arXiv:1112.1448v1[math.CT]6Dec2011
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS
JOHN BOURKE AND RICHARD GARNER
Abstract. We introduce the notion of pie algebra for a 2-monad, these bearing
the same relationship to the flexible and semiflexible algebras as pie limits do
to flexible and semiflexible ones. We see that in many cases, the pie algebras
are precisely those “free at the level of objects” in a suitable sense; so that, for
instance, a strict monoidal category is pie just when its underlying monoid of
objects is free. Pie algebras are contrasted with flexible and semiflexible algebras
via a series of characterisations of each class; particular attention is paid to the
case of pie, flexible and semiflexible weights, these being characterised in terms
of the behaviour of the corresponding weighted limit functors.
1. Introduction
One category-theoretic approach to universal algebra is based on the theory
of monads. Single-sorted (possibly infinitary) algebraic theories correspond with
monads on the category of sets, and models of a theory with algebras for the associated
monad; this justifies our regarding monads on other categories as generalised
algebraic theories, and many basic aspects of classical universal algebra may be reconstructed
in this broader context. A further generalisation is obtained on passing
from the study of monads on categories to that of 2-monads on 2-categories, which
yields a kind of “two-dimensional universal algebra”. The simplest case studies
2-monads on Cat, which encode many familiar structures that may be borne by a
category: thus there are 2-monads whose algebras are monoidal categories, or categories
with finite products, or distributive categories, or cocomplete categories,
and so on.
In the passage from 1- to 2-dimensional monad theory, a number of new phenomena
come into being. As well as strict algebras for a 2-monad, we also have
pseudo and lax ones, for which the algebra axioms have been weakened to hold only
up to coherent 2-cells, invertible in the former case but not in the latter. Likewise,
between the algebras for a 2-monad we have not only the strict morphisms, but
also pseudo and lax ones, which preserve the algebra structure in correspondingly
weakened manners. The interplay between strict, pseudo and lax in 2-dimensional
monad theory provides an abstract setting for the study of coherence problems of
the kind exemplified by Mac Lane’s famous result [26] that every monoidal category
is monoidally equivalent to a strict one. The study of coherence from the
standpoint of 2-monad theory was championed by Max Kelly, who initiated the
Date: December 8, 2011.
2000 Mathematics Subject Classification. Primary: 18D05, 18C15.
The first author acknowledges the support of the Eduard ˇCech Center for Algebra and Geometry,
grant number LC505. The second author acknowledges the support of an Australian
Research Council Discovery Project, grant number DP110102360.
1
2 JOHN BOURKE AND RICHARD GARNER
programme in [8, 9] and with his collaborators, brought it to a particularly fine
expression in [3].
A subtle and crucial notion in Kelly’s framework is that of flexibility; this first
arose in [9], where was introduced the notion of a flexible 2-monad. One of the
most important properties of flexible 2-monads, as described in [14, Theorem 3.3],
is that strict algebra structure may be transported along equivalences: which is
to say that if A ≃ B, and A bears strict algebra structure for a flexible 2-monad,
then so does B. This is the case, for example, with the 2-monad on Cat whose
algebras are monoidal categories, but not for the 2-monad whose algebras are strict
monoidal categories; and indeed, the former 2-monad is flexible, whilst the latter is
not. This particular good behaviour of flexible 2-monads is a consequence of a more
fundamental one: that every pseudoalgebra for a flexible 2-monad is isomorphic
to a strict one. Intuitively, we think that flexibility of a 2-monad expresses a
certain “looseness” in the structure it imposes on its algebras, and this intuition
has been expressed for 2-monads on Cat in the following way: that such a 2monad
is flexible if its algebras can be presented as categories equipped with basic
operations and with natural transformations between derived operations, satisfying
equations between derived natural transformations but with no equations being
imposed between derived operations themselves.
The scope of the concept of flexibility was vastly expanded in [3], where was
introduced the notion of a flexible algebra for a 2-monad with rank on a complete
and cocomplete 2-category. With suitable cardinality constraints, 2-monads may
themselves be viewed as algebras for such a 2-monad, so that flexible 2-monads are
an instance of the more general notion. The flexible algebras exhibit the same kind
of “looseness” as we saw above, this now being manifested in the result that each
pseudomorphism out of a flexible algebra is isomorphic to a strict one; which is in
fact the same property that allows pseudoalgebras to be replaced by isomorphic
strict ones for a flexible 2-monad. We have the intuitive picture that an algebra for
a 2-monad on Cat is flexible if it admits a presentation in which no equalities are
forced at the level of objects: so that, for example, a monoidal category is flexible
if it may be obtained from a free one through the addition of new morphisms and
new equations between morphisms, but without any new equations being added
between objects.
The notion of flexibility is natural from a 2-categorical perspective, but also
from a homotopy-theoretic one; Lack showed in [20] that, for a 2-monad T with
rank on a locally presentable 2-category C, the 2-category of strict T-algebras
and strict algebra morphisms bears a Quillen model structure whose cofibrant
objects are precisely the flexible algebras. Many results concerning flexibility can
be understood from this homotopy-theoretic perspective, describing as they do
certain good properties of cofibrant objects.
We have above discussed the notion of flexibility in pragmatic terms, showing
that it is strong enough to imply certain desirable properties, and broad enough
to admit all the examples falling under our intuitive picture of flexibility. Yet in
doing so we have glossed over a gap in our understanding. On the one hand, it was
shown in [3, Theorem 4.7] that the ability to replace pseudomorphisms by strict
ones in the manner described above characterises not the flexible algebras, but the
larger class of semiflexible algebras; these being the closure of the flexible algebras
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 3
amongst all algebras under equivalences. On the other, the algebras answering to
our intuitive picture of flexibility, as being those which are “free at the level of
objects”, need not comprise the totality of the flexible algebras, since the flexibles
are closed under retracts, whilst those those which are free in the manner just
mentioned may not be. We thus have three classes of algebras, each containing the
next: the semiflexibles, the flexibles, and a third class, as yet unnamed, comprising
those algebras “free at the level of objects”; and whilst there is a substantial body
of results concerning the flexibles, we do not have a good grasp of what is lost
from these results on moving to the larger class of semiflexibles, or of what is
gained on passing to the smaller third class in which many examples of flexible
algebras reside. One of the two main objectives of this paper is to rectify this,
by proving a number of theorems which completely characterise the algebras in
the three classes. Each of these theorems considers a particular kind of good
behaviour that an algebra may have, and gives three increasingly strong forms of
that behaviour which are respectively equivalent to the algebra’s lying in the first,
second or third of our classes. We will also describe closure properties by which
we can recognise that an algebra constructed in a particular way must lie in one
of these classes, and consider the extent to which these closure properties, in turn,
completely characterise the given classes.
Yet before we can do any of these things, we need to give a precise definition
for the last of our three classes. Clearly, to speak of an algebra for a 2-monad’s
being “free at the level of objects” is vague; and even if made precise, it would be
insufficiently general, limiting us to 2-monads on Cat and similarly well-behaved
2-categories. The second main objective of this paper, then, is to give a fully
general definition of this class, and to analyse its scope for a range of 2-monads of
interest. The algebras in this class will be the pie algebras of our title; and we will
see that in many cases, they capture precisely our intuitive notion of an algebra
“free at the level of objects”.
The motivation for the name pie algebra comes from the theory of 2-categorical
limits. The relevant limits in this context are the weighted limits of [11, Chapter
3], in specifying which one gives not only a diagram D: J → C over which a
universal cone is to be constructed, but also a weight W ∈ [J, Cat] specifying the
nature of the cones amongst which the universal one is to be sought. For a fixed
J, the 2-category of weights [J, Cat] can be viewed as the 2-category of algebras
for a 2-monad on [ob J, Cat], so allowing us to speak of flexible and semiflexible
weights—those which are flexible or semiflexible as algebras—and weighted
by these, semiflexible and flexible limits. These last were studied in [2], where it
was shown that, amongst other things, the flexible limits are precisely those constructible
from products, inserters, equifiers and splittings of idempotents; see [12]
for the definitions of these limit-types.
Pie limits are, by definition, those constructible from products, inserters and
equifiers alone. Many interesting 2-categorical limits are pie—for example, comma
objects, inverters, descent objects, Eilenberg-Moore objects of monads, and pseudo,
lax and oplax limits—and experience shows that pie limits are characterised by
the property that limiting cones force no equations between 1-cells. When this
property of pie limits is re-expressed as a property of the defining weights, it becomes
the statement that the pie weights are those which are “free at the level
4 JOHN BOURKE AND RICHARD GARNER
of objects”, in a sense which was made precise in [29, Corollary 3.3]. In other
words, when we view weights as algebras for a 2-monad, those answering to the
intuitive description of our third class are precisely the pie weights; so motivating
our naming this third class of algebras, the pie algebras. Of course, for this to be a
consistent notation, we must ensure that the pie algebras for the weight 2-monad
are precisely the pie weights. This was in fact done in [24]; we give an alternative
proof in Section 3 below.
Let us now give a more detailed account of the content of this paper. We
begin in Section 2 by defining the notion of pie algebra for a 2-monad T with
rank on a complete and cocomplete 2-category C. For such a T, it was shown
in [3] that the inclusion of T-Algs, the 2-category of strict T-algebras and strict
algebra morphisms, into T-Alg, the 2-category of strict T-algebras and algebra
pseudomorphisms, admits a left adjoint Q: T-Alg → T-Algs; and it is in terms of
this Q that the authors of [3] defined their notions of flexibility and semiflexibility.
We recast these definitions in terms of the 2-comonad on T-Algs induced by Q
and its right adjoint. We will see that an algebra is semiflexible just when it is
a pseudocoalgebra for this 2-comonad, and flexible just when it is a normalised
pseudocoalgebra: and this leads us to define an algebra to be pie when it admits
strict coalgebra structure. The remainder of Section 2 is devoted to further analysis
of the notion of pie algebra, leading to our first main theorem, which provides
conditions under which our intuitive picture of the pie algebras, as those which
are “free at the level of objects”, is confirmed.
We make use of this theorem in Section 3, where we describe the pie algebras for
a range of 2-monads of interest. In particular, we see that a weight W ∈ [J, Cat]
is a pie algebra for the weight 2-monad on [ob J, Cat] just when it is a pie weight,
so confirming the consistency of our terminology with its motivating case. Further
examples show that a monoidal category is pie just when its monoidal structure
is free at the level of objects, that a 2-category is pie just when its underlying
category is free on a graph, and so on.
In Section 4, we turn to the second of our main objectives, that of clarifying
the relationship between the semiflexible, flexible and pie algebras for a 2-monad.
As explained above, we do so through a number of theorems that characterise
the algebras in these three classes in terms of certain good behaviours they may
possess. In fact, we shall give special consideration to the case of weights, providing
alternate forms of our results which characterise the semiflexible, flexible and pie
weights in terms of the behaviour of the corresponding weighted limit functor. Let
us now give a brief overview of the theorems we will prove.
The first of these builds on the characterisation of the semiflexible algebras described
above: an algebra is semiflexible just when every pseudomorphism out of
it may be replaced by an isomorphic strict one. The corresponding results for flexible
and pie algebras require that this replacement should be done in increasingly
well-behaved ways; whilst the corresponding results for semiflexible, flexible and
pie weights concern the manner in which weighted pseudocones may be replaced by
strict ones. As an application of this result, we show that the semiflexible weights
are precisely those whose corresponding limits are also bilimits.
Our second characterisation result takes its most intuitive form for the pie
weights: we show that a weight W ∈ [J, Cat] is pie just when the limit 2-functor
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 5
{W, –}: [J, K] → K admits an extension to a 2-functor Ps(J, K) → K for every
complete 2-category K; here Ps(J, K) denotes the 2-category of 2-functors, pseudonatural
transformations and modifications. The corresponding results for flexible
and semiflexible weights ask for correspondingly weaker extensions of {W, –} to
Ps(J, K); whilst those concerning an algebra A for a general 2-monad deal with
extensions of the hom 2-functor T-Algs(A, –): T-Algs → Cat to T-Alg.
Our third result also characterises each class of algebras in terms of the behaviour
of the hom 2-functor T-Algs(A, –); this time, with respect to certain kinds
of weak equivalence in T-Algs. The flexible case says that the algebras in this
class are precisely the cofibrant objects of the model structure on T-Algs—as was
shown in [20, Theorem 4.12]—whilst in the pie case, it becomes the statement
that these algebras are the algebraically cofibrant objects in the sense of [30].
The corresponding results for weights concern the behaviour of the limit 2-functor
{W, –}: [J, K] → K with respect to certain kinds of pointwise diagram equivalence.
In Section 5, we study closure properties of the three classes of algebras. We
prove that the semiflexibles, flexibles and pie algebras each contain the frees, and
are each closed under the corresponding class of weighted colimits; this gives sufficient
conditions by which to check that an algebra lies in one of these classes. In
the flexible case it is known that this sufficient condition is also necessary—every
flexible algebra is a flexible colimit of frees—and it is therefore natural to ask if
the same is true for the semiflexible or pie algebras. For the semiflexibles, we do
not completely resolve the question; but for the pie algebras, we give a much more
complete answer. We will see that in general, not every pie algebra need lie in the
closure of the frees under pie colimits, but that this will be the case under certain
additional assumptions, satisfied in many of our examples.
We conclude the paper in Section 6 with an extended application of the results
developed in it. We use it to provide a necessary and sufficient characterisation of
the 2-monads on Cat which admit a presentation of the form described above; that
is, one by operations and natural transformations in which no equations are imposed
between derived operations, only between derived natural transformations.
We do not, in fact, consider the most general kind of 2-monad on Cat, but restrict
attention to the strongly finitary 2-monads of [13]: those whose generating operations
are all of the form Cn → C for some natural number n. The 2-category of
such 2-monads is 2-monadic over the product 2-category CatN
, and we show that
with respect to the induced 2-monad on CatN
, the pie algebras are precisely the
strongly finitary 2-monads which admit a presentation of the good form described
above.
2. Pie algebras for a 2-monad
In this section, we introduce the notion of pie algebra for a 2-monad, and give
an analysis showing that in many cases, the pie algebras are precisely the algebras
“free at the level of objects”, in a sense we shall make precise. We start this
section, however, by recalling some necessary 2-categorical preliminaries.
A map f : X → Y of a 2-category C is said to be fully faithful just when for
each A ∈ C, the functor C(A, f): C(A, X) → C(A, Y ) is fully faithful; which is to
say that every 2-cell α: fh ⇒ fk: A → Y in C is of the form fβ for a unique
6 JOHN BOURKE AND RICHARD GARNER
β : h ⇒ k: A → X. It is easy to see that any right adjoint 2-functor preserves
fully faithfuls; and if C is finitely complete, then we can say a little more. For then
2-cells α and β as above correspond to generalised elements of the comma object
f ↓ f and the cotensor product X2, respectively, so that f is fully faithful just
when the canonical arrow X2 → f ↓ f is an isomorphism. We conclude that for
finitely complete C, any finite-limit-preserving 2-functor C → D will preserve fully
faithfuls, and will moreover reflect them if it is conservative.
A map f : X → Y in a 2-category C is called an equivalence if there exists a
map g: Y → X and invertible 2-cells g ◦ f ∼= 1 and f ◦ g ∼= 1. It may be that g
can be chosen such that f ◦ g = 1, or such that g ◦ f = 1; in which case we call
f a surjective or an injective equivalence, respectively. It is easy to see that f is
an equivalence just when it is fully faithful and admits a pseudo-section: a map
g: Y → X with f ◦ g ∼= 1; and similarly, that f is a surjective equivalence just
when it is fully faithful and admits a genuine section.
Throughout the paper, T will denote a 2-monad with rank on a complete and cocomplete
2-category C; recall that a 2-monad has rank if its functor part preserves
λ-filtered colimits for some regular cardinal λ. These hypotheses on C and on T will
always be assumed to be in effect, though we will repeat them for emphasis from
time to time throughout the paper. Following [3], we write T-Algs and T-Alg for
the 2-categories whose objects are, in both cases, the strict T-algebras, and whose
morphisms are, respectively, the strict T-algebra morphisms, and the T-algebra
pseudomorphisms; recall that a pseudomorphism of T-algebras (A, a) (B, b) is
given by a morphism f : A → B between the underlying objects together with
an invertible 2-cell ¯f : b ◦ Tf ∼= f ◦ a satisfying two coherence axioms. We write
F : C ⇆ T-Algs : U for the free-forgetful adjunction induced by the 2-monad T,
and will abuse notation by writing U also for the extension of the forgetful functor
to the category of pseudomorphisms T-Alg. Recall also the notion of pseudoalgebra
for T; this being given by an object A of C equipped with an action a: TA → A
which verifies compatibility with the unit and multiplication of T only up to given
coherent isomorphisms. Though we shall not make direct use of the notion of pseudoalgebra
here, we shall have cause to consider the dual notion of pseudocoalgebra
for a 2-comonad.
We now recall from [3] the notions of flexible and semiflexible T-algebra. Under
our standing hypotheses on C and on T, the central result of [3]—contained in its
Theorem 3.13 and Remark 3.14—assures us that the evident inclusion 2-functor
ι: T-Algs → T-Alg has a left 2-adjoint Q: T-Alg → T-Algs. At an algebra A,
the unit and counit of the adjunction Q ⊣ ι are respectively pseudo and strict
morphisms of algebras qA : A QA and pA : QA → A satisfying the triangle
equations:
A
qA
GGGoGoGoGo
1
55❍❍❍❍❍❍❍❍❍❍ QA
pA

A
and
QA
QqA
GG
1
66❏❏❏❏❏❏❏❏❏❏ Q2A
pQA

A
where we, as is standard, omit to write the action of the identity on objects
inclusion ι: T-Algs → T-Alg. The first triangle equation asserts that the unit
qA is a section of the counit in T-Alg; in fact, as was shown in [3, Theorem 4.2],
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 7
pA : QA → A is a surjective equivalence in T-Alg with equivalence inverse qA. In
particular, pA is fully faithful in T-Alg: but since ι is locally fully faithful, it follows
that pA is also fully faithful in T-Algs.
It sometimes happens that pA : QA → A admits a strict section, a section in
T-Algs, and in this case, the algebra A is said to be flexible. Since pA is fully
faithful, the section a: A → QA is an equivalence inverse of pA, which therefore
becomes a surjective equivalence in T-Algs: so the flexible algebras are equally
those A for which pA is a surjective equivalence in T-Algs. This is [3, Theorem 4.4],
which furthermore characterises A as being flexible just when it is a retract of QB
for some B. On the other hand, it may be that pA does not admit a section
in T-Algs but does admit a pseudo-section: and in this situation the algebra A
is said to be semiflexible. Since pA is fully faithful, such a pseudo-section is an
equivalence inverse for pA, so that A is semiflexible if and only if pA : QA → A is
an equivalence in T-Algs. Another characterisation, given in [3, Theorem 4.7], is
that A is semiflexible just when it is equivalent in T-Algs to some QB.
In order to motivate the definition of pie algebra, we now provide another description
of the flexible and semiflexible algebras: one given in terms of coalgebra
structure for the 2-comonad that is induced on T-Algs by the adjunction
Q: T-Alg ⇆ T-Algs : ι. This comonad should rightly be called Qι, but omitting
to write the ι as above, we will refer to it simply as Q.
Proposition 1. A T-algebra A is semiflexible just when it admits pseudo-Qcoalgebra
structure, and flexible just when it admits normalised pseudocoalgebra
structure.
Proof. Suppose first that A is semiflexible; so we have a: A → QA and an isomorphism
α0 : pA ◦ a ∼= 1A. Since pA is fully faithful, there is a unique 2-cell
β : a ◦ pA
∼= 1QA with pA ◦ β = α−1
0 ◦ pA and now (a, α0): A ⇆ QA: (pA, β) is
an adjoint equivalence in T-Algs. The T-algebra QA underlies the cofree strict
Q-coalgebra (QA, ∆A); and so by Theorem 6.1 of [14] we may transport this
coalgebra structure along the adjoint equivalence with A to obtain the required
pseudo-Q-coalgebra structure on A. Direct calculation shows the induced pseudocoalgebra
structure to have coaction map a: A → QA and counitality constraint
α0 : pA ◦ a → 1A; now if A is flexible, then α0 may be chosen to be the identity,
so that the induced pseudocoalgebra structure is normalised. This proves one direction
of the proposition. For the other, if A admits pseudocoalgebra structure,
then it is equivalent to QA, and hence semiflexible; whilst if the pseudocoalgebra
structure is normalised, then A is a retract of QA and hence flexible.
With this result in mind, it is natural to define a pie algebra to be a T-algebra
which admits strict Q-coalgebra structure. As discussed in the introduction, the
name is motivated by the situation where algebras are weights W ∈ [J, Cat], for
which we have the notion of pie weight, a weight which defines a pie limit. In order
for the naming to be consistent, we must verify that the weights which are pie as
algebras are precisely the pie weights. This is, in fact, Theorem 6.12 of [24]; we
give an alternative proof, not relying on the theory of [24], as Proposition 9 below.
We now wish to give a concrete description of the pie algebras for a range
of 2-monads of interest. Rather than doing so in an ad hoc manner, we will
prove a general result which, under additional hypotheses on T and on C, gives a
8 JOHN BOURKE AND RICHARD GARNER
precise and practical characterisation of the pie T-algebras; this result will then
be employed in giving our examples. In order to state the extra hypotheses that
our result requires, we need the following notion. A morphism f : X → Y of a
2-category C is called objective if for every fully faithful g: W → Z in C, the square
C(Y, W)
C(Y,g)
GG
C(f,W )

C(Y, Z)
C(f,Z)

C(X, W)
C(X,g)
GG C(X, Z)
is a pullback in Cat. Being a pullback at the level of objects says that g is orthogonal
to all fully faithful morphisms in C; at the level of morphisms, it asserts
a two-dimensional aspect to this orthogonality. One consequence of the definition
is that any left adjoint 2-functor preserves objectives, since any right adjoint 2functor
preserves fully faithfuls; another is that the objectives are closed in the
arrow 2-category C2 under 2-dimensional colimits, in particular under retracts.
Due to their orthogonality, objective and fully faithful morphisms will form
a factorisation system on C so long as every morphism does in fact decompose
as an objective followed by a fully faithful. This is the case, for example, when
C = [J, Cat], with the objectives and fully faithfuls being the maps which are
respectively pointwise bijective on objects and pointwise fully faithful. For our
characterisation result, we will suppose that C is a 2-category admitting (objective,
fully faithful) factorisations, and that they lift to T-Algs. By this we mean
that T-Algs should itself admit such factorisations, with a map f of T-Algs being
objective or fully faithful precisely when Uf is correspondingly so in C. The
following result gives some alternative characterisations.
Proposition 2. If C admits (objective, fully faithful) factorisations, then the following
are equivalent:
(1) (Objective, fully faithful) factorisations lift to T-Algs;
(2) T preserves objective morphisms;
(3) Every map of T-Algs admits a factorisation f = g ◦ h where Uh is objective
and Ug is fully faithful.
Proof. (1) ⇒ (2) is trivial: for if (1) holds, then U preserves objectives, as does
its left adjoint F; whence T = UF will also preserve objectives. We consider next
(2) ⇒ (3). Given a T-algebra map f : (A, a) → (C, c), we factorise it in C as an
objective h: A → B followed by a fully faithful g: B → C. Now in the diagram
TA
Th GG
h◦a

TB
c◦Tg
b
||③
③
③
③
③
B g
GG C
the top edge is objective and the bottom edge fully faithful, whence there is a
unique diagonal filler b as indicated. It is easy to verify that this b makes B into a
T-algebra, with respect to which g and h are strict T-algebra maps. So (2) ⇒ (3).
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 9
Finally we show that (3) ⇒ (1). Since U preserves limits and is conservative,
it preserves and reflects fully faithfuls; to obtain (1), it will suffice to show that
U also preserves and reflects objectives. For the preservation, let f : X → Y be
objective in T-Algs. By assumption, we have a factorisation f = gh with Ug fully
faithful and Uh objective. Since U reflects fully faithfuls, g is fully faithful in
T-Algs and so in the diagram
X
f
GG
h

Y
1
k
~~⑦
⑦
⑦
⑦
⑦
Z g
GG Y
there is a unique filler k as indicated. This makes f a retract of h in T-Alg2
s ;
whence Uf is a retract of the objective Uh in C2, and thus itself objective. So U
preserves objectives; we must finally show that it also reflects them. Let f : X → Y
in T-Algs with Uf objective, and consider the commuting diagram:
FUFUX
F UǫX
GG
ǫF UX
GG
F UF Uf

FUX
ǫX
GG
F Uf

X
f

FUFUY
F UǫY
GG
ǫF UY
GG FUY ǫY
GG Y .
Both rows are coequalisers in T-Algs, and so the whole diagram is a coequaliser
in T-Alg2
s . But since Uf is objective, and both U and F preserve objectives, the
two left-hand columns are objective, and so their coequaliser f is as well.
As we have said, our characterisation result will assume that C has (objective,
fully faithful) factorisations which lift to T-Algs; it then aims to show that the pie
T-algebras are the algebras “free at the level of objects”. In order to give meaning
to this last phrase, we will make a further assumption on C, ensuring that every
object of C has a universal cover by one which is discrete in a suitable sense.
Then the 2-monad T will induce an ordinary monad Td on the sub-1-category of
C spanned by these discrete objects, and each T-algebra structure on X ∈ C gives
rise to a Td-algebra structure on X’s discrete cover. Now in saying that a T-algebra
is “free at the level of objects”, we mean to say that this induced Td-algebra is free.
The most obvious notion of discrete object in a 2-category is the following: we
say that X ∈ C is representably discrete when C(A, X) is a discrete category for
each A ∈ C. However, for our purposes it is a different notion which will be relevant.
We call an object X projectively discrete if the covariant hom-functor C(X, –)
preserves objective morphisms. In elementary terms, this says that each morphism
f : X → Z of C admits a unique factorisation through each objective morphism
g: Y → Z, or more briefly: X sees objective morphisms as bijective. In Cat, the
projectively discrete objects are precisely the discrete categories; similarly, when
J is a locally discrete 2-category, the projectively discrete objects of [J, Cat] are
those functors taking their image in discrete categories. In both cases, an object is
projectively discrete if and only if it is representably discrete, but this need not always
be so. If J is a small 2-category which is not locally discrete, then in [J, Cat],
10 JOHN BOURKE AND RICHARD GARNER
all representables are projectively discrete, but not all are representably discrete.
This projective discreteness of representables is an instance of a more general phenomenon:
if a right adjoint 2-functor C → D preserves objective morphisms, then
the corresponding left adjoint will preserve projective discretes. By the preceding
proposition, this is so for the free/forgetful adjunction F : C ⇆ T-Algs : U when C
admits (objective, fully faithful) factorisations and T preserves objectives, so that
in this case free algebras on projective discretes are again projectively discrete.
A 2-category C will be said to have enough discretes if every X ∈ C admits an
objective morphism from a projective discrete. It is easy to see that this is the case
when C = Cat, or when C = [J, Cat] for some small and locally discrete J; as we
will explain shortly, it is in fact true of [J, Cat] for any small 2-category J. If C is
a 2-category with enough discretes, then on choosing for each X ∈ C an objective
morphism λX : DOX → X from a projective discrete, we obtain a coreflection
Cd
D GG
⊥ C0
O
oo (2.1)
with pointwise objective counit; here C0 is the underlying category of C, and Cd is
its full subcategory on the projective discretes. In this circumstance it is easy to
see that the maps inverted by O are precisely the objective morphisms.
Note that, for example, Catd is the full subcategory of Cat spanned by the
discrete categories; in practice, we would rather work with the equivalent category
of sets and functions. In our examples, we will therefore allow Cd to denote a
category which is merely equivalent to the full subcategory of C0 on the projective
discretes; this does not change the properties of the adjunction (2.1), these being
equivalence-invariant, but does afford us the convenience of taking Catd = Set,
or [J, Cat]d = [J0, Set] for locally discrete J, and so on; henceforth, we will do so
without further comment.
In our examples, we only need to know that [J, Cat] has enough discretes for J
locally discrete; but in fact, as remarked above, every presheaf 2-category [J, Cat]
has enough discretes, again with [J, Cat]d = [J0, Set]. As the theory we develop will
apply also in this situation, let us briefly justify this less obvious claim. Consider
the adjunction
[J0, Set]
D GG
⊥ [J0, Cat]0
O
oo
F GG
⊥ [J, Cat]0
U
oo
obtained by composing that which exhibits [J0, Set] as [J0, Cat]d with that given
by restriction and left Kan extension along the inclusion 2-functor J0 → J. Clearly
U preserves objectives, and so F preserves projective discretes; thus objects in the
image of FD are projectively discrete, and so we will be done if we can show the
counit ǫ: FDOU → 1 to be pointwise objective. Since U preserves and reflects
objectives and O inverts precisely the objectives, this is equally to show that OUǫ is
invertible, which will follow if we can show the unit η: 1 → OUFD to be invertible.
Now, the component of η at a representable J0(X, –) is the image under O of the
objective morphism J0(X, –) → J(X, –) of [J0, Cat]0, and so invertible; but since
each of O, U, F and D preserve (1-dimensional) colimits, we conclude that every
component of η is invertible. Thus OUǫ is invertible, ǫ is pointwise objective, and
so [J, Cat] has enough discretes with [J, Cat]d = [J0, Set] as claimed.
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 11
Suppose now that C is a 2-category with enough discretes, and T a 2-monad
on it. Underlying the free-forgetful adjunction of T, we have the ordinary adjunction
F0 : C0 ⇆ (T-Algs)0 : U0 generating the monad T0 on C0; and quite
clearly T0-Alg = (T-Algs)0. Composing (2.1) with this adjunction, we induce
a monad Td = OT0D on Cd, and now have the canonical comparison functor
j : T0-Alg → Td-Alg making both triangles commute in:
T0-Alg
j
GG
OU
66❏❏❏❏❏❏❏❏❏❏❏❏
Td-Alg .
Ud
zztttttttttttt
Cd
F D
dd❏❏❏❏❏❏❏❏❏❏❏❏
Fd
XXtttttttttttt
(2.2)
Explicitly, j’s action on objects is given by:
(a: TA → A) → (OTDOA
OTλA
−−−−→ OTA
Oa
−−→ OA) .
It is in terms of this functor j that we may speak of a T-algebra’s being “free at
the level of objects”; and we are thus ready to state our characterisation result.
Theorem 3. Let C have enough discretes and (objective, fully faithful) factorisations
lifting to T-Algs. If Cd has, and the induced monad Td preserves, coreflexive
equalisers, then a T-algebra A is pie if and only if jA is a free Td-algebra.
The remainder of this section gives the proof of this theorem. To begin with,
we assume only our standing hypotheses that C be complete and cocomplete and
that T have rank. We then have the adjunction ι: T-Algs ⇆ T-Alg: Q as before,
and underlying each unit map qA : A QA a morphism UqA : UA → UQA in C.
Transposing this under the free-forgetful adjunction F : C ⇆ T-Algs : U yields a
map in T-Algs, which we denote by ρA : FUA → QA.
Lemma 4. Each ρA is an objective morphism of T-Algs.
Proof. One way of proving this would be to analyse Q in terms of isocodescent
objects, as was done in [18]; however, it is just as easy to proceed directly. For the
one-dimensional aspect of ρA’s objectivity, we must show that for any square
FUA
ρA
GG
h

QA
{{✇
✇
✇
✇
✇
k

C
f
GG D
in T-Algs with f fully faithful, there is a unique map QA → C as indicated making
both induced triangles commute. Equivalently, after transposing under adjunction,
we must show that for any pseudomorphism (k, ¯k): A D, each factorisation of
k: UA → UD through Uf is the image under U of a unique factorisation of (k, ¯k)
through f. Thus given k = Uf ◦ h we seek a unique ¯h such that (h, ¯h): A C is
a pseudomorphism with (k, ¯k) = f ◦ (h, ¯h). This last condition asserts a pasting
12 JOHN BOURKE AND RICHARD GARNER
equality
UFUA
UF k GG
¯kUǫA

UFUD
UǫD

UA
k
GG UD
=
UFUA
UF h GG
UǫA

¯h
UFUC
UF Uf
GG
UǫC

=
UFUD
UǫD

UA
h
GG UC
Uf
GG UD ;
but as f is fully faithful in T-Algs, Uf is fully faithful in C, whence there is
exactly one 2-cell ¯h for which this is so. It is now straightforward to check that
this ¯h does indeed make (h, ¯h) into a pseudomorphism A C. This verifies the
one-dimensional aspect of ρA’s objectivity, and the two-dimensional aspect follows
similarly.
Lemma 5. The ρA’s are the components of a comonad morphism ρ: FU → Q;
moreover, Q is, up to isomorphism, the unique comonad on T-Algs that has pointwise
fully faithful counit and admits a pointwise objective comonad morphism from
FU.
Proof. It is easy to see that ρ is 2-natural, since q is, and also that the axiom
p ◦ ρ = ǫ: FU → 1 expressing compatibility with the counits of Q and FU is
satisfied. We must further show that ∆ ◦ ρ = ρρ ◦ FηU, where ∆ = Qq is the
comultiplication of Q. The map ∆◦ρ corresponds under adjunction to U∆◦Uq =
UQq ◦ Uq = UqQ ◦ Uq, but also ρρ ◦ FηU = ρQ ◦ FUρ ◦ FηU corresponds to
UqQ ◦ Uρ ◦ ηU = UqQ ◦ Uq, so that ∆ ◦ ρ = ρρ ◦ FηU as required.
Now given any comonad (R, e, d) on T-Algs which has pointwise fully faithful
counit e and admits a pointwise objective comonad morphism σ: FU → R, we
have a commutative square in [T-Algs, T-Algs] as on the left of
FU
ρ
GG
σ

Q
p

α
~~⑤
⑤
⑤
⑤
⑤
⑤
R e
GG 1
FU
ρ
GG
F ηU

Q
}}④
④
④
④
④
④
④
α

FUFU
σσ 
RR
eQ
GG R .
Both sides of this square being (objective, fully faithful) factorisations of ǫ, there is
a unique isomorphism α: Q → R as indicated making both triangles commute. By
construction, this α is compatible with the counits of Q and R; whilst compatibility
with the comultiplications follows by observing that the square on the right above
has, by orthogonality, a unique diagonal filler, and that both d ◦ α and αα ◦ ∆ are
such fillers.
In the statement of the following result, we write T0-Alg/ffA for the full subcategory
of T0-Alg/A spanned by the fully faithful maps into A.
Lemma 6. If C has enough discretes, and T preserves objectives, then for each
T-algebra A, the functor
jA : T0-Alg/ffA → Td-Alg/jA ,
induced by slicing the comparison functor j of (2.2), is fully faithful.
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 13
Proof. Given a T-algebra (A, a) and objects (B, b)
m
−→ (A, a) and (C, c)
n
−→ (A, a)
of T0-Alg/ff(A, a), we must show that each map jA(m) → jA(n) in Td-Alg/jA is
the image under jA of a unique map m → n. To give a map jA(m) → jA(n) is to
give a map g: OB → OC in Cd making the left-hand diagram below commute:
OTDOB
OTDg
GG
OTλB

OTDOC
OTλC

OTB
Ob

OTC
Oc

OB
Om 77❑❑❑❑❑❑❑❑❑❑ g
GG OC
Onyyssssssssss
OA
DOB
λC ◦Dg

λB
GG B
m

k
||②
②
②
②
②
②
C n
GG A .
Now the commutativity of the lower triangle on the left is equivalent to that of
the right-hand square in C0; and since λB is objective and n is fully faithful, there
is a unique diagonal filler k as indicated. Commutativity of the upper triangle so
induced asserts that λC ◦Dg = k ◦λB = λC ◦DOk which, transposing through the
adjunction, is equally well the assertion that g = Ok. Thus it will follow that k is
the desired unique lifting of g so long as we can show that it is in fact a T-algebra
map (B, b) → (C, c), or in other words, that k ◦ b = c ◦ Tk: TB → C.
By orthogonality, it will suffice to verify this equality on postcomposition with a
fully faithful morphism and on precomposition with an objective one. On the one
hand, we have the fully faithful n: C → A, and calculate that n ◦ k ◦ b = m ◦ b =
a ◦Tm = a ◦Tn ◦Tk = n ◦c ◦Tk as required. On the other, we have the objective
λTB : DOTB → TB, and showing that k ◦ b ◦ λTB = c ◦ Tk ◦ λTB is equivalent
to showing that Ok ◦ Ob = Oc ◦ OTk. Since Ok = g, we have from the left-hand
commutative diagram above that Ok ◦ Ob ◦ OTλB = Oc ◦ OTλC ◦ OTDOk =
Oc ◦ OTk ◦ OTλB. But since λB is objective, so also is TλB, whence OTλB is
invertible; so that Ok ◦ Ob = Oc ◦ OTk as required.
Using this, we may now prove a direct predecessor of our main Theorem 3:
Proposition 7. If C has enough discretes and (objective, fully faithful) factorisations
lifting to T-Algs, then a T-algebra A is pie if and only if jA admits FdUdcoalgebra
structure.
Proof. We first observe that j : T0-Alg → Td-Alg inverts objective morphisms.
Indeed, if f is objective in T0-Alg, then Uf is objective in C0, whence OUf is
invertible in Cd. But OUf = Udjf and Ud is conservative, and so jf is invertible
in Td-Alg as claimed.
Now let Q0 denote the comonad on T0-Alg underlying the 2-comonad Q. By
Lemmas 4 and 5, we have a comonad morphism ρ0 : F0U0 → Q0 which is pointwise
objective; since j inverts objectives, we obtain from this a natural isomorphism
jρ0 : jF0U0 → jQ0. We also have a comonad morphism F0λU0 : F0DOU0 → F0U0,
which is again pointwise objective, since λ is so and F0 preserves objective morphisms,
and thus we obtain a natural isomorphism jF0λU0 : jF0DOU0 → jF0U0.
14 JOHN BOURKE AND RICHARD GARNER
Now by composing inverses we obtain a natural isomorphism
φ := jQ0
(jρ0)−1
−−−−−−−−→ jF0U0
(jF0λU0)−1
−−−−−−−−−−−→ jF0DOU0 = FdUdj
which may be shown to equip j with the structure of a comonad morphism
(T0-Alg, Q0) → (Td-Alg, FdUd). Thus if A is a pie algebra—hence admitting
Q-coalgebra structure—then jA is a FdUd-coalgebra. Conversely, suppose that jA
is a FdUd-coalgebra, with structure map g: jA → FdUdjA. Then the left-hand
diagram commutes in:
jA
j(1)
88▲▲▲▲▲▲▲▲▲▲▲▲
g
GG FdUdjA
ǫjA

φ−1
GG jQA
j(pA)
xxrrrrrrrrrrr
jA
A
1
44❊❊❊❊❊❊❊❊❊
a GG QA .
pA
||②②②②②②②②
A
Since both 1: A → A and pA : QA → A are fully faithful T-algebra maps, we
have, by the preceding lemma, a T-algebra morphism a: A → QA as on the right
with j(a) = φ−1 ◦ g. We will show that this a satisfies the comultiplication axiom,
so making A into a Q-coalgebra and hence a pie algebra. So observe that both
triangles in the diagram:
A
1
66■■■■■■■■■■■
∆A◦a
GG
Qa◦a
GG QQA
ppA
zz✉✉✉✉✉✉✉✉✉
A
are commutative; moreover, both diagonal maps are fully faithful and so to show
that the two horizontal maps are equal, as we must, it is enough by the preceding
lemma to do so after applying j: which follows by an entirely straightforward
calculation.
Finally, we obtain:
Proof of Theorem 3. Comparing this theorem’s statement with that of Proposition
7, we see that the gap between the two is the gap between the free Td-algebras
and the FdUd-coalgebras. This gap is certainly closed if the adjunction Fd ⊣ Ud
is comonadic as well as monadic, but this is more than is needed. A necessary
and sufficient condition is that for each coalgebra a: A → FdUdA, the parallel pair
(ηUdA, Uda): UdA ⇒ UdFdUdA in Cd should admit an equaliser which is preserved
by Fd; for then it follows that the equaliser map E ֌ UdA exhibits A as free on E.
This condition will certainly be satisfied if the category Cd has, and Fd preserves,
all coreflexive equalisers; but since Ud creates limits, this is equally to ask, as in the
statement of this theorem, that Cd have and Td preserve coreflexive equalisers.
3. Examples of pie algebras
We now apply the results developed in the previous section to the analysis of
the pie algebras for some 2-monads of interest. In each of our examples, we shall
verify the conditions of Theorem 3 and use it to give an explicit description of the
class of pie algebras.
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 15
3.1. Categories with structure. As we commented in the introduction, many
natural examples of 2-monads reside on the 2-category Cat, their algebras being
certain kinds of structured category. We have already seen that Cat has enough
discretes and (objective, fully faithful) factorisations, and for any given 2-monad
on Cat it is relatively straightforward to verify whether the remaining hypotheses
of Theorem 3 are satisfied. We consider in detail one example: the 2-monad T for
strict monoidal categories.
First observe that if F : A → C is a strict monoidal functor between strict
monoidal categories, and F = GH a (bijective on objects, fully faithful) factorisation
of its underlying functor, then the interposing category B admits a strict
monoidal structure with respect to which both G and H are strict monoidal; the
reason that this is possible is that the arities required to describe strict monoidal
structure, the discrete categories 0, 1, 2 and 3, are projectively discrete. It therefore
follows from Proposition 2 that (objective, fully faithful) factorisations lift to
T-Algs.
Now consider the induced monad Td on Catd = Set. This sends a set X to the set
of objects of the free strict monoidal category on X; it is thus the monoid monad
on Set, and the comparison functor j : T0-Alg → Td-Alg sends a strict monoidal
category to its underlying monoid of objects. Finally, because the monoid monad
n∈N(–)n is a coproduct of representable functors, it preserves all connected limits,
and so in particular coreflexive equalisers. We therefore conclude from Theorem 3
that:
Proposition 8. A strict monoidal category is pie just when its underlying monoid
of objects is free; that is, just when it is a (many-sorted) PRO in the sense of [27].
Arguments of an entirely analogous form characterise the pie algebras for many
other 2-monads on Cat; we list a few below as a representative sample.
• Symmetric strict monoidal categories: (objective, fully faithful) factorisations
again lift to T-Algs, and the induced monad Td on Set is again the monoid
monad. Thus pie algebras are symmetric strict monoidal categories whose
underlying monoid of objects is free; in other words, they are (many-sorted)
PROPs in the sense of [27].
• Categories with strictly associative and unital finite products: arguing exactly
as in the preceding two examples, we conclude that in this case the pie algebras
are (many-sorted) Lawvere theories.
• (Symmetric) monoidal categories: this time the induced monad on Set is the
monad for pointed magmas: these being sets equipped with a binary and a
nullary operation, satisfying no laws. The underlying functor of this monad,
like that of any monad free on a signature, is a coproduct of representables, and
so preserves coreflexive equalisers as before; thus by Theorem 3, a (symmetric)
monoidal category is pie just when its underlying pointed magma of objects
is free.
This last example was particularly easy to analyse by virtue of the induced
monad Td’s being free on a signature. This property is shared by many other 2monads
on Cat, including those whose algebras are categories with finite products,
16 JOHN BOURKE AND RICHARD GARNER
or distributive categories, or categories equipped with a monad, or categories with
finite products and monoidal structure, or linearly distributive categories, and so
on. We will consider such 2-monads in more detail in Section 6 below; in particular,
Proposition 37 shows that the above characterisation of the pie monoidal categories
is equally valid for the algebras of any such 2-monad.
3.2. Weights. Given some small 2-category J, we may consider the adjunction
F : [ob J, Cat] ⇆ [J, Cat]: U given by restriction and left Kan extension along the
inclusion of objects ob J → J. This adjunction is strictly 2-monadic, so allowing
us to identify [J, Cat] with the 2-category of algebras for the induced 2-monad
T = UF. As in the introduction, objects of [J, Cat] are to be thought of as weights
for limits or colimits; and our goal is to show that a weight W is pie as a T-algebra
just when it determines a pie limit—that is, one constructible from products,
inserters and equifiers. The pie limits were the primary object of study of [29],
and Corollary 3.3 of that paper shows that a weight W ∈ [J, Cat] determines a pie
limit just when the presheaf
J0
W0
−−→ Cat0
ob
−→ Set
is a coproduct of representables. Our goal, then, is to show that a weight W has
this property just when it is pie as a T-algebra. As mentioned before, this was done
in [24, Theorem 6.12]; however, it is quite straightforward to give an alternative
proof using our Theorem 3.
Observe first that [ob J, Cat] has (objective, fully faithful) factorisations which
lift to T-Algs = [J, Cat], as in both 2-categories a morphism is objective or fully
faithful just when it is pointwise so. As for the induced monad Td on [ob J, Cat]d =
[ob J, Set] we may calculate this to be given by the formula
(TdX)(j) = k∈J J0(k, j) × Xk .
This shows Td to be pointwise a coproduct of representable functors; as such it
preserves connected limits and in particular coreflexive equalisers. It furthermore
shows that Td-Alg ∼= [J0, Set], and now under this identification, the comparison
functor T0-Alg → Td-Alg becomes the functor [J, Cat]0 → [J0, Set] sending
W : J → Cat to the presheaf ob◦W0. We therefore conclude from Theorem 3 that:
Proposition 9. A weight W : J → Cat is a pie algebra for the weight 2-monad
on [ob J, Cat] just when its underlying presheaf ob ◦ W0 : J0 → Set is free, i.e., a
coproduct of representables; that is, just when W determines a pie limit.
Let us remark that our proof of the above proposition was not self-contained
but required an application of Corollary 3.3 of [29]. In fact, we could equally have
appealed to our Proposition 31 below, which can be seen as a generalisation of
Power and Robinson’s result from weights to more general algebras.
3.3. 2-categories and related structures. For our third example, we consider
2-categories and related structures as algebras for a 2-monad, and determine which
of these algebras are pie. We begin with the case of 2-categories.
By a category-enriched graph, we mean a diagram A1 ⇒ A0 of categories with
A0 discrete. We write P for the parallel pair category • ⇒ •, and write Cat-Gph for
the full and locally full sub-2-category of [P, Cat] spanned by the category-enriched
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 17
graphs. On the mere category [P, Cat]0, we have the monad whose algebras are
categories internal to Cat—thus double categories—and it is easy to enrich this to
a 2-monad S on [P, Cat], and then to restrict this to a 2-monad T on Cat-Gph.
The algebras for T are 2-categories; the strict algebra morphisms are 2-functors,
and the pseudomorphisms are pseudofunctors. As for the algebra 2-cells, these are
the icons of [23, 22]; recall that an icon between two 2-functors G, H : A ⇒ B is an
oplax natural transformation α: G ⇒ H whose 1-cell components are all identities.
We write Icon for the 2-category of 2-categories, 2-functors and icons; it is thus
T-Algs for the above-defined T, and our task is to identify the pie T-algebras.
It turns out that the results of Theorem 3 are not directly applicable in this
situation, as although Cat-Gph has (objective, fully faithful) factorisations, these
do not lift to Icon. We must therefore follow a more roundabout route to our
characterisation. We will reduce the problem to that of characterising the pie
2-categories with a fixed object set X, seen as algebras for a 2-monad on categoryenriched
graphs with object set X; the point being that in this situation, (objective,
fully faithful) factorisations do lift, with the other hypotheses of Theorem 3 being
similarly satisfied.
Thus given some set X, we let IconX denote the locally full sub-2-category of
Icon spanned by the 2-categories with object set X and the identity on objects
2-functors between them. There is a corresponding sub-2-category Cat-GphX of
Cat-Gph, and the free-forgetful adjunction F : Cat-Gph ⇆ Icon: U restricts to an
adjunction FX : Cat-GphX ⇆ IconX : UX which is again strictly 2-monadic. If we
call the induced 2-monad TX , then our claim is that a 2-category A with object
set X is pie as a T-algebra just when it is pie as a TX -algebra. This will follow
from:
Lemma 10. The comonad Q on Icon associated with T may be so chosen as to
restrict to IconX for each X, there yielding the comonad QX associated with TX.
For indeed, if the above lemma holds then certainly each QX -coalgebra yields a
Q-coalgebra structure. Conversely, if A ∈ IconX admits a Q-coalgebra structure
a: A → QA then pA’s being the identity on objects ensures that so too is a which,
as a morphism of IconX , thereby exhibits A as a QX-coalgebra.
In proving this lemma, we will need a characterisation of the fully faithful maps
both in Icon and IconX , and some understanding of the objectives. With regard
to Icon, note that both the forgetful 2-functor Icon → Cat-Gph and the inclusion
Cat-Gph → [P, Cat] are limit-preserving and conservative, whence a 2-functor
G: A → B is fully faithful in Icon just when its underlying morphism
A1
 
G1
GG B1
 
A0
G0
GG B0
(3.1)
is so in [P, Cat]. This happens just when G1 is a fully faithful functor, and G0 an
injective function, so that the fully faithful maps in Icon are the 2-functors which
are injective on objects and locally fully faithful.
18 JOHN BOURKE AND RICHARD GARNER
In the case of IconX we have the limit-preserving and conservative forgetful
2-functor IconX → Cat-GphX , so that a morphism G: A → B is fully faithful
in IconX just when its underlying morphism is so in Cat-GphX. This latter 2category
is equivalent to [X × X, Cat], and from our understanding of the fully
faithful morphisms therein, we deduce that G is fully faithful in IconX just when
its underlying morphism (3.1) has G1 fully faithful; which is to say that G is a
locally fully faithful 2-functor.
Now IconX admits a (bijective on 1-cells, locally fully faithful) factorisation
system; since the right class comprises the fully faithfuls in IconX, it follows that
the left class comprises the objectives. Considering the inclusion IconX → Icon,
it is immediate that it preserves fully faithfuls; but it also preserves objectives,
since in Icon, a 2-functor which is the identity on objects and 1-cells is orthogonal
to any locally fully faithful 2-functor, and so certainly objective. Let us take
this opportunity also to remark that the objectives in Cat-GphX are again the
morphisms bijective on 1-cells, so that (objective, fully faithful) factorisations in
IconX are in fact lifted from Cat-GphX .
Proof of Lemma 10. If A is a 2-category with object set X, then by Lemma 5, we
have an (objective, fully faithful) factorisation
FUA
ǫA
−→ A = FUA
ρA
−→ QA
pA
−→ A , (3.2)
where ǫ is the counit of the comonad FU on Icon. Since ǫA : FUA → A is the
identity on objects, it may be viewed as a map of IconX; seen as such, it is in
fact the counit component at A of the comonad FX UX on IconX , and we therefore
have in IconX an (objective, fully faithful) factorisation
FUA
ǫA
−→ A = FUA
(ρX )A
−−−−→ QXA
(pX )A
−−−−→ A . (3.3)
The above discussion shows this also to be an (objective, fully faithful) factorisation
in Icon, and we conclude that QA is uniquely isomorphic to QX A for each object
A ∈ IconX. Transporting along these unique isomorphisms, we may take it that
in fact QA = QXA, ρA = (ρX )A and pA = (pX )A for each A ∈ IconX.
On doing so, we have the endofunctor Q and the counit p: Q → 1 both the identity
on objects; because pQ ◦ ∆ = 1, it follows from this that the comultiplication
∆: Q → QQ is also the identity on objects, whence the comonad Q restricts from
Icon to IconX for each set X. But by construction, the factorisation (3.2) restricts
to give the factorisation (3.3), so that by Lemma 5, the restricted comonad is
indeed QX .
Using this lemma, our problem is thus reduced to that of characterising the pie
TX -algebras. As previously noted the (objective, fully faithful) factorisations of
Cat-Gph lift to IconX
∼= TX -Algs and since Cat-GphX is equivalent to [X×X, Cat]
it has enough discretes. We may take (Cat-GphX )d to be [X × X, Set]: now the
induced monad (TX )d is the monad for categories with object set X, and the
comparison functor j : (TX )0-Alg → (TX )d-Alg sends a 2-category with object set
X to its underlying ordinary category. To see that (TX )d preserves coreflexive
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 19
equalisers, observe that it is given by
(TX )d(A)(x, y) =
x=x0,...,xn=y∈X
A(x0, x1) × · · · × A(xn−1, xn) .
It is thus pointwise a coproduct of representables and so preserves connected limits,
in particular coreflexive equalisers. We conclude from Theorem 3 that:
Proposition 11. A 2-category is pie just when its underlying ordinary category
is free on a graph.
The arguments given above may be generalised in two directions. The first is to
consider bicategories in place of 2-categories. In this case, we begin by considering
the 2-category of bicategories, strict homomorphisms and icons, seeing that this
is again 2-monadic over Cat-Gph, and now continue the argument as before. Our
conclusion will differ in the nature of the underlying structure which is required to
be free. As in [19], we define a compositional graph to be a directed graph X1 ⇒ X0
equipped with identity and composition operations, subject to no axioms: thus a
category is a compositional graph in which the unit and associativity laws are
verified. The objects and 1-cells of any bicategory form a compositional graph,
the underlying compositional graph of the bicategory. Now our result is that:
Proposition 12. A bicategory is pie if and only if its underlying compositional
graph is free.
The second direction of generalisation concerns double categories. This generalisation
is in fact also a simplification, as in this case Theorem 3 will be immediately
applicable. Recall from above that we formed the 2-category Cat-Gph
as a full sub-2-category of [P, Cat], and the 2-monad T for 2-categories thereon
as the restriction and corestriction of a 2-monad S on [P, Cat]. This latter 2monad
we constructed as a 2-dimensional enrichment of the free category monad
on [P, Cat]0; as such the strict S-algebras are categories internal to Cat, that is,
double categories. The corresponding strict and pseudo algebra morphisms are
double functors and pseudo double functors, respectively, whilst the algebra 2-cells
are horizontal transformations; our terminology throughout is that of [7].
The 2-category [P, Cat] on which S resides has (objective, fully faithful) factorisations;
what distinguishes this case from that of 2-categories is that these
factorisations lift. The easiest way of seeing this is to observe that a strict double
functor G, when seen as an internal functor in Cat, admits a factorisation
A1
 
G1
GG B1
 
A0
G0
GG B0
=
A1
 
H1
GG C1
 
K1
GG B1
 
A0
H0
GG C0
K0
GG B0
into a pair of internal functors for which H0 and H1 are bijective on objects, and K0
and K1 are fully faithful; now by applying Proposition 2, the claim follows. The
2-category [P, Cat] has enough discretes, with [P, Cat]d = [P, Set], the category
of graphs; the induced monad Sd on [P, Set] is the free category monad, and
the comparison functor j : S0-Alg → Sd-Alg assigns to each double category its
underlying category of objects and vertical arrows. Finally, as the free category
20 JOHN BOURKE AND RICHARD GARNER
monad is pointwise a coproduct of representables, it preserves connected limits
and in particular coreflexive equalisers; thus we conclude from Theorem 3 that:
Proposition 13. A double category is pie if and only if its underlying vertical
category is free on a graph.
We may combine the two generalisations given above to obtain a fourth characterisation
result for the pseudo double categories of [7]; it states that a pseudo
double category is pie just when its underlying compositional graph of objects and
vertical arrrows is free.
4. Characterisation theorems for algebras and weights
We now turn to the second main objective of the paper, that of clarifying the
relationship between the semiflexible, flexible and pie algebras for a 2-monad; we
do so under our standing assumptions that the 2-monad in question should have
rank and should reside on a complete and cocomplete 2-category. We prove three
theorems, each describing a property which, when given in its weakest form, characterises
completely the semiflexible algebras; when strengthened slightly, characterises
the flexible algebras; and when strengthened further still, characterises the
pie algebras.
In fact, we pay special attention to the case of the semiflexible, flexible and
pie weights, and so each of our characterisation theorems will have a second form
dealing with these. So as to be able to give this second form, we now establish some
notational conventions concerning weighted limits. For any weight W ∈ [J, Cat]
and any 2-category K, we have a 2-functor
ConeW : Kop
× [J, K] → Cat
(X, D) → [J, Cat](W, K(X, D–)) ;
we call an object α ∈ ConeW (X, D) a W-weighted cone from X to D, and denote
it by α: X ˙→D. We write β ◦ α and α ◦ f for its postcomposition with a 2-natural
transformation β : D → D′ and its precomposition with a map f : X′ → X. For
some given D: J → K, a limit of D weighted by W is a representation for the
2-functor ConeW (–, D): Kop → Cat. Of course, if the 2-category K admits Wweighted
limits for all D ∈ [J, K], then on making a choice of such, we obtain a
2-functor {W, –}: [J, K] → K.
Given W ∈ [J, Cat] and a 2-category K, we can also consider the 2-functor
PsConeW : Kop
× Ps(J, K) → Cat
(X, D) → Ps(J, Cat)(W, K(X, D–)) ,
whose elements α ∈ PsConeW (X, D) we call W-weighted pseudocones α: X ˙ D;
now given some D: J → K, a pseudolimit of D weighted by W is a representation
for PsConeW (–, D). If the 2-category K admits W-weighted pseudolimits
for all D: J → K, then on making a choice of such we obtain a 2-functor
{W, –}ps : Ps(J, K) → K.
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 21
4.1. First characterisation. As mentioned in the introduction, the semiflexible
algebras are characterised by the property that any algebra pseudomorphism out
of one may be replaced by an isomorphic strict morphism; this was proven in [3,
Theorem 7.2]. What we will now show is that the flexibles and pies may be further
distinguished by the increasingly well-behaved manner in which this replacement
can be done. The corresponding version of this result for weights concerns the
manner in which pseudocones may be replaced by strict ones.
Theorem 14. A T-algebra A is semiflexible, flexible, or pie, just when, respec-
tively:
(a) There is a function assigning to each pseudomorphism f : A B a strict
morphism R(f): A → B which is isomorphic to f in T-Alg(A, B);
(b) Moreover, this R may be chosen so that R(f) = f when f is strict, and so that
R(g ◦ f) = g ◦ R(f) for each f : A B and g: B → C;
(c) Moreover, this R may be chosen so that R(g ◦ R(f)) = R(g ◦ f) for each
f : A B and g: B C.
In fact, we will only prove one direction of this implication here, namely that
the semiflexibles, flexibles and pies satisfy clauses (a), (b) and (c) respectively.
The other direction will be obtained through a cycle of implications which will be
completed in the proofs of Theorems 17 and 20 below.
Proof. If A is semiflexible, then pA : QA → A admits a pseudoinverse a: A → QA
in T-Algs. Now the inclusion T-Algs(A, –) → T-Alg(A, ι–) is equally the composite
T-Algs(A, –)
T-Algs(pA,–)
−−−−−−−−→ T-Algs(QA, –)
∼=
−→ T-Alg(A, ι–)
which consequently has pseudoinverse
R: T-Alg(A, ι–)
∼=
−→ T-Algs(QA, –)
T-Algs(a,–)
−−−−−−−→ T-Algs(A, –)
in [T-Algs, Cat]. This R assigns to each pseudomorphism f : A B a strict
morphism R(f) which is isomorphic to f, as required.
If now A is flexible, then the a above may be chosen to be a section of pA, so
that the corresponding R satisfies R(f) = f for any strict f : A → B. That also
R(g ◦ f) = g ◦ R(f) for any f : A B and g: B → C is a direct consequence of
R’s 2-naturality.
Finally, if A is pie, then the a above may be chosen so as to exhibit A as a
Q-coalgebra; now given pseudo maps f : A B and g: B C, corresponding to
strict maps ¯f : QA → B and ¯g : QB → C, we calculate that
R(g ◦ R(f)) = (¯g ◦ Q( ¯f ◦ a)) ◦ a = ¯g ◦ Q ¯f ◦ (Qa ◦ a)
= ¯g ◦ Q ¯f ◦ (∆A ◦ a) = (¯g ◦ Q ¯f ◦ ∆A) ◦ a = R(g ◦ f) .
The version of this result for weights is now:
Theorem 15. A weight W ∈ [J, Cat] is semiflexible, flexible, or pie, just when,
respectively:
22 JOHN BOURKE AND RICHARD GARNER
(a) For every 2-category K and every X ∈ K, there is a function assigning to
each W-weighted pseudocone α: X ˙ D a strict cone R(α): X ˙→D which is
isomorphic to α in PsConeW (X, D);
(b) Moreover, these R may be chosen so that R(α) = α whenever α is a strict cone,
so that R(β ◦ α) = β ◦ R(α) for each α: X ˙ D and 2-natural β : D → D′, and
so that R(α ◦ f) = R(α) ◦ f for each α: X ˙ D and f : X′ → X ∈ K;
(c) Moreover, these R may be chosen so that R(β ◦ R(α)) = R(β ◦ α) for each
α: X ˙ D and pseudonatural β : D D′.
Again, we prove only the forward implication, with the other direction now
following from the proofs of Theorems 18 and 21 below.
Proof. Viewing W as an algebra for the weight 2-monad on [ob J, Cat], the strict
and pseudo cones from X to D are now strict and pseudo algebra morphisms with
domain W. This being the case, we see that each clause of the present theorem has
a corresponding clause in Theorem 14 of which it is an immediate consequence; the
only exception being that in (b) above we also demand that R(α ◦ f) = R(α) ◦ f
for each α: X ˙ D and each f : X′ → X in K. However, the pseudocone α ◦ f is
given by the composite
W
α GGGoGoGoGo K(X, D–)
K(f,D–)
GG K(X′, D–) ,
and since K(f, D–) is 2-natural and occurs to the right of α, we see that this clause
too follows from Theorem 14(b).
Recall that, given W ∈ [J, Cat] and D: J → K, a bilimit of D weighted by
W is a birepresentation for the 2-functor PsConeW (–, D); that is, a pseudocone
α: U ˙ D, composition with which induces an equivalence of categories K(X, U) →
PsConeW (X, D) for each X ∈ K. Now given a limit {W, D} existing in a 2-category
K, we may ask whether its limiting cone α: {W, D} ˙→D also exhibits {W, D} as
the W-weighted bilimit of D; which is equally to ask that the composite
K(X, {W, D}) ∼= ConeW (X, D) → PsConeW (X, D) (4.1)
should be an equivalence for each X ∈ K. If this is so for every W-weighted limit
existing in every 2-category, let us then say that limits weighted by W are bilimits.
Corollary 16. A weight W is semiflexible if and only if limits weighted by W are
bilimits.
Proof. If W ∈ [J, Cat] is a semiflexible weight, then by Theorem 15(a), each inclusion
ConeW (X, D) → PsConeW (X, D) is essentially surjective on objects; since
these inclusions are always fully faithful, we conclude that they are equivalences.
Thus for every suitable K, D and X, the functor (4.1) is the composite of two
equivalences and so itself an equivalence.
For the converse, first observe that if both the limit {W, F} and pseudolimit
{W, F}ps of a diagram F exist in some 2-category then the limit {W, F} is a
bilimit just when the canonical map {W, F} → {W, F}ps is an equivalence. Now
letting X, D and K be arbitrary as before, both the limit and pseudolimit in Cat of
the diagram K(X, D–): J → Cat do exist and the canonical map {W, K(X, D–)} →
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 23
{W, K(X, D–)}ps is precisely the inclusion ConeW (X, D) → PsConeW (X, D). Thus
if limits weighted by W are bilimits, this map is an equivalence for any choice of
X, D and K; in particular, it is essentially surjective on objects so that by Theorem
15(a), W is semiflexible.
4.2. Second characterisation. We now give our second characterisation result;
as for our other results, it will have six versions, one for each of the semiflexible,
flexible and pie algebras, and one for each corresponding class of weights. It
takes its most intuitive form in the case of the pie weights: here it states that
a weight W is pie just when for every complete category K, the weighted limit
2-functor {W, –}: [J, K] → K may be extended to a 2-functor Ps(J, K) → K.
That such extensions exist for pie weights was shown as a special case of [24,
Proposition 5.8]; what was not shown there is that such extensions exist only for
pie weights. The corresponding result for a general pie algebra A is that the hom 2functor
T-Algs(A, –): T-Algs → Cat may be extended to a 2-functor T-Alg → Cat,
whilst in the flexible and semiflexible situations, we obtain similar extensions, but
of successively weaker kinds.
Let us now give these results, first for algebras and then for weights:
Theorem 17. A T-algebra A is semiflexible, flexible, or pie, just when, respec-
tively:
(a) The hom 2-functor T-Algs(A, –) admits an extension
T-Algs
ι GG
T-Algs(A,–) 88▲▲▲▲▲▲▲▲▲▲
θ +3
T-Alg
Hyy yW yW yW yW yW yW
Cat
(4.2)
where H is a pseudofunctor and θ an invertible icon;
(b) This extension may moreover be chosen so that θ is an identity, and so that H’s
pseudofunctoriality constraints Hg ◦ Hf → H(g ◦ f) are identities whenever
either f or g is strict;
(c) This extension may moreover be chosen so that H is a 2-functor.
The notion of icon in clause (a) of the statement of this theorem is as it was
in Section 3.3, though now we are concerned only with invertible icons: these are
pseudonatural transformations all of whose 1-cell components are all identities.
Proof. We continue our cycle of implications; thus we must show that clauses (a),
(b) and (c) of Theorem 14 imply the corresponding clauses of the present theorem.
In each case this will be achieved by establishing, successively stronger, properties
of the 2-natural transformation
T-Algs
ι GG
T-Algs(A,–) 88▲▲▲▲▲▲▲▲▲▲
+3
T-Alg
T-Alg(A,–)yyssssssssss
Cat
(4.3)
24 JOHN BOURKE AND RICHARD GARNER
induced by ι. These properties of (4.3) will feed into the corresponding clause of
Lemma 19 below, which is concerned with the construction of extensions, and the
theorem will follow directly.
Assume first that Theorem 14(a) holds; then the function R given there witnesses
(4.3) as pointwise essentially surjective on objects. Since (4.3) is always
pointwise fully faithful, it is therefore in this case a pointwise equivalence. Clause
(a) of the present theorem now follows directly upon application of Lemma 19(a).
In the situation of Theorem 14(b) the function R is also assumed to satisfy
Rf = f whenever f is strict; such an R determines a family of pointwise sections
ρB : T-Alg(A, B) → T-Algs(A, B) of (4.3). The further condition imposed on R
in Theorem 14(b) amounts to the assertion that the family ρ is 2-natural; thus
a section of (4.3), which is consequently an injective equivalence in [T-Algs, Cat].
Clause (b) of the present theorem now follows directly from Lemma 19(b).
Clause (c) of Theorem 14 imposes one further condition on R, which, rephrased
in terms of ρ, asserts precisely that the diagram:
T-Alg(A, B)
T-Alg(A,g)

ρB
GG T-Algs(A, B)
ι GG T-Alg(A, B)
T-Alg(A,g)

T-Alg(A, C) ρC
GG T-Algs(A, C) T-Alg(A, C)ρC
oo
is commutative for all g: B C in T-Alg; and now clause (c) of the present
theorem follows directly from Lemma 19(c).
Theorem 18. A weight W ∈ [J, Cat] is semiflexible, flexible, or pie, just when,
respectively:
(a) For all complete 2-categories K, the limit 2-functor {W, –}: [J, K] → K admits
an extension
[J, K]
ι GG
{W,–}
66■■■■■■■■■■
θ +3
Ps(J, K)
H
yy yW yW yW yW yW yW
K
(4.4)
where H is a pseudofunctor and θ an invertible icon;
(b) This extension may moreover be chosen so that θ is an identity, and so that H’s
pseudofunctoriality constraints Hg ◦ Hf → H(g ◦ f) are identities whenever
either f or g is strict;
(c) This extension may moreover be chosen so that H is a 2-functor.
Proof. We continue our cycle of implications by showing that clauses (a), (b) and
(c) of Theorem 15 imply the corresponding clauses of the present theorem. This
time, we do so by establishing, for a given complete 2-category K, successively
stronger properties of the 2-natural transformation
[J, K]
ι GG
{W,–}
66■■■■■■■■■■ γ
+3
Ps(J, K)
{W,–}ps
yyrrrrrrrrrrr
K
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 25
induced by the canonical comparison maps {W, D} → {W, D}ps; these properties
fed into the corresponding clauses of Lemma 19 then imply the present theorem.
Let us first show, assuming Theorem 15(a), that γ is a pointwise equivalence.
This will be the case precisely if K(X, γ) is a pointwise equivalence for each X ∈ K.
But K(X, γ) is isomorphic to the inclusion ConeW (X, –) → PsConeW (X, –), which
is always pointwise fully faithful, and is pointwise essentially surjective as witnessed
by the function R of Theorem 15(a). Now applying Lemma 19(a) yields clause (a)
of the present theorem.
Suppose next that Theorem 15(b) holds. To give a section of γ in [[J, K], K]
is, by the Yoneda lemma, to give sections of each K(X, γ) which are 2-natural in
X ∈ K; which is equally well to give sections of the isomorphic
ConeW (X, –) → PsConeW (X, –) ,
2-naturally in X. The witnessing function R of Theorem 15(b) provides just such a
section; assuming this we therefore obtain a section ρ of γ, which is consequently an
injective equivalence in [[J, K], K]. Clause (b) of the theorem now follows directly
from Lemma 19(b).
The additional property of R specified in Theorem 15(c) now translates, in terms
of ρ, to the commutativity of
{W, A}ps
{W,g}ps

ρA
GG {W, A}
γA
GG {W, A}ps
{W,g}ps

{W, B}ps ρB
GG {W, B} {W, B}psρB
oo
for all pseudonatural transformations of diagrams g: A B; and clause (c) of the
present theorem now follows directly from Lemma 19(c).
We now give the lemma which was used in the proof of the above two results.
Lemma 19. Consider a diagram
A
I GG
F
11❄❄❄❄❄❄❄❄❄❄❄
α +3
B
G
⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧
C
of 2-categories, 2-functors and a pseudonatural transformation α.
(a) If I is bijective on objects and α a pointwise equivalence, then this diagram
admits a factorisation as
A
I GG
F
11❄❄❄❄❄❄❄❄❄❄❄
α1
+3
α2
+3
B
H
c c c
 c c c
GmmC
where H is a pseudofunctor, α1 an invertible icon and α2 a pseudonatural
equivalence.
26 JOHN BOURKE AND RICHARD GARNER
(b) If in the situation of (a), α is in fact 2-natural, and moreover admits a retraction
β in [A, C], then the α1 of this factorisation may be chosen to be the
identity; it then follows that H strictly preserves composition with maps from
A, in the sense that the pseudofunctoriality constraint Hg ◦ Hf → H(g ◦ f) is
an identity whenever either f or g is in the image of I.
(c) If in the situation of (b), the retraction β may be chosen so that the diagram
GIB
Gg

βB
GG FB
αB
GG GIB
Gg

GIC
βC
GG FC GIC
βC
oo
(4.5)
commutes for every g: IB → IC in B, then H may be taken to be a 2-functor.
Proof. We start with (a). Without loss of generality we may assume that I is
in fact the identity on objects. Let Hom(B, C) denote the 2-category of pseudofunctors,
pseudonatural transformations and modifications from B to C. We
have the forgetful 2-functor U : Hom(B, C) → [ob B, C] and it is well-known that
this admits the lifting of adjoint equivalences. Now for each X ∈ ob A = ob B
we have the map αX : FX → GIX = GX which by assumption is an equivalence;
so choosing an adjoint pseudoinverse βX for each αX , we obtain an adjoint
equivalence (α, r): ob F ⇆ UG: (β, s) in [ob B, C]. Lifting this along U we obtain
a pseudofunctor H : B C and pseudonatural equivalence α2 : H G with
Uα2 = α. Explicitly, H has the same action on objects as F, and on morphisms
sends g: A → B in B to the composite
FA
αA
−−→ GA
Gg
−−→ GB
βB
−−→ FB
obtained by conjugating Gg by the adjoint equivalences. The action on 2-cells
is similar, whilst the unit and composition coherence constraints are obtained
from the unit and counit maps rA and sA in the evident manner. As for the
pseudonatural transformation α2, this has its 1-cell components being those of α
and its 2-cell component at g: A → B in B being given by the pasting composite
FA
αA
GG
αA

GA
Gg
GG GB
βB
GG
1
44❉❉❉❉❉❉❉❉❉❉❉❉ FB
sB
 αB

GA
Gg
GG GB .
This completes our description of H and α2, and we now turn to α1. As indicated
above, its 1-cell components are identities; whilst at a morphism f : A → B of A
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 27
its 2-cell component is given by the top row of the following pasting diagram:
FA
F f
GG
αf

FB
αB

1 GG
rB

FB
FA αA
GG
αA

GA
GIf
GG GB βB
GG
1
44❉❉❉❉❉❉❉❉❉❉❉❉ FB
sB
 αB

GA
GIf
GG GB .
The bottom row of this diagram is (α2I)f and so the entire pasting constitutes
the value of (α2I ◦ α1)f ; which, since rB and sB cancel by the triangle identities,
is equal to αf , so showing that α2I ◦ α1 = α as required. Note that we have not
checked the coherence axioms for α1 but since its composite with the pseudonatural
α2I is pseudonatural, the same is true of α1: pseudonatural equivalences detect
pseudonaturality.
This completes the proof of (a); as for (b), suppose now that α is in fact 2natural,
and that the pseudoinverses βX chosen for each αX above constitute a
retraction for α in [A, C]. Then each 2-cell αf is the identity, as is each rB : 1 →
βBαB, so that each 2-cell component of α1, being a composite of such 2-cells, is
itself the identity. Now given maps f : A → B in B and g: B → C in A, we must
show that the pseudofunctoriality constraint HIg◦Hf → H(Ig◦f) is the identity,
or in other words, that the whiskering
FA
αA
GG GA
Gf
GG GB
1
cc
βB
GG FB
αB
GG GB
GIg
GG GC
βC
GG FC
sB

is the identity: which is so as βC ◦ GIg ◦ sB = Fg ◦ βb ◦ sB = 1F g. The argument
for the dual case, where f : A → B in A and g: B → C in B, is similar.
Finally, we prove (c). Given a retraction β as before, let us suppose that its
components make each diagram (4.5) commute; we will show that H is then a
2-functor. Since F is a 2-functor and HI = F, it already follows that H preserves
identities strictly. As for binary composition, we must show that for every f : A →
B and g: B → C in B the whiskering
FA
αA
GG GA
Gf
GG GB
1
cc
βB
GG FB
αB
GG GB
Gg
GG GC
βC
GG FC
sB

is an identity 2-cell; we will show in fact that the composite 2-cell βC ◦ Gg ◦ sB
on the right of the whiskering is an identity. Commutativity in (4.5) asserts that
βC ◦ Gg ◦ sB is an endomorphism; to show that it is the identity, we observe that
βC ◦ Gg ◦ sB ◦ αB = 1βC◦Gg ◦ αB by the triangle identities, and conclude that
βC ◦ Gg ◦ sB = 1βC ◦Gg since αB is an equivalence.
28 JOHN BOURKE AND RICHARD GARNER
4.3. Third characterisation. We now turn to our final characterisation result,
which is most easily motivated by the following question concerning limits: for
which weights W ∈ [J, Cat] does the weighted limit 2-functor {W, –}: [J, K] → K
send pointwise equivalences to equivalences for every complete category K? A
closely related question was considered by Par´e [28]; his notion of persistent limit
concerns weights which display this good behaviour, but with respect to a doublecategorical,
and not 2-categorical, notion of pointwise diagram equivalence. It was
shown by Verity in [31] that the persistent weights are precisely the flexible ones.
However, for our purely 2-categorical question, the class of weights answering to
it turns out to be the larger class of semiflexibles; amongst which the flexibles are
characterised by their also sending pointwise surjective equivalences to surjective
equivalences.
As is well-known, pointwise equivalences or surjective equivalences in a functor
category [J, K] need not be genuine ones; were this the case, then every weight
would answer to the above characterisations, since {W, –}, like any 2-functor, preserves
such equivalences. Yet the inclusion ι: [J, K] → Ps(J, K) sends pointwise
equivalences or surjective equivalences to genuine ones: and so by asking that
the 2-functor {W, –}: [J, K] → K admit a suitable extension to Ps(J, K), as in
Theorem 18, it will follow that {W, –} sends pointwise equivalences or pointwise
surjective equivalences, as appropriate, to genuine ones; this is the core of our
characterisation.
In order to give the corresponding results for semiflexible and flexible algebras,
we need a notion corresponding to that of pointwise diagram equivalence. In the
case that K is cocomplete, we may view [J, K] as the algebras for a 2-monad on C =
[ob J, K], and now the pointwise equivalences therein are equally well the algebra
maps which become equivalences under the forgetful 2-functor U : T-Algs → C.
We may consider this same class of algebra maps for a general 2-monad, calling
them U-equivalences: and the basic form of our result is now that an algebra
A is semiflexible just when T-Algs(A, –) sends U-equivalences to genuine ones.
Similarly, we have the notion of U-surjective equivalence—algebra maps which
become surjective equivalences on applying U—and with respect to these, the
flexibles have a corresponding characterisation.
Given a 2-monad T with rank on a locally presentable 2-category C, it is shown
in [20] that the U-equivalences and the U-surjective equivalences are the weak
equivalences and trivial fibrations of a naturally-arising Quillen model structure
on T-Algs. The corresponding cofibrant objects are the flexible algebras, and our
characterisation of them as those objects for which T-Algs(A, –) sends U-surjective
equivalences to surjective equivalences is now contained in Proposition 2.3 of that
paper.
We have so far not discussed the pie case; this can be understood in terms of
the algebraic model structures of [30]. The notion of algebraic model structure
strengthens the classical one in a number of respects; one of these is that cofibrant
replacement becomes a comonad. The model structure on T-Algs described in
the previous paragraph can be made algebraic in such a way that the comonad in
question is the Q of our considerations: and now the coalgebras for this comonad,
our pie algebras, are the algebraically cofibrant objects of this model structure.
These may be characterised by their bearing a coherent choice of liftings against
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 29
the algebraic trivial fibrations; and for the algebraic model structure on T-Algs, an
algebraic trivial fibration is composed of a U-surjective equivalence f together with
a chosen section of Uf—thus a witness to f’s being a U-surjective equivalence. It
is to such maps that the pie case of our characterisation theorem for algebras will
refer.
Theorem 20. A T-algebra A is semiflexible, flexible, or pie, just when, respec-
tively:
(a) The hom 2-functor T-Algs(A, –) sends U-equivalences to equivalences;
(b) The hom 2-functor T-Algs(A, –) sends U-surjective equivalences to surjective
equivalences;
(c) The hom 2-functor T-Algs(A, –) coherently transports U-split surjective equivalences
to split surjective equivalences.
In clauses (a) and (b), the U-equivalences and U-surjective equivalences are as
before those algebra maps which U sends to equivalences or surjective equivalences;
but these are equally well, as in Proposition 4.10 of [20], those maps whose image
under ι: T-Algs → T-Alg is an equivalence or surjective equivalence. As for clause
(c) we say that a 2-functor F : T-Algs → C coherently transports U-split surjective
equivalences to split surjective equivalences if it sends U-surjective equivalences to
surjective equivalences and if, furthermore, there exists a function ϕ which to each
U-surjective equivalence f : B → C in T-Algs and each splitting k: UB → UA for
Uf assigns a splitting ϕf (k): FC → FB for Ff, all subject to two axioms: firstly,
that ϕf (Uk) = Fk whenever k is a splitting for f in T-Algs, and secondly, that
ϕgf (kℓ) = ϕf (k) ◦ ϕg(ℓ) whenever this makes sense.
Proof. We will show that clauses (a), (b) and (c) of Theorem 17 imply the corresponding
clauses of the current result; then we show that these clauses in turn
imply that an algebra is respectively semiflexible, flexible or pie. This will complete
our cycle of implications.
As remarked above, if f : B → C is a U-equivalence in T-Algs then ι(f) is an
equivalence in T-Alg. If f is a U-surjective equivalence with splitting k: UC → UB
in C, fully faithfulness of Uf ensures that there exists a unique pseudomorphism
structure on this k making it into a section ¯k: C B for ιf. It follows from the
uniqueness of this correspondence that the function ϕf (k) = ¯k exhibits the inclusion
ι: T-Algs → T-Alg as coherently transporting U-split surjective equivalences
to split surjective equivalences.
Now suppose that Theorem 17(a) holds, so that T-Algs(A, –) admits an extension
(H, θ) as in (4.2). If f : B → C is a U-equivalence in T-Algs, then ι(f) is an
equivalence in T-Alg, and hence Hι(f) is one in Cat. But T-Algs(A, f) ∼= Hι(f)
via θf and so T-Algs(A, f) is an equivalence in Cat also. This verifies clause (a)
of the current result.
Next suppose that Theorem 17(b) holds; we must show that T-Algs(A, –) sends
each U-surjective equivalence f : B → C to a surjective equivalence. Certainly
T-Algs(A, f) is an equivalence by the case just proved, and it remains to show
that it admits a section. Let Uf admit a section k: UC → UB with ¯k: C B
as above the corresponding section for ιf. We now calculate that Hιf ◦ H¯k =
30 JOHN BOURKE AND RICHARD GARNER
H(ιf ◦ ¯k) = H(1C) = Hι(1C) = T-Algs(A, 1C ) = 1, so that H¯k is a section for
Hιf = T-Algs(A, f), as required.
Finally, suppose that Theorem 17(c) holds; we must then show that T-Algs(A, –)
coherently transports U-split surjective equivalences to split surjective equivalences.
The inclusion ι: T-Algs → T-Alg does so, with associated function ϕf (k) =
¯k, and it follows, as with any 2-functor based on T-Alg, that Hι = T-Algs(A, –)
has the same transport property, now with associated function ϕf (k) = H¯k.
It remains to show that clauses (a), (b) and (c) of the current result imply
that an algebra A is semiflexible, flexible or pie respectively. So suppose first
that T-Algs(A, –): T-Algs → Cat sends U-equivalences to equivalences. Observe
that pA : QA → A is a U-equivalence, since UqA : UA → UQA is a pseudoinverse
for it in C; and hence T-Algs(A, pA): T-Algs(A, QA) → T-Algs(A, A) is an
equivalence in Cat. In particular its essential surjectivity means that we can find
a ∈ T-Algs(A, QA) with pA ◦ a ∼= 1A; whence A is semiflexible.
Suppose now that T-Algs(A, –) sends U-surjective equivalences to surjective
equivalences. The map pA : QA → A is in fact a U-surjective equivalence, since
UqA is a splitting for UpA, and so T-Algs(A, pA) is in this case a surjective equivalence
in Cat. Now surjectivity allows us to find a: A → QA with pA ◦ a = 1A, so
that A is flexible.
Finally, suppose that T-Algs(A, –) comes equipped with a function ϕ which
coherently transports U-split surjective equivalences in T-Algs to split surjective
equivalences in Cat. Then to the splitting UqA : UA → UQA of UpA we have
assigned a splitting ϕ(UqA): T-Algs(A, A) → T-Algs(A, QA) of T-Algs(A, pA),
and we claim that the morphism a = ϕ(UqA)(1A): A → QA exhibits A as a
strict Q-coalgebra. Clearly we have pA ◦ a = 1A and so it remains to prove
coassociativity. To obtain this, consider the following two equalities of U-split
surjective equivalences in T-Algs:
QQA pQA
GG
ÑÑ
UqQA
eÑ
Y{ Su Go Ai 5™
)“
QA pA
GG
ÔÔ
UqA
gÓ
b~ Tv Go @h 2–
'‘
A
=
QQA
QpA
GG
ÑÑ
UQqA
QA pA
GG
ÔÔ
UqA
gÓ
b~ Tv Go @h 2–
'‘
A
and
QQA pQA
GG
ÑÑ
UqQA
eÑ
Y{ Su Go Ai 5™
)“
QA pA
GG
ÔÔ
Ua
A
=
QQA
QpA
GG
ÑÑ
UQa
QA pA
GG
ÔÔ
UqA
gÓ
b~ Tv Go @h 2–
'‘
A
where the left pointing arrows are the splittings in C, and drawn strictly if they
underlie a morphism of T-Algs. Applying the transport function ϕ to each of these,
and using its coherence, we obtain from the first equality the commutative square
below on the left, and from the second, that on the right:
T-Algs(A, A)
ϕ(UqA)

ϕ(UqA)
GG T-Algs(A, QA)
ϕ(UqQA)

T-Algs(A, A)
ϕ(UqA)

a◦(–)
oo
T-Algs(A, QA)
QqA◦(–)
GG T-Algs(A, QQA) T-Algs(A, QA)
Qa◦(–)
oo
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 31
Now evaluating both commutative squares at 1A ∈ T-Algs(A, A) we obtain from
the left one the equality QqA ◦a = ϕ(UqQA)(a) and from the right one the equality
Qa ◦ a = ϕ(UqQA)(a); whence Qa ◦ a = QqA ◦ a = ∆A ◦ a and so a: A → QA is a
strict coalgebra as required.
Finally, we give the corresponding version of the above theorem for the case of
weights. In the third clause, a pointwise split surjective equivalence is a pointwise
surjective equivalence equipped with a chosen section of each component. We note
that if K is cocomplete, so that [J, K] is T-Algs for a 2-monad T on [ob J, K], then
such maps are precisely the U-split surjective equivalences for this T. The notion of
coherent transport of pointwise split surjective equivalences may be transcribed directly
from the corresponding transport notion for U-split surjective equivalences,
and is again a special case of it when K is cocomplete.
Theorem 21. A weight W ∈ [J, Cat] is semiflexible, flexible, or pie, just when,
respectively:
(a) For all complete 2-categories K, the 2-functor {W, –}: [J, K] → K sends pointwise
equivalences to equivalences;
(b) For all complete 2-categories K, the 2-functor {W, –}: [J, K] → K sends pointwise
surjective equivalences to surjective equivalences;
(c) For all complete 2-categories K, the 2-functor {W, –}: [J, K] → K coherently
transports pointwise split surjective equivalences to split surjective equivalences.
Proof. The argument that Theorem 18’s clauses (a), (b) and (c) imply those of the
present result is exactly as in the preceding proof. That these in turn imply that
a weight W ∈ [J, Cat] is semiflexible, flexible or pie is now a direct consequence
of Theorem 20. Indeed, if we view [J, Cat] as T-Algs for the weight 2-monad T
on [ob J, Cat], then the pointwise equivalences and their two variants become for
this T the U-equivalences and their corresponding variants. Moreover, there is
a natural isomorphism {W, –} ∼= [J, Cat](W, –): [J, Cat] → Cat, and so clauses
(a), (b) and (c) of the present theorem may be identified for this T with the
corresponding clauses of Theorem 20. Thus assuming each to hold in turn, we
apply that result to deduce that W is, respectively, a semiflexible, flexible or pie
algebra; that is, a semiflexible, flexible or pie weight.
Let us conclude this section by remarking upon some further properties of flexible
algebras and weights which do not quite fit into our series of characterisations.
In Theorem 20(b) we characterised the flexibles as those algebras A for which the
hom 2-functor T-Algs(A, –) sends U-surjective equivalences to surjective equivalences.
The argument by which we proved a flexible algebra A to have this property
required only that T-Algs(A, –) admit an extension H : T-Algs → Cat preserving
identities strictly and satisfying H(g ◦ f) = H(g) ◦ H(f) whenever g is strict. But
in fact Theorem 17(b) also asserts that H(g ◦ f) = H(g) ◦ H(f) whenever f is
strict; and thus by a corresponding argument we conclude that if A is flexible then
the 2-functor T-Algs(A, –) sends U-injective equivalences to injective equivalences.
It seems unlikely that the converse of this implication holds in general; but it does
at least in the case of weights, as the following theorem shows.
32 JOHN BOURKE AND RICHARD GARNER
Theorem 22. A weight W ∈ [J, Cat] is flexible if and only if for all complete
2-categories K, the 2-functor {W, –}: [J, K] → K sends pointwise injective equivalences
to injective equivalences.
Proof. If W is flexible then for each complete K the limit 2-functor {W, –}: [J, K] →
K admits, by Theorem 18, an extension H : Ps(J, K) → K preserving identities
strictly and satisfying H(g ◦f) = H(g) ◦H(f) whenever f or g is strict. Just as in
the preceding discussion, an extension of this form ensures that {W, –}: [J, K] → K
sends pointwise injective equivalences to injective equivalences.
Conversely suppose the limit 2-functor {W, –}: [J, K] → K has this property
for each complete K. This is easily seen to be equivalent to the assertion that
the colimit 2-functor W ⋆ –: [Jop, K] → K sends pointwise surjective equivalences
to surjective equivalences for each cocomplete K. Let K be the cocomplete 2category
[J, Cat] and consider the comonad Q thereon, induced by the inclusion
ι: [J, Cat] → Ps(J, Cat) and its left adjoint. That left adjoint exists on replacing
Cat by any cocomplete 2-category; in particular, on replacing Cat by Catop
, from
which it follows that ι admits also a right 2-adjoint; this was observed in [5,
Section 4.2]. Thus the comonad Q is a composite of left adjoint 2-functors and as
such is cocontinuous.
Now by definition a component of the counit p: Q → 1 at a weight is a surjective
equivalence just when that weight is flexible. Each representable is flexible so that
precomposing p by the Yoneda embedding Y : Jop → [J, Cat] yields a pointwise
surjective equivalence p ◦ Y : Q ◦ Y → Y in [Jop, [J, Cat]]. Taking the image of
p ◦ Y under W ⋆ –: [Jop, [J, Cat]] → [J, Cat] we consequently obtain a surjective
equivalence W ⋆ (Q ◦ Y ) → W ⋆ Y . On the one hand the Yoneda lemma asserts
that W ⋆ Y ∼= W; on the other hand cocontinuity of Q ensures that W ⋆ (Q ◦
Y ) ∼= Q(W ⋆ Y ) ∼= QW. Combining these isomorphisms with the above surjective
equivalence now yields a surjective equivalence QW → W; whence W, as a retract
of QW, is flexible.
5. Closure properties
In this section, we continue our study of the semiflexible, flexible and pie algebras
for a 2-monad T by considering the closure properties of these three classes of
algebras. Once again, we remind the reader of our standing assumptions that T
be a 2-monad with rank on a complete and cocomplete 2-category C.
Proposition 23. The semiflexible, flexible and pie algebras each contain the frees,
and are each closed in T-Algs under the corresponding class of weighted colimits.
Proof. The free-forgetful adjunction F : C ⇆ T-Algs : U generates the comonad
FU on T-Algs, and clearly every free T-algebra bears FU-coalgebra structure. But
as in Corollary 5, we have a comonad map FU → Q, whence every free T-algebra
bears strict Q-coalgebra structure, and is thus pie, flexible and semiflexible.
As for the closure under colimits, we consider only the semiflexible case, the
arguments in the other two cases being identical in form. So let W ∈ [J, Cat] be a
semiflexible weight, and D: Jop → T-Algs a 2-functor taking values in semiflexible
algebras; we must show that the colimit W ⋆ D in T-Algs is again semiflexible.
By Theorem 20, it suffices for this to show that T-Algs(W ⋆ D, –): T-Algs → Cat
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 33
takes U-equivalences to equivalences. Now by definition of weighted colimit, we
have a 2-natural isomorphism
T-Algs(W ⋆ D, –) ∼= [J, Cat](W?, T-Algs(D?, –)) ,
and so it is enough to show that the 2-functor on the right-hand side takes
U-equivalences to equivalences. But as each DX is semiflexible, the functor
T-Algs(D?, –): T-Algs → [J, Cat] takes U-equivalences to pointwise equivalences;
and as W is semiflexible, the functor [J, Cat](W, –): [J, Cat] → Cat takes pointwise
equivalences to equivalences. Combining these two facts yields the result.
Specialising to the case of weights yields the following corollary. In light of
Proposition 9, the case dealing with pie weights is tautologous; the one for flexibles
is Theorem 4.9 of [2], whilst the semiflexible case appears to be new. What is
certainly new is the possibility of treating all three cases in a uniform manner.
Corollary 24. Each of the classes of semiflexible, flexible and pie weights is sat-
urated.
Proof. As in [1], to say that a class Φ of weights is saturated is to say that the full
subcategory of [J, Cat] spanned by the weights in Φ with domain J contains the
representables and is closed under Φ-colimits. If we view [J, Cat] as the category of
algebras for the weight 2-monad on [ob J, Cat], then every representable weight is
free as a T-algebra; whence the result follows from the preceding proposition.
The preceding result can be restated as saying that, in the case of the 2-monad
for weights, the semiflexible, flexible and pie algebras in [J, Cat] are precisely the
closure of the free algebras in T-Algs under the corresponding class of colimits. In
the remainder of this section, we consider the extent to which this remains true on
replacing weights by algebras for a general 2-monad T. For the flexible algebras,
the answer is straightforward; the following was observed in Remark 5.5 of [20].
Proposition 25. The flexible algebras are the closure of the frees in T-Algs under
flexible colimits.
Proof. As in [18], every algebra of the form QA may be presented as a pie colimit
of free algebras. Now every flexible algebra is a retract of one of the form QA, and
thus the splitting of an idempotent on some QA. Since the colimit which splits an
idempotent is flexible, every flexible algebra is a flexible colimit of frees.
There is an equally straightforward statement concerning the semiflexibles.
Proposition 26. The semiflexible algebras are the closure of the frees in T-Algs
under bicolimits.
Proof. First observe that the semiflexibles are closed under bicolimits. For indeed,
if X is a W-weighted bicolimit of a diagram D of semiflexible algebras, then X is
equivalent to W ⋆ps D, the W-weighted pseudocolimit of D; since semiflexibles are
closed under equivalence it is therefore enough to show that W ⋆ps D is semiflexible.
But W ⋆ps D ∼= QW ⋆ D is a semiflexible colimit of semiflexibles, and so indeed
semiflexible. Every free algebra is semiflexible, and so to complete the proof it
suffices to show that every semiflexible is a bicolimit of frees. As we observed
above, any algebra of the form QA is a pie colimit of frees; hence a semiflexible
34 JOHN BOURKE AND RICHARD GARNER
colimit of frees, and so, by (the dual of) Corollary 16, a bicolimit of frees. An
algebra is semiflexible just when it is equivalent to one of the form QA, and any
object equivalent to a bicolimit of frees is again a bicolimit of frees.
We have not attempted to ascertain whether or not the semiflexible algebras
are, in fact, the closure of the frees in T-Algs under semiflexible colimits; our
investigations have shown only that there seems to be no obvious proof of this
fact. We have, however, considered in some detail the corresponding question for
the pie algebras, and found it to be false: the pie algebras for an arbitrary 2-monad
T may comprise a strictly larger class than the closure of the frees in T-Algs under
pie colimits. We will describe a 2-monad exhibiting this divergence in the proof of
Proposition 33 below.
However, it turns out that the two classes just mentioned will coincide under
certain additional assumptions on our 2-monad T and our 2-category C. The
remainder of this section will be given over to the analysis underlying this result.
The key step will be to interpose between the pie algebras and the pie colimits of
the frees a third class of algebras: the objective quotients of the frees. Here, we
call an object B of a 2-category an objective quotient of an object A if there exists
some objective morphism A → B.
Proposition 27. Considering the following three classes of T-algebras:
(1) The closure of the frees under pie colimits;
(2) The objective quotients of the frees; and
(3) The pie algebras,
we have inclusions (1) ⊆ (2) ⊆ (3).
In the statement of this result, and in what follows, when we speak of colimits or
of objective quotients, it is to be understood that these are being taken in T-Algs.
Proof. To prove (1) ⊆ (2) it is clearly enough to show that the objective quotients
of the frees are closed under pie colimits. For closure under coproducts, suppose
that (Ai | i ∈ I) is a collection of T-algebras which are objective quotients of frees,
and let qi : FXi → Ai witness this fact for each i ∈ I. Now Σiqi : ΣiFXi → ΣiAi
is objective, since objective morphisms are closed under colimits in the arrow
category, and ΣiFXi is free on ΣiXi, whence ΣiAi is an objective quotient of a
free. For closure under coinserters, let f, g: A ⇒ B in T-Algs, with A and B
objective quotients of frees. Now the coinserter morphism h: B → C, like any
coinserter morphism, is objective; whence C is an objective quotient of B, and
hence also an objective quotient of a free. The case of coequifiers is identical on
observing that coequifier morphisms, too, are objective.
As for (2) ⊆ (3), we saw in Proposition 23 above that every free algebra is a
pie algebra; the proof is now completed by the following proposition, which we
separate out for its independent interest.
Proposition 28. Any objective quotient of a pie algebra is a pie algebra.
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 35
Proof. Let A ∈ T-Algs admit strict Q-coalgebra structure a: A → QA, and let
f : A → B be an objective morphism. We must show that B admits strict Qcoalgebra
structure. So consider the square on the left in the diagram:
A
Qf◦a

f
GG B
1B
b}}⑤
⑤
⑤
⑤
QB pB
GG B
A
∆◦Qf◦a

f
GG B
b

QQB pQB
GG QB .
Since f is objective, and pB fully faithful, there exists as indicated a unique diagonal
filler b: B → QB making both triangles commute. We claim b equips
B with Q-coalgebra structure. The lower-right triangle asserts the counit axiom
pB ◦ b = 1B; as for the comultiplication axiom, it is easy to verify that both Qb ◦ b
and ∆ ◦ b are diagonal fillers for the right-hand square above, and so must be
equal.
We now turn our attention to finding conditions under which the two inclusions
of Proposition 27 become equalities. It seems that obtaining necessary and
sufficient conditions is a hard problem; we shall therefore content ourselves with
giving sufficient conditions that are broad enough to encompass the classes of
examples that we studied in Section 3. We first consider when the inequality
(2) ⊆ (3) becomes an equality; this is closely related to our Theorem 3, and in
what follows, we make free use of the definitions and notational conventions established
for that result. In particular, we recall from (2.2) the comparison functor
j : T0-Alg → Td-Alg, existing for a C with enough discretes, that assigns to each
T-algebra its induced structure “at the level of objects”.
Proposition 29. If C has enough discretes and (objective, fully faithful) factorisations
lifting to T-Algs, then for each T-algebra A, the following are equivalent:
(i) A is an objective quotient of a free;
(ii) A is an objective quotient of a free on a (projective) discrete;
(iii) jA is a free Td-algebra.
Proof. First we prove (i) ⇔ (ii). For the non-trivial direction, if q: FX → A
exhibits A as an objective quotient of a free, then q ◦ FλX : FDOX → A exhibits
it as an objective quotient of a free on a discrete: observe that FλX is objective
in T-Algs since λX is objective in C and left adjoints preserve objectivity.
Now we prove (ii) ⇔ (iii). For any X ∈ Cd, since jFDX = FdX, the action of
j on homs gives a function T0-Alg(FDX, A) → Td-Alg(FdX, jA); this is in fact a
bijection, with inverse given by the composite
Td-Alg(FdX, jA) ∼= Cd(X, UdjA) = Cd(X, OUA) ∼= T0-Alg(FDX, A) .
We claim that under this bijection, objective morphisms FDX → A correspond
with isomorphisms FdX → jA; this will complete the proof. For this, we observe
that q: FDX → A is objective just when OUq = Udjq is invertible in A; but Ud is
conservative, so this happens just when jq is invertible in Td-Alg, as desired.
Combining this result with Theorem 3, we obtain:
36 JOHN BOURKE AND RICHARD GARNER
Corollary 30. Let C have enough discretes and (objective, fully faithful) factorisations
lifting to T-Algs. If Cd has, and the induced monad Td preserves, coreflexive
equalisers, then every pie T-algebra is an objective quotient of a free on a discrete.
As we observed in Section 2, a free algebra on a projective discrete is again
projectively discrete. Thus the preceding corollary ensures that, in particular, the
full sub-2-category of T-Algs spanned by the pie algebras has enough discretes,
and that, in fact, the projectively discrete objects of this 2-category are precisely
the algebras free on projective discretes.
We now consider when the inequality (1) ⊆ (2) of Proposition 27 becomes an
equality; we shall do so in a slightly less general situation than that we have
considered so far.
Proposition 31. Let J be a small 2-category, and T an objective-preserving 2monad
on [J, Cat]. The following are equivalent properties of the T-algebra A:
(i) A is an objective quotient of a free;
(ii) A is an objective quotient of a free on a discrete;
(iii) A lies in the closure of the frees under pie colimits;
(iv) A lies in the closure of the frees on discretes under pie colimits.
Proof. We have (iv) ⇒ (iii) trivially, (iii) ⇒ (i) by Proposition 27 and (i) ⇒
(ii) by Proposition 29. It remains to show that (ii) ⇒ (iv). First note that by
Proposition 2, since T preserves objectives so too does U : T-Algs → [J, Cat]. Thus
it is clearly sufficient that we prove:
If q0 : X0 → A in T-Algs has Uq0 pointwise bijective on objects, and X0 is a pie
colimit of frees on discretes, then so is A. To see this, form the comma object in
[J, Cat] as on the left of the diagram:
Y
d GG
c

γ

UX0
Uq0

UX0
Uq0
GG UA
FDOY
¯d GG
¯c

δ
X0
i0

X0
i0
GG X1 .
Composing d and c with the discrete cover λY : DOY → Y and taking transposes,
we obtain maps ¯d, ¯c: FDOY ⇒ X0; we now form their coinserter as on the right
above. The transpose of the 2-cell γ ◦λY under adjunction constitutes a coinserter
cocone under ¯d, ¯c and so we have an induced map q1 : X1 → A making the triangle
X0
i0
GG
q0
22❆❆❆❆❆❆❆❆ X1
q1
~~⑥⑥⑥⑥⑥⑥⑥⑥
A
commute. Now i0 is objective, being a coequifier morphism, and so Ui0, like Uq0, is
pointwise bijective on objects; which in turn implies that Uq1 is pointwise bijective
on objects. We claim that it is also pointwise full. Indeed, given objects x and y
in X1(j) and a morphism f : q1x → q1y in A(j), we have, since Ui0 is pointwise
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 37
bijective on objects, a cone as on the left in the diagram:
J(j, –)
i−1
0 x
GG
i−1
0 y

f
UX0
Uq0

UX0
Uq0
GG UA
J(j, –)
¯f
GG DOY
η
GG UFDOY
U(i0
¯d)
@@
U(i0¯c)
TTUδ UX1 .
This cone corresponds to a map J(j, –) → Y which, since J(j, –) is projectively
discrete, factors through the objective λY : DOY → Y as ¯f : J(j, –) → DOY , say.
So we have a 2-cell as on the right above; now this in turn corresponds to a map
g: x → y in X1(j), and it is easy to verify that q1g = f as required. Thus we have
shown that Uq1 is pointwise bijective on objects and full, and so to complete the
proof, it will suffice to show that:
If q1 : X1 → A in T-Algs has Uq1 pointwise bijective on objects and full, and
X1 is a pie colimit of frees on discretes, then so is A. To this end, let us write 2
for the arrow category, and P for the category • ⇒ •, as in Section 3.3. Taking
K : P → 2 to be the unique bijective-on-objects functor, we now form in C the
limit of the arrow Uq1 weighted by K ∈ [2, Cat], as on the left of the diagram:
Z
u
99
v
UU
σ τ UX1
Uq1
GG UA FDOZ
¯u
99
¯v
UU¯σ ¯τ X1
i1
GG X2 .
Thus Uq1 ◦ σ = Uq1 ◦ τ and (Z, u, v, σ, τ) are universal amongst such data; we
call them the equi-kernel of Uq1. Whiskering with λZ : DOZ → Z and taking
transposes, we obtain ¯u, ¯v, ¯σ and ¯τ as on the right; let i1 as displayed be their
coequifier. Since clearly q1◦¯σ = q1◦¯τ, we obtain an induced morphism q2 : X2 → A
making the triangle
X1
i1
GG
q1
22❆❆❆❆❆❆❆❆ X2
q2
~~⑥⑥⑥⑥⑥⑥⑥⑥
A
commute. Now since i1 is a coequifier morphism it is objective; as q1 is objective
so also is q2 and thus Uq2 is pointwise bijective on objects. We claim that it is
also a split epimorphism. To see this, observe first that the equality i1 ◦ ¯σ = i1 ◦ ¯τ
transposes under adjunction to give Ui1 ◦ σ ◦ λZ = Ui1 ◦ τ ◦ λZ; now as λZ is
pointwise bijective on objects it is cofaithful, and thus Ui1 ◦σ = Ui1 ◦τ. Since Uq1
is pointwise bijective on objects and full, it is by an easy calculation the coequifier
of its own equi-kernel (σ, τ), and so there is a unique map g: UA → UX2 with
g ◦ Uq1 = Ui1. Now Uq2 ◦ g ◦ Uq1 = Uq2 ◦ Ui1 = Uq1 and hence Uq2 ◦ g = 1UA,
since Uq1 is pointwise bijective on objects and full, and thus epimorphic. Thus Uq2
is a split epimorphism as claimed; since it is also pointwise bijective on objects,
it is therefore pointwise full. But now also UFUq2 is a split epimorphism, and
also pointwise bijective on objects, since T = UF preserves objectives; whence
UFUq2 is also pointwise bijective on objects and full. To complete the proof it
will therefore suffice to show that:
38 JOHN BOURKE AND RICHARD GARNER
If q2 : X2 → A in T-Algs has both Uq2 and UFUq2 pointwise bijective on objects
and full, and X2 is a pie colimit of frees on discretes, then so is A. To prove this,
we form the equi-kernel of Uq2, as on the left of the diagram:
W
w
99
z
UU
φ ψ UX2
Uq2
GG UA FDOW
¯w
99
¯z
UU
¯φ
¯ψ X2
q2
GG A .
Whiskering with λW : DOW → W and taking transposes, we obtain ¯w, ¯z, ¯φ and ¯ψ
as on the right; we now claim that q2 is a coequifier of ¯φ and ¯ψ, which will complete
the proof. Observe first that q2 is epimorphic in T-Algs, since U is faithful and
Uq2 is epimorphic in [J, Cat]; so it suffices to show that any map g: X2 → B of
T-algebras satisfying g ◦ ¯φ = g ◦ ¯ψ factors through q2.
For such a g, by taking adjoint transposes we see that Ug◦φ◦λW = Ug◦ψ◦λW ,
and since λW is cofaithful, it follows that Ug ◦ φ = Ug ◦ ψ. Since Uq2 is bijective
on objects and full, it is as before the coequifier in [J, Cat] of φ and ψ, and hence
Ug admits a factorisation Ug = h ◦ Uq2. We will be done if we can show that
h: UA → UB is in fact an algebra map; or in other words, that the square on the
right of the diagram
UFUX2
UǫX2

UF Uq2
GG UFUA
UǫA

UF h GG UB
UǫB

UX2
Uq2
GG UA
h
GG UB
is commutative. But since UFUq2 is epimorphic, the commutativity of the righthand
square follows from that of the outside and of the left-hand square.
Taking together Theorem 3, Corollary 30 and Proposition 31, we obtain the
main result of this section:
Theorem 32. Let J be a small 2-category and T an objective-preserving 2-monad
on [J, Cat] for which the induced monad Td on [J0, Set] preserves coreflexive equalisers.
Then the following classes of T-algebras coincide:
(1) The closure of the frees (on discretes) under pie colimits;
(2) The objective quotients of the frees (on discretes);
(3) The pie algebras;
and may be characterised as the T-algebras A for which jA is a free Td-algebra.
Before continuing, let us make a few observations concerning the proof of Proposition
31. In that proof we only made use of the fact that the base 2-category was
[J, Cat] in two places: in determining that the objective λW is cofaithful, and in
determining that Uq1 is full. Regarding the first of these, objectives are in fact
cofaithful in any sufficiently complete 2-category; as for the second, we only exploited
the fullness of Uq1 in later asserting it to be the coequifier of its equikernel,
and this assertion makes sense in any 2-category. It is possible then to at least
state the argument of this proof over other bases, and it would be valid so long as
the base 2-category in question had enough exactness properties (and discretes).
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 39
Since there is no concerted account of 2-categorical exactness properties in print,
we have chosen to work in the more concrete setting of a presheaf 2-category.
There is however another set of conditions, more directly related to exactness
notions, under which the pie colimits of frees and the objective quotients of frees
can be shown to coincide. These were investigated in the first author’s [4]; they
allow C to be more general than a presheaf 2-category but impose stronger conditions
on T than its preserving objectives. Since these conditions are more difficult
to verify in our examples, we have found no cause to give the arguments in any
detail here; but let us nonetheless break off briefly to discuss them.
Given a map of sets f : A → B, we may obtain its (strong epi, mono) factorisation
by first forming the kernel-pair of f, and then the coequaliser f1 : A → C
of that. Since f also coequalises this kernel-pair, it must factor through f1 as
f = f2 ◦ f1; now f1 is regular epi, hence strong epi, and it turns out that f2 is
mono, so giving the required factorisation. We have a similar limit-colimit description
of the (objective, fully faithful) factorisation of a functor f : A → B. By
taking the comma object f|f and appropriate pullbacks one may form the higher
kernel of f, which is an internal category in Cat as on the left below:
f|f|f f|f A B .
C
d GG
ioo
c
GG
p
GG
m GG
q
GG
f1 66❏❏❏❏❏❏❏
f2
XXttttttt
f
GG
A codescent cocone under this higher kernel is given by a 1-cell k: A → X
together with a 2-cell γ : kd ⇒ kc satisfying cocycle conditions. In Cat, there is
a universal such cocone (f1 : X → C, γ), with C called the codescent object of the
given data, and f1 a codescent morphism. Now f itself forms part of a codescent
cocone under its higher kernel, whence there is an induced factorisation f = f2 ◦f1
as on the right above. Just as each regular epi is strong epi, so every codescent
morphism is objective, so in particular f1 is objective; moreover, it turns out that
in Cat, the induced f2 is always fully faithful, so that we obtain an (objective,
fully faithful) factorisation of f as claimed.
One can emulate this same argument in any 2-category C which admits higher
kernels and codescent objects of higher kernels, so long as when one factorises
f = f2 ◦ f1 in C as above, the arrow f2 is known always to be fully faithful; this
is an exactness property of the 2-category C, analogous to regularity in the onedimensional
setting. In such a C we have (codescent, fully faithful) factorisations,
and it follows that the objectives and codescent morphisms in C coincide. Now
given a 2-monad T on such a C preserving codescent objects of higher kernels the
factorisation described above lifts, so that objectives and codescent morphisms in
T-Algs again coincide; and under these assumptions, together with our standing
ones, any objective quotient of a free T-algebra admits a canonical presentation as
a codescent object, which exhibits it as belonging to the closure of the frees under
pie colimits. For the details of this argument, see [4, Chapter 9].
We conclude this section with an example showing, as promised, that the classes
of algebras in Proposition 27 need not always coincide.
Proposition 33. Not every pie algebra need lie in the closure of the frees under
pie colimits.
40 JOHN BOURKE AND RICHARD GARNER
Proof. Consider the 2-monad T on Cat whose action on objects is given by T(0) =
0 and TC = C + 1 whenever C is non-empty. An algebra is either a pointed
category, or the empty category, and an algebra map is either a functor preserving
the point, or the unique functor out of the empty category. Clearly T preserves
objectives, which are just the bijections on objects; thus the objectives in T-Algs
are exactly the bijective on objects algebra maps. By Proposition 27 each Talgebra
A lying in the closure of the frees under pie colimits admits an objective
morphism FC → A from some free T-algebra; thus the sets of objects of A and
FC have the same cardinality. By definition of T we know that the underlying
category of FC has either no objects (if C is the empty category) or at least two;
in particular the underlying category of such an A cannot have a single object.
Now the terminal category 1, which does have a single object, admits a unique
T-algebra structure; so we will be done if we can show that this T-algebra is pie.
By Proposition 7, this will be so if and only if j(1), the 1-element set with its
unique point, admits an FdUd-coalgebra structure for the induced monad Td on
Set. Arguing as before, the algebras for Td are either pointed sets or the empty
set; and amongst these, j(1) is initial with respect to all but the empty one.
As FdUdj(1) is non-empty, having cardinality 2, there exists a unique Td-algebra
morphism j(1) → FdUdj(1). To check that this equips j(1) with a coalgebra
structure involves checking the commutativity of diagrams with codomains j(1)
and (FdUd)2j(1), both of which are non-empty algebras; since j(1) is initial with
respect to such algebras both diagrams necessarily commute, so that j(1) admits
FdUd-coalgebra structure; whence 1 with its point is a pie algebra.
6. Strongly finitary 2-monads
In this final section, we give an extended application of the results developed
in the rest of the paper. We will use them to characterise, amongst the 2-monads
on Cat of a certain class, those whose algebras can be presented as categories
equipped with basic operations and basic natural transformations between derived
operations, satisfying equations between derived natural transformations but with
no equations being imposed between derived operations themselves. Let us agree
to call such a 2-monad pie-presentable; a precise definition will be given shortly.
At various places in the literature can be found observations to the effect that
each pie-presentable 2-monad is a flexible 2-monad in the sense of [9]; see, for
example, [8, Proposition 4.3], [14, Remark 6.3] or [20, Section 7.5]. Under suitable
cardinality constraints, the notion of flexible 2-monad in question can, as in the
introduction, be seen as a particular case of the general one: the 2-category of
monads with rank κ on Cat is 2-monadic over the corresponding 2-category of
endofunctors, and the flexible algebras for the induced 2-monad are precisely the
flexible 2-monads.
In this situation, we have also the corresponding pie algebras, hence a notion of
pie 2-monad. It turns out that every pie-presentable 2-monad is not just flexible
but also pie, but that this still does not characterise the pie-presentables amongst
all 2-monads. For example, consider the 2-monad on Cat for which an algebra is a
category C equipped with a strictly commutative binary operation θ: C ×C → C.
This 2-monad T is free on the endofunctor Sym2
of Cat which sends a category C to
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 41
the coequaliser of the identity and symmetry maps C ×C → C ×C; thus T is a pie
2-monad by Proposition 23, but it is not pie-presentable by virtue of the equation
between operations θ(–, ?) = θ(?, –) which must be imposed. In characterising
the pie-presentable 2-monads we must therefore make one further refinement. We
consider 2-monads of our chosen class as 2-monadic over a 2-category of signatures,
and call the pie algebras for the induced 2-monad the strongly pie 2-monads. Our
result now states that, amongst the class of 2-monads we will consider, a 2-monad
is pie-presentable if and only if it is strongly pie.
We will prove this result using the tools developed in the previous section; and
in order for these to be applicable, we must ensure that when we view our class of
2-monads as 2-monadic over signatures, the induced 2-monad preserves objectives.
This would not be the case if, for example, we were to consider all finitary 2-monads
on Cat; and for this reason, we shall instead be concerned with the strongly finitary
2-monads of [13], which are roughly speaking those whose algebras may be defined
only with reference to functors of the form Cn → C for various natural numbers n,
and to natural transformations between such functors. Whilst this rules out such
2-monads as that for categories with finite limits—which requires, amongst other
things, an operation of the form C·⇒· → C to express the taking of equalisers—
we still retain an acceptably large class of examples including the 2-monads for
monoidal categories, categories with finite products, distributive categories, and
so on. The characterisation result we will prove, then, is that a strongly finitary
2-monad on Cat is pie-presentable if and only if it is strongly pie. Of course, there
is a corresponding version of this result where “finitary” has been replaced by “of
rank κ” for some regular cardinal κ; we leave its formulation to the reader.
We now define precisely the notions appearing in the statement of our result.
Let F denote the full sub-2-category of Cat spanned by the categories 0, 1, 2, . . .
discrete on the corresponding number of elements. The inclusion F → Cat induces
by left Kan extension a 2-fully faithful functor [F, Cat] → End(Cat), left adjoint
to restriction; and as in [13], an endofunctor of Cat is called strongly finitary
when it is in the essential image of this 2-functor. The 2-category Endsf(Cat)
of strongly finitary endofunctors is thus 2-equivalent to [F, Cat], and so locally
finitely presentable. It was shown in [13] that the strongly finitary endofunctors are
closed under composition so that the inclusion 2-functor Endsf(Cat) → End(Cat)
is strict monoidal. It is also a left adjoint, since [F, Cat] → End(Cat) is so, and
thus by [9] is the left adjoint part of a monoidal adjunction: as such, it lifts to
a left adjoint Mndsf(Cat) → Mnd(Cat) between the corresponding 2-categories
of monoids. On the right is the 2-category of 2-monads on Cat; on the left is
the 2-category of strongly finitary 2-monads, which are equally just the 2-monads
whose underlying endofunctor is strongly finitary. There is a forgetful 2-functor
W : Mndsf(Cat) → Endsf(Cat) which has a left adjoint and is finitarily 2-monadic;
it follows that Mndsf(Cat), like Endsf(Cat), is locally finitely presentable, and in
particular, complete and cocomplete.
Let us write N for the discrete sub-2-category of Cat spanned by the categories
0, 1, 2, . . . ; by a signature, we mean an object of the presheaf 2-category [N, Cat].
There is a forgetful 2-functor V : Endsf(Cat) → [N, Cat], given by restriction along
the inclusion N ֒→ Cat, and this too has a left adjoint and is finitarily 2-monadic.
42 JOHN BOURKE AND RICHARD GARNER
So we have a pair of finitarily 2-monadic adjunctions as on the left in
[N, Cat]
G GG
⊥ Endsf(Cat)
V
oo
H GG
⊥ Mndsf(Cat)
W
oo [N, Cat]
K GG
⊥ Mndsf(Cat) .
Z
oo
Now taking Z = V W and K = HG we obtain a further finitary adjunction as
on the right; this is not a priori 2-monadic, but turns out to be so by a direct
application of [17, Theorem 2]. Consequently, if presented with a strongly finitary
2-monad on Cat, we may regard it either as an WH-algebra or a ZK-algebra; and
in either guise may ask whether, as an algebra, it is semiflexible, flexible, or pie.
The counit of the adjunction G ⊣ V yields a morphism of 2-comonads KZ =
HGV W → HW. From this fact, together with Lemma 4, we deduce the existence
of a morphism of 2-comonads from the Q associated with ZK to that associated
with WH; so that if a strongly finitary 2-monad is semiflexible, flexible or pie as
a ZK-algebra, then it is correspondingly so as a HW-algebra. This justifies our
calling a 2-monad strongly semiflexible, flexible or pie in the former case, with the
corresponding unqualified name serving in the latter one.
We now describe what is meant in saying that a strongly finitary 2-monad is
pie-presentable. Such 2-monads have been defined only vaguely so far, and by
reference to their algebras, which are to be categories equipped with various kinds
of structure: operations and transformations, both basic and derived and with
equations imposed between derived transformations. To make precise our notion
we will define these latter terms; showing that the strongly finitary 2-monads
whose algebras are categories so equipped are exactly those admitting a specific
colimit presentation in terms of free monads. The possibility of presenting monads
in this way was first discussed in detail in [15]; the 2-monad case was considered
in [20, Section 6.4] and [21, Section 5].
To draw the correspondence between colimits of free 2-monads and their algebras
one uses the endomorphism 2-monad C, C ∈ Mnd(Cat) of a category C,
which has value [CD, D] at a category D. The key property is that 2-monad maps
T → C, C correspond with T-algebra structures on C; it follows that one can
understand the algebras for a colimit of 2-monads in terms of the algebras for
its constituent parts. As the inclusion Mndsf(Cat) → Mnd(Cat) has a right adjoint
the same holds for colimits of strongly finitary 2-monads, except we now use
the strongly finitary coreflection C, C sf of C, C , which in particular has value
[Cn, C] for n ∈ N.
Given a discrete signature Σ1—that is, a discrete object of [N, Cat]—we say
that a category C has basic Σ1-operations if it is equipped with a functor Cn → C
for each object of Σ1(n). To so equip C is to give a signature morphism Σ1 →
Z C, C sf and so by adjointness a monad morphism c: KΣ1 → C, C sf, equally
amounting to a KΣ1-algebra structure on C.
In this situation, we have underlying the monad morphism c a signature morphism
Zc: ZKΣ1 → Z C, C sf which equips C with basic ZKΣ1-operations. Precomposing
with the (monic) unit map η: Σ1 → ZKΣ1 we re-find amongst these
the specified basic Σ1-operations on C; but also further operations, obtained by
substituting and reindexing these basic ones, that are necessarily present because
ZKΣ1 underlies a 2-monad. Let us agree to call these derived Σ1-operations on
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 43
C: thus for each t ∈ KΣ1(n) we have the derived Σ1-operation t : Cn → C, the
value of t under cn : KΣ1(n) → C, C sf (n) = [Cn, C].
Now suppose we are given another discrete signature Σ2 and a pair of signature
morphisms s, t: Σ2 ⇒ ZKΣ1. A category C with basic Σ1-operations will be said
to have basic Σ2-transformations if it comes equipped with a natural transformation
s(x) ⇒ t(x) : Cn → C between derived Σ1-operations for each x ∈ Σ2(n).
To equip a category C with basic Σ1-operations and basic Σ2-transformations is
to give a 2-cell of [N, Cat] as on the left in:
ZKΣ1 Zc
88◆◆◆◆
Σ2
s VV♣♣♣♣♣
t
88◆◆◆◆◆ Z C, C sf
ZKΣ1
Zc
VV♣♣♣♣
KΣ1 c
88◆◆◆◆
KΣ2
s VV♣♣♣♣
t
88◆◆◆◆ C, C sf
KΣ1
c
VV♣♣♣♣
KΣ1 q
88▲▲▲▲▲
γ
KΣ2
s VVrrrr
t
88▲▲▲▲ R ;
KΣ1
q
VVrrrrr
or equally, by adjointness, a 2-cell in Mndsf(Cat) as in the middle; or equally, a
morphism d: R → C, C sf in Mndsf(Cat) out of the coinserter of s and t as on
the right; or equally, an R-algebra structure on C.
Now the coinserter map q: KΣ1 → R, like any coinserter map, is objective;
and we will see below that the 2-monad ZK preserves objectives, so that by
Proposition 2, Zq is also objective, which is to say, pointwise bijective on objects.
Thus objects of R(n) coincide with objects of KΣ1(n), that is, with derived Σ1operations,
a fact which would no longer be true in the world of finitary 2-monads
on Cat. On the other hand, R(n) will almost certainly have different morphisms
to KΣ1(n), and these will be the concern of our next definition.
For a category C with basic Σ1-operations and Σ2-transformations, amounting
to an R-algebra structure as above, we define a derived Σ2-transformation
α : s ⇒ t : Cn → C between derived Σ1-operations to be the image under the
monad map d: R → C, C sf of some morphism α: s → t of R(n). By precomposing
with the adjoint transpose of the coinserter 2-cell γ, we re-find amongst such
derived transformations the basic Σ2-transformations with which C was equipped;
but also others, obtained by substituting and composing together the basic ones,
that are necessarily present because R is a 2-monad.
Finally, let us suppose given a third discrete signature Σ3 and a parallel pair of
2-cells α, β as on the left of:
Σ3
h
99
k
WWα β ZR KΣ3
h
77
k
WWα β R KΣ3
h
77
k
WWα β R
r GG T .
A category C with basic Σ1-operations and Σ2-transformations is said to satisfy
Σ3-equations if αx = βx : h(x) ⇒ k(x) for each x ∈ Σ3(n). Forming the
adjoint transposes in Mndsf(Cat) of α and β, as in the centre above, this is equally
well to ask that the monad map d: R → C, C sf encoding C’s R-algebra structure
should satisfy d ◦ α = d ◦ β; or equally, that C should bear algebra structure for
T, the coequifier in Mndsf(Cat) of α and β as on the right above.
Thus basic operations on a category C are encoded by its being an algebra for
the free monad KΣ1 on a discrete signature Σ1, basic transformations between
44 JOHN BOURKE AND RICHARD GARNER
derived operations by its being an algebra for the coinserter R of a pair of maps
KΣ2 ⇒ KΣ1 with Σ2 discrete, and equations between derived transformations
by its being an algebra for the coequifier T of a pair of 2-cells between maps
KΣ3 ⇒ R with Σ3 also discrete. We define a strongly finitary 2-monad T to be
pie-presentable when it admits such a colimit presentation.
With this in place, we are in a position to state our main result:
Theorem 34. A strongly finitary 2-monad on Cat is pie-presentable if and only
if it is strongly pie.
One direction is easy: for by construction, any pie-presentable strongly finitary
2-monad lies in the closure of the free ZK-algebras under pie colimits, and is
therefore strongly pie by Proposition 27. For the other direction, we will apply
the result of the previous section; in preparation for which, we will now analyse
the action of the 2-monad ZK further.
Given a signature Σ ∈ [N, Cat], the free 2-monad KΣ on it is equally well
the free 2-monad on the endofunctor GΣ = n∈N Σ(n) × (–)n. To describe this,
consider Σ-Alg, the 2-category of algebras for the endofunctor GΣ; its objects are
categories C equipped with functors Σ(n)×Cn → C for each n ∈ N, and its 1- and
2-cells are the evident structure-preserving maps. There is a forgetful 2-functor
Σ-Alg → Cat; this has a left 2-adjoint, and by the argument of [10, Proposition
22.2], the induced 2-monad on Cat is precisely KΣ.
We now describe the left 2-adjoint of Σ-Alg → Cat, thus the free Σ-algebra on
a category C. The description is essentially standard, and can be found at various
levels of generality in [16, Section 2.2], [25, Appendix D] or [6, Theorem 24], for
example. First, let Ω be the set inductively defined by the following clauses:
• ⋆ ∈ Ω;
• Whenever n ∈ N and α1, . . . , αn ∈ Ω, then also (α1, . . . , αn) ∈ Ω.
Next, we recursively associate to each element α ∈ Ω a natural number |α| and an
object ˆα ∈ [N, Cat], as follows:
|α| =
1 if α = ⋆;
|α1| + · · · + |αn| if α = (α1, . . . , αn);
and ˆα =
0 if α = ⋆;
ˆα1 + · · · + ˆαn + yn if α = (α1, . . . , αn).
Here, yn ∈ [N, Cat] is the representable at n. Now the free Σ-algebra on C will
have underlying category
C∗
=
α∈Ω
[N, Cat](ˆα, Σ) × C|α|
; (6.1)
to give its Σ-algebra structure, we must give functors Σn × (C∗)n → C∗ for each
n. Unfolding the definition (6.1), this is equally to give a functor
Σn × [N, Cat]( ˆα1, Σ) × · · · × [N, Cat]( ˆαn, Σ) × C|α1|
× · · · × C|αn|
→ C∗
for every n ∈ N and α1, . . . , αn ∈ Ω. On defining α = (α1, . . . , αn) ∈ Ω, we observe
that the domain of this functor is isomorphic to [N, Cat](ˆα, Σ) × C|α|, so that we
may take it to be the coproduct injection at α. This makes C∗ into a Σ-algebra,
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 45
which may now be shown to be the free Σ-algebra on C. It follows that (6.1) gives
the value at C of the free 2-monad KΣ on Σ, and we conclude that:
Proposition 35. The 2-functor ZK on [N, Cat] is given by:
(ZKΣ)(n) =
α∈Ω
[N, Cat](ˆα, Σ) × n|α|
. (6.2)
This proposition exhibits ZK as pointwise a coproduct of hom-functors of the
form [N, Cat](ˆα, –). Since each ˆα is projectively discrete in [N, Cat], every such
hom-functor will preserve objective morphisms, whence so also will ZK itself. In
fact, more is true: every such hom-functor, and hence also ZK, will preserve maps
which are pointwise bijective on objects and full. We will use this fact shortly.
Finally, we observe that the monad (ZK)d induced by ZK on [N, Cat]d =
[N, Set] sends an object Σ to α∈Ω[N, Set](ˆα, Σ) × n|α|; it is thus also pointwise a
coproduct of representable functors, and so preserves connected limits, in particular
coreflexive equalisers. So all of the hypotheses of Theorem 32 are satisfied,
which allows us to conclude that every pie ZK-algebra—that is, every strongly pie
strongly finitary 2-monad—is a pie colimit of frees on discrete signatures. However,
this is not quite enough to show that every such 2-monad is pie-presentable, as we
must to complete the proof of Theorem 34; for that, we need to produce a colimit
presentation of the specific form demanded in the definition of pie-presentability.
We will do so by adapting the proof of Proposition 31.
Proof of Theorem 34. We are to show that a strongly pie strongly finitary 2-monad
T is pie-presentable. By Theorem 32, we know that such a T admits an objective
morphism f : KΣ1 → T in Mndsf(Cat), where Σ1 is a discrete signature. We now
trace through the proof of Proposition 31 to derive from this a pie-presentation of
T. Arguing as in the first part of that proof, we form the comma object in [N, Cat]
as on the left in:
Y
d GG
c

γ

ZKΣ1
Zf

ZKΣ1
Zf
GG ZT
KΣ2
f
GG
g

δ
KΣ1
q

KΣ1 q
GG R .
We set Σ2 = DOY , the discrete coreflection of Y , and take f, g: KΣ2 ⇒ KΣ1 to
be the transposes of d ◦ λY and c ◦ λY . Now on forming the coinserter of f and
g as on the right above, the argument of Proposition 31 yields an induced map
r: R → T in Mndsf(Cat) for which Zr is pointwise bijective on objects and full.
But now ZKZr is also pointwise bijective on objects and full, since as we observed
above, ZK preserves such morphisms. So to complete the construction, we argue
as in the final part of the proof of Proposition 31. We form the equi-kernel of Zr,
as on the left of:
W
w
99
z
WWφ ψ ZR
Zr GG ZT KΣ3
h
77
k
WWα β R
r GG T .
Now we set Σ3 = DOW, and on whiskering with λW : DOW → W and taking
transposes, obtain data h, k, α, β as on the right; now the argument of the last
46 JOHN BOURKE AND RICHARD GARNER
part of Proposition 31 shows that r exhibits T as the coequifier of α and β, so that
T is pie-presentable as required.
Let us now relate the notion of strongly pie 2-monad with our intuition that
the pie algebras are those which are “free at the level of objects”. If T is a
strongly finitary 2-monad on Cat, then we have, as before, an induced monad Td
on Catd = Set; and it is easy to see that this Td is finitary. Let us say that a
finitary monad on Set is free on a signature if it is in the essential image of the
left adjoint of the (monadic) forgetful functor Mndf(Set) → [N, Set].
Proposition 36. A strongly finitary 2-monad T on Cat is strongly pie just when
the induced finitary monad Td on Set is free on a signature.
For an earlier result in a similar spirit to this one, see [8, Proposition 4.3].
Proof. An analysis identical in form to the one leading to Proposition 35 yields a
description of the free finitary monad on Set generated by a signature Σ ∈ [N, Set],
and from this we deduce that there is a commuting diagram of left and right
adjoints as on the left in:
Mndsf(Cat)0
(−)d
GG
OZ
66■■■■■■■■■■■
Mndf(Set)
zz✉✉✉✉✉✉✉✉✉✉✉
[N, Set]
KD
dd■■■■■■■■■■■
XX✉✉✉✉✉✉✉✉✉✉✉
Mndsf(Cat)0
(–)d
GG

Mndf(Set)

ZK0-Alg
j
GG ZKd-Alg .
Consider now the square on the right above, in which the unlabelled vertical arrows
are the canonical equivalences. Each vertex admits an adjunction with [N, Set],
whilst each morphism commutes with both the left and the right adjoint parts of
these adjunctions. As there is a unique functor to a category of algebras commuting
with both adjoints it follows that the square commutes. Thus to say of T ∈
Mndsf(Cat) that Td is free on a signature is equally well to say that jAT is a free
Td-algebra; and so equally, by Theorem 3, that T is a strongly pie 2-monad.
For our final result, we fulfil a promise made at the end of Section 3.1 by giving
a general characterisation of the pie algebras for any strongly pie 2-monad.
Proposition 37. If T is a strongly pie strongly finitary 2-monad on Cat, then the
following classes of T-algebras coincide:
(1) The closure of the frees (on discretes) under pie colimits;
(2) The objective quotients of the frees (on discretes);
(3) The pie algebras;
and may be characterised as the T-algebras A for which jA is a free Td-algebra.
Proof. To say that T : Cat → Cat is strongly finitary is to say that it is the left
Kan extension along the inclusion F ֒→ Cat of some D: F → Cat. Thus T ∼=
n∈F
Dn × Cat(n, –) is a colimit of functors of the form Cat(n, –), and since every
such functor preserves objectives, so too does T. Moreover, since T is strongly
pie, the induced monad Td on Set is free on a signature. Any such monad is
a coproduct of representables, and as such preserves all connected limits, and
in particular coreflexive equalisers. Thus all the hypotheses of Theorem 32 are
satisfied and the result follows.
ON SEMIFLEXIBLE, FLEXIBLE AND PIE ALGEBRAS 47
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Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2,
Brno 60000, Czech Republic
E-mail address: bourkej@math.muni.cz
Department of Computing, Macquarie University, NSW 2109, Australia
E-mail address: richard.garner@mq.edu.au