A = B(Z W ) ;! B is a distributive adjoint string of length 3.
Proof. The distributive law T L ;! LT provides a restriction of L to the EilenbergMoore object for T , Y : B ;! C. The restriction, Lj, is given by Lj = X LY =
XUV Y
= X U and the Eilenberg-Moore object for X U is
= JP
= J 1A P
= JP IP
J : A ' BL ;! B. Similarly, T L ;! LT provides the restriction T j = V T U =
V Y XU
= IP JP
= I 1A P
= IP
= V Y of T to the Eilenberg-Moore object for L,
T ;! B. Now from 11]
U : B ;! C. The Eilenberg-Moore object for V Y is I : A ' B
it can be inferred that
'

j

j

A

I

;
'

J

?
B

Y

-B
U

- C?

Theory and Applications of Categories,

Vol. 1, No. 6

133

is a bi-pullback. In particular, we have UI
= Y J as displayed. Taking right adjoints, we

have also QV = PZ .
The distributive law GL ;! LG, expands to Y ZUV ;! UV Y Z . An application of
Lemma 4.7 gives the
rst isomorphism in ZU
= ZUV Y ZU
= ZUIPZU
= ZY JQV U
=

1B JQ1B = JQ.
The distributive law GH ;! HG, expanded and with V (;)U applied to it, gives
V Y ZWXU ;! V WXY ZU , which by fully faithfulness of W and Y gives an arrow
V Y ZWXU ;! ZU . Substituting along the isomorphisms we have derived yields an
arrow IPIQJP ;! JQ, which can be rewritten I 1A QJP ;! J 1A Q
= JPIQ. Finally,
we have IQJP ;! JPIQ, which is distributivity for the UIAO J a P a I .

It is instructive to picture some aspects of the proof of Theorem 4.14 in terms of
Lawvere's cylinders as mentioned in the paragraph preceding Theorem 4.6. Prior to the
proof of Theorem 4.14 and the assumption of further distributivity for the string B ;! C,
we had already constructed a UIAO A ;! B so that we knew B to have the structure
of a directed cylinder. The functors U , W and Y thus provided for three copies of the
cylinder B, fully faithfully in C. Adjointness further provided for two \cylinders", where
the top and bottom of the \cylinders" each had the shape of cylinder B. Inversion of
U ;! W by J showed that the cylinders U and W are glued at their tops in C while
inversion of W ;! Y by I showed that cylinders W and Y are glued at their bottoms.
In establishing the isomorphism UI
= Y J in the proof above, which explicitly used the
higher order distributivity, we were joining the bottom of cylinder U to the top of cylinder
Y , as suggested in the picture below.




 






UJ
= WJ

TT

TT

T

T

T

T

T

T

T

T
T
T
T
T
T
U
T - TTW
T
T
T
T
T
T
T
T
T
Tw
/
T
T
T 
T



T
T


UI  ?:TT Y I
= WI





Y


YJ


T

Note that the \triangle" structure bounded by cylinders, that this provides for C, is
not \hollow".

Theory and Applications of Categories,

Vol. 1, No. 6

134

5. Longer Adjoint Strings

Given a distributive adjoint string from B to C we consider now the construction of a
longer adjoint string, with domain C. Our \lengthening" construction will in fact be left
adjoint to the \shortening" construction for distributive adjoint strings that we described
in the last section.
In particular, suppose that we have a fully faithful : B ;! C, a distributive adjoint
string of length 1. As before, write for the composite . Now Axiom 3 ensures the
coalescence of
nite sums and
nite products in M. Thus we use direct sum notation
below and there is a profunctor, : C  C - C  C, where is the following matrix:
!
1
1 1
in which the 1's denote 1C : C ;! C. (Here and elsewhere in this section the th
entry of such a matrix denotes an arrow from the th summand of the domain to the
th summand of the codomain.) Recall from 21] that Axiom 3 ensures that the hom
categories of M have
nite sums and that matrix multiplication, using this additive
structure, provides for composition of profunctors given by matrices. In particular, the
identity on C  C is:
!
1 0
0 1
where the 0's denote the initial object of M(C C). Recall that a transformation 1CC ;!
is a matrix of transformations, given componentwise. Evidently such is provided by:
!
1 !
! 1
where the !'s denote the unique transformation, in each case, with domain 0. Thus is
a pointed endo-arrow of M. The composite
: C  C - C  C is given by the
matrix:
!
1+
+
1+1 +1
where the +'s denote binary sum in the hom categories of M. Consider the transformation
;! given by:
!
11
1G 1G
11 11
11
where we have used \row vectors", bracketed by and to display transformations out
of local sums.
5.1. Lemma. The transformations
;! ; 1 introduced above provide a monad
structure on .
It should be noted that a detailed proof of Lemma 5.1 must take into account the
associativity isomorphisms of the bicategory M and the further isomorphisms that are
Y

G

YZ

M

M

G

i
j

i

j




M

M

MM

G

G

G

G

MM

M

<

<


 >




>

<




< 


<

MM

M

M

>

>

>

Theory and Applications of Categories,

Vol. 1, No. 6

135

implicit in using categorical sum, + and 0 in the matrices above, for a matrix calculus.
However, note too that Lemma 5.1 holds given only that is well-augmented by .
Now Axiom 4 provides for a Kleisli opalgebra, : C  C ;! (C  C)M in K and by
Axiom 3 the arrow is a 2 by 1 \column vector":
G



K

K

!

J
I

with and in K. Writing D for (C  C)M , we have functors
: C ;! D. The
opalgebra action
;! can be analyzed by
rst computing the matrix :
J

I

J
I

KM

K

J

+
+

IG

J

KM

!

I

and examining the unitary and associativity requirements in terms of the components.
The unitary requirement says that two of the four components are identities so that
;! amounts to, say, : ;! and : ;! . By associativity these satisfy:
KM

K



IG

J









G

=

 =


J

I

J

I :

However, the transformation corresponds, by adjointness, to a transformation :
;! . It is a simple calculation to show that is the the inverse of precisely
when the two equations above hold.
Recall that for any arrow : B ;! C in a bicategory, the coinvertee of is a
transformation, : ;! : C ;! D, with an isomorphism and which is moreover
(bi-)universal with this property. The notion of coinvertee does not seem to have been
explicitly studied to the same extent as the dual notion, invertee, of an arrow. As an
example, the coinvertee in CAT of 0 ;! 1 is 0  1 : 1 ;! 2.
5.2. Lemma. For : B ;! C fully faithful, the transformation : ;! : C ;!
(C  C)M = D above is a coinvertee in M with universality restricting to K.
Proof. Observe that the data and equational considerations in the discussion above
apply to any opalgebra for . The universality of the Kleisli opalgebra provides the
universality required of a coinvertee.
Of course a particular transformation with domain C that is inverted by : B ;! C
is 11C : 1C ;! 1C : C ;! C, so universality ensures a functor : D ;! C and
'
' 1 ,
isomorphisms, which we may elect to direct as : 1C ;!
and :
;!
C
satisfying  = ;1 . Similar considerations produce a transformation : ;! 1D
satisfying = ;1 and =  and a transformation : 1D ;! satisfying
=  and = ;1. From these equations it follows that we have adjunctions
: a and
: a . Moreover, invertibility of and provides that a a
: C ;! D is a UIAO.


IY



JY



Y

Y



J

Y

I

Y

Y



J

I

M

Y

P





J

J



I

I

J



P

P

PJ






PI



J

I



I



J



JP

IP

I

P

I





J

P

Theory and Applications of Categories,

Vol. 1, No. 6

136

5.3. Theorem. For : B ;! C fully faithful, the adjoint string above, a a :
C ;! D is distributive.
Proof. Observe that : ;! 1C : C - C is also a transformation inverted by
, since =
and is the counit for a . This gives rise to a : D - C,
not necessarily
in K'because is not necessarily in K, and compatible isomorphisms,
'
;! and 1C ;!' . Direct computation shows that a and, by Theorem 4.3,
the isomorphism
;! and idempotence of shows that the UIAO in question is
distributive.
It is interesting to note that the arrow components of , respectively , as determined
by the universal property of D, constitute the
rst, respectively second, column of the
matrix . This will be elaborated upon elsewhere.
Of course, given a functor, : B ;! C, and a transformation, : ;! : C ;! D,
we can ask both whether is the coinvertee of and whether is the inverter of .
When both conditions hold we have a form of 2-dimensional exactness. The \lengthening"
procedure for a fully faithful as described above does indeed produce an exact diagram
in this sense.
5.4. Corollary. For : B ;! C fully faithful, the lengthening construction under
consideration, followed by the shortening construction that precedes Theorem 4.6, recovers
. Moreover, satises the stronger property of being Eilenberg-Moore in M for the
comonad .
We now consider the problem of generating, from a distributive UIAO a a
: B ;! C, a distributive adjoint string, C ;! D, of length 5. As in the previous
section, we write for
, for
and for . Consider the profunctor :
CCC
C  C  C where is the following matrix:
1
0
1
CA
B
@1 1
1 1
with conventions as above. There is an evident pointing, 1CCC ;! , so consider
, the matrix:
1
0
1+ +
+ +
+
+
CA
B@ 1 + 1 +
+1+
+ +
+1+
+1+1
+ +1
and the transformation
;! given by:
1
0
1 

CA
B@
11 
1

11


1
where we have written 1 for 11, for 1H etc.. The calculations required to prove the next
Lemma are straightforward but lengthy and tedious.
Y

J

P

I






Y

G

YZ

G



Y

Z

Q

G

QJ

G

QI

QJ

I

G

Q

G

P

Q

M

Y





Y

J

I

Y



Y

Y

Y

Y

QJ

W

X

Y

H

WX

T

YX

G

YZ

M

M

H

HG
G

T

M

MM

H

T HG

H

T

HT

MM





<




Y Z




H

TG

T

<

H

HG

HG

G

GH

HG

H GT

G

HG
G

G

M

T G >

< H
H
H  >

Y Z >

< 


< T

T >

< 

H

< H G

H G >


 >

W  X





< G
G
G >

>

< 

W X

H T



>

Theory and Applications of Categories,

Vol. 1, No. 6

137

5.5. Lemma. The transformations
;! ; 1 introduced above provide a monad
structure on .
The Kleisli opalgebra for is a 3 by 1 matrix, say:
0 1
B@ CA
MM

M

M

M

K
J
I

with domain C  C  C and codomain (C  C  C)M . We will denote the latter by D,
so that we have functors
: C ;! D. To analyze the opalgebra action we compute
the composite of the above 3 by 1 matrix and to be:
1
0
+ +
CA
B
+ +
@
+ +
K
J
I

M

K

JH

K

IHG

J

KT

IG

J

I

and, again, examine the unitary and associativity requirements in terms of the components. The unitary requirement says that all components of the form ;! are identities
so that the data amounts to
;! ,
;! , ;! , ;! , ;!
and ;! . The
rst, second, fourth and
fth of these are equivalent, by adjointness,
to transformations
;! , ;! , ;! and
;! , respectively. In terms of these, associativity' states that the data consists of a transformation
;! and' an isomorphism
;! ,'a transformation ;! and an isomorphism ;! and an isomorphism
;! . This last will provide, as explained
in the closing paragraphs of Section 4, the glue to join a chain of three linked cylinders
into a triangle.
5.6. Lemma. For a distributive UIAO, a a : B ;! C, the functors , and
and transformations described above are universal in M with universality restricting to
K.
Proof. Again, the data and equational considerations apply to any opalgebra for the
monad .
Now consider the
rst column of matrix , remembering that the 1's are 1C's. Trivially, we have the transformation 11C : 1C ;! 1C inverted by . We have the unit
for , ' : 1C ;! which, since = , is inverted
by . We have an isomorphism
'
;
1
1C ;!
because =
and
:
;! 1B . Construing this data as an
opalgebra de
nes a functor : D ;! C.
Similarly, examining the second column of , we consider : ;! 1 and 1 : 1 ;! 1.
The necessary isomorphisms for an opalgebra structure are easily found and so we have
a functor : D ;! C.
Finally, consideration of the third column suggests :
;! and : ;! 1.
Inversion of the
rst by is equivalent to invertibility of
and this follows from the
considerations of Section 4, in particular from the proof of Theorem 4.6. Inversion of by
x

JH

J

IHG

K

K

J

IG

J

KT

I

I

JW

K

K

x

KW

J

IW

KY

KW

JY

IY

IY

JY

KY

JW

J

KY

W

IW

I

IW

X

Y

K

J

I

M

M

W

T

Y



T

TW

T

T

YX

YX



Y

XW

Q

M



H

P

G

W

HG

G



G

Z W



Theory and Applications of Categories,

Vol. 1, No. 6

138

'
follows simply from the de
nition of . An isomorphism ( ) ;!
1C is found
by noting that
=
. The opalgebra de
nes a profunctor : D - C
If we note carefully the compatibility isomorphisms and equations that universality
further provides in the de
nitions of , and , as we did in the simpler case preceding
Theorem 5.3, then we can prove a a a a a . For example, in the de
nition
= 1, = 1 and
of we have isomorphisms
= . The
rst of these provides
a unit for a . It is an isomorphism so the adjoint string of length 5 in K is fully
faithful.
= , arising from the de
nition of , establishes, since is
The isomorphism
an idempotent, that the constructed string is at least distributive in the sense we de
ned

rst for a UIAO.
5.7. Theorem. For a a : B ;! C a distributive UIAO, the adjoint string of
length 5 constructed above, a a a a : C ;! D, is distributive.
Proof. The higher order distributivity required here is the isomorphic distributivity,

( )( )
= gives also, by taking right adjoints,
= ( )( ). The isomorphism
= . We have noted = above
= . The de
nition of gives
=

=

is familiar. Assembling these we have
and
=
=
=
=
.
5.8. Corollary. For a a : B ;! C a distributive UIAO, the lengthening
construction followed by the shortening construction recovers a a .
For the cases that we have considered, evident de
nitions of arrows between adjoint
strings allow us to say, \lengthening is fully faithful and left adjoint to shortening".
Y

G

HG

W XY Z

K

Q

P

I

R

T

QI

KQ

P

X

Y

K

Q

J

P

IR

R

RK

IW

HG

QI

WZ

HG

H

I

KY

XR

I H GQ

I RK Q

T

IW ZQ

K QI R

KT R

KY XR

J

QJ

H

W

ZQ

R

Q

PK

KQ

P

Q

QK

IR

W

R

Q

K

HG Y

W

X

Y

W

X

Y

6. Generalizing the Construction of

For an adjoint string of comonads and monads,    a a a
following \table" of distributive laws:
R

!
!
!
!
!
!
!
!
!
!
!


L

GG

GG

TG

GT

GT

TT

TT

HG

GH

GH

HG

HT

TH

TH

HT

SG

GS

GS

SG

HH

HH

ST

TS

TS

ST

LG

GL

GL

LG

SH

HS

HS

SH

LT

TL

TL

LT

RG

GR

SS

SS

LH

HL

HL

LH

RT

TR

TR

RT

LS

SL

SL

LS

RH

HR

HR

RH

LL

LL

RS

SR

SR

RS

RL

LR

LR

RL

RR

RR

!
!
!
!
!
!
!
!
!


TG

!
!
!
!
!
!
!


!
!
!
!
!


!
!
!


S

a a , consider the

H

T

!


GR

RG

G



It is to be understood that in each column the entries correspond via adjointness.
Moreover, let us assume that the (1 1) entry is 1GG : ;!
and that the
rst non


GG

GG

Theory and Applications of Categories,

Vol. 1, No. 6

139

blank entry in column , entry ( ), is the inverse of entry ( ; 1). Thus, by Lemmas
3.6 and 4.12, e