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Sel. Math. New Ser. (2018) 24:2791–2830
https://doi.org/10.1007/s00029-017-0361-3
Selecta Mathematica
New Series
Relative pseudomonads, Kleisli bicategories,
and substitution monoidal structures
M. Fiore1·N. Gambino2·M. Hyland3·
G. Winskel1
Published online: 20 November 2017
© The Author(s) 2017. This article is an open access publication
Abstract We introduce the notion of a relative pseudomonad, which generalizes the
notion of a pseudomonad, and define the Kleisli bicategory associated to a relative
pseudomonad. We then present an efficient method to define pseudomonads on the
Kleisli bicategory of a relative pseudomonad. The results are applied to define several
pseudomonads on the bicategory of profunctors in an homogeneous way and provide a
uniform approach to the definition of bicategories that are of interest in operad theory,
mathematical logic, and theoretical computer science.
Mathematics Subject Classification 18D05 ·18C20 ·18D50
1 Introduction
Just as classical monad theory provides a general approach to study algebraic structures
on objects of a category (see [5] for example), 2-dimensional monad theory offers an
elegant way to investigate algebraic structures on objects of a 2-category [10,29,34,
37,54,56]. Even if the strict notion of a 2-monad is sufficient to develop large parts of
BN. Gambino
n.gambino@leeds.ac.uk
M. Fiore
Marcelo.Fiore@cl.cam.ac.uk
M. Hyland
m.hyland@dpmms.cam.ac.uk
G. Winskel
Glynn.Winskel@cl.cam.ac.uk
1Computer Laboratory, University of Cambridge, Cambridge, UK
2School of Mathematics, University of Leeds, Leeds, UK
3DPMMS, University of Cambridge, Cambridge, UK
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2792 M. Fiore et al.
the theory, the strictness requirements that are part of its definition are too restrictive
for some applications and it is necessary to work with the notion of a pseudomonad
[12] instead, in which the diagrams expressing the associativity and unit axioms for
a 2-monad commute up to specified invertible modifications, rather than strictly. In
recent years, pseudomonads have been studied extensively [14,39,49,50,52,53,62].
Our general aim here is to develop further the theory of pseudomonads. In particu-
lar, we introduce relative pseudomonads, which generalize pseudomonads, define the
the Kleisli bicategory associated to a relative pseudomonad, and describe a method
to extend a 2-monad on a 2-category to a pseudomonad on the Kleisli bicategory of
a relative pseudomonad. We use this method to show how several 2-monads on the
2-category Cat of small categories and functors can be extended to pseudomonads
on the bicategory Prof of small categories and profunctors (also known as bimod-
ules or distributors) [8,42,60]. This result has applications in the theory of variable
binding [22,24,55,61], concurrency [13], species of structures [23], models of the
differential λ-calculus [21], and operads and multicategories [15–17,25,27].
For these applications, one would like to regard the bicategory of profunctors as
a Kleisli bicategory and then use the theory of pseudo-distributive laws [34,49,50],
i.e. the 2-dimensional counterpart of Beck’s fundamental work on distributive laws [6]
(see [58] for an abstract treatment). In order to carry out this idea, one is naturally
led to try to consider the presheaf construction, which sends a small category Xto its
category of presheaves P(X)=def [Xop ,Set], as a pseudomonad. Indeed, a profunctor
F:X→Y, i.e. a functor F:Yop ×X→Set, can be identified with a functor
F:X→P(Y). However, the presheaf construction fails to be a pseudomonad for
size reasons, since it sends small categories to locally small ones, making it impossible
to define a multiplication. Although some aspects of the theory can be developed
restricting the attention to small presheaves [20], which support the structure of a
pseudomonad, some of our applications naturally involve general presheaves and thus
require us to deal not only with coherence but also size issues.
In order to do so, we introduce the notion of a relative pseudomonad (Definition 3.1),
which is based on the notions of a relative monad [1, Definition 2.1] and of a no-
iteration pseudomonad [53, Definition 2.1]. These notions are, in turn, inspired by
Manes’ notion of a Kleisli triple [47], which is equivalent to that of a monad, but better
suited to define Kleisli categories (see also [51,64]). For a pseudofunctor between
bicategories J:C→D(which in our main example is the inclusion J:Cat →CAT
of the 2-category of small categories into the 2-category of locally small categories), the
core of the data for a relative pseudomonad Tover Jconsists of an object TX ∈D
for every X∈C, a morphism iX:JX →TX for every X∈C, and a morphism
f∗:TX →TY for every f:JX →TY in D. This is as in a relative monad, but
the equations for a relative monad are replaced in a relative pseudomonad by families
of invertible 2-cells satisfying appropriate coherence conditions, as in a no-iteration
pseudomonad. As we will see in Theorem 4.1, these conditions imply that every
relative pseudomonad Tover J:C→Dhas an associated Kleisli bicategory Kl(T),
defined analogously to the one-dimensional case. In our main example, the presheaf
construction gives rise to a relative pseudomonad over the inclusion J:Cat →CAT
in a natural way and it is then immediate to identify its Kleisli bicategory with the
bicategory of profunctors. It should be noted here that the presheaf construction is
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Relative pseudomonads, Kleisli bicategories, and… 2793
neither a no-iteration pseudomonad (because of size issues) nor a relative monad
(because of strictness issues).
As part of our development of the theory of relative pseudomonads, we show
how relative pseudomonads generalize no-iteration pseudomonads (Proposition 3.3)
and hence (by the results in [53]) also pseudomonads. We then introduce relative
pseudoadjunctions, which are related to relative pseudomonads just as adjunctions are
connected to monads. In particular, we show that every relative pseudoadjunction gives
rise to a relative pseudomonad (Theorem 3.8) and that the Kleisli bicategory associated
to a relative pseudomonad fits in a relative pseudoadjunction (Theorem 4.4).
Furthermore, we introduce and study the notion of a lax idempotent relative pseu-
domonad, which appears to be the appropriate counterpart in our setting of the notion
of a lax idempotent 2-monad (often called Kock–Zöberlein doctrines) [37,38,65] and
pseudomonad [48,52,60]. In Theorem 5.3 we give several equivalent characterizations
of lax idempotent relative pseudomonad and combine this result with the analogous
one in [52] to show that a pseudomonad is lax idempotent as a pseudomonad in the
usual sense only if it is lax idempotent as a relative pseudomonad in our sense. This
notion is of interest since it allows us to exhibit examples of relative pseudomonads
by reducing the verification of the coherence axioms for a relative pseudomonad to
the verification of certain universal properties. In particular, the relative pseudomonad
of presheaves can be constructed in this way.
We then consider the question of when a 2-monad on the 2-category Cat of small
categories can be extended to a pseudomonad on the bicategory Prof of profunctors.
Rather than adapting the theory of distributive laws to relative pseudomonads along the
lines of what has been done for no-iteration monads [51], which would involve complex
calculations with coherence conditions, we establish directly that, for a pseudofunctor
J:C→Dof 2-categories, a 2-monad S:D→Drestricting to Calong Jin a suitable
way,and a relative pseudomonad Tover J,ifTadmits a lifting to 2-categories of strict
algebras or pseudoalgebras for S, then Sadmits an extension to the Kleisli bicategory
of T(Theorem 6.3). We do so bypassing the notion of a pseudodistributive law in a
counterpart of Beck’s result.
This result is well-suited to our applications, where the structure that manifests itself
most naturally is that of a lift of the relative pseudomonad of presheaves to various
2-categories of categories equipped with algebraic structure, often via forms of Day’s
convolution monoidal structure [18,31]. In particular, our results will imply that the
2-monads for several important notions (categories with terminal object, categories
with finite products, categories with finite limits, monoidal categories, symmetric
monoidal categories, unbiased monoidal categories, unbiased symmetric monoidal
categories, strict monoidal categories, and symmetric strict monoidal categories) can
be extended to pseudomonads on the bicategory of profunctors. A reason for interest
in this result is that the compositions in the Kleisli bicategories of these pseudomonads
can be understood as variants of the substitution monoidal structure that can be used
to characterize notions of operad [36,57].
As an illustration of the applications of our theory, we discuss our results in the
special case of the 2-monad Sfor symmetric strict monoidal categories, showing how
it can be extended to a pseudomonad on the bicategory of profunctors. This result
is the cornerstone of the understanding of the bicategory of generalized species of
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2794 M. Fiore et al.
structures defined in [23] as a ‘categorified’ version of the relational model of linear
logic [28,30] and leads to a proof that the substitution monoidal structure giving rise
to the notion of a coloured operad [4] is a special case of the composition in the Kleisli
bicategory. The results presented here are intended to make these ideas precise by
dealing with both size and coherence issues in a conceptually clear way.
Organization of the paper
Section 2reviews some background material on 2-monads, pseudomonads and their
algebras. Our development starts in Sect. 3, where we introduce relative pseudomon-
ads, relate them to no-iteration pseudomonads and ordinary pseudomonads, introduce
relative pseudoadjunctions and establish a connection between relative pseudoadjunc-
tions and relative pseudomonads. Section 4defines the Kleisli bicategory associated
to a relative pseudomonad and discusses some of its basic properties. In Sect. 5we
introduce and study lax idempotent relative pseudomonads. Section 6shows that an
extension of a relative pseudomonad Tto 2-categories of strict algebras or pseudoal-
gebras for a 2-monad Sinduces an extension of Sto the Kleisli bicategory of T,as
well as a composite relative pseudomonad TS. We conclude the paper in Sect. 7by
discussing applications of our theory and showing how several 2-monads on Cat can
be extended to pseudomonads on Prof.
2 Background
2-categories and 2-monads
We assume that readers are familiar with the fundamental aspects of the theory of
2-categories and of bicategories (as presented, for example, in [7,11,40]) and confine
ourselves to review some facts that will be used in the following and to fix notation
and conventions.
For a 2-category Cand a pair of objects X,Y∈C, we write C[X,Y]for the hom-
category of morphisms f:X→Yand 2-cells between them, which we denote with
lower-case Greek letters, φ:f→f. Two parallel morphisms f,f:X→Yare
said to be isomorphic if they are isomorphic as objects of C[X,Y], and we write
f∼
=fin this case. We write CAT for the 2-category of locally small categories,
functors, and natural transformations. Its full sub-2-category spanned by small cat-
egories will be written Cat. We then have an inclusion J:Cat →CAT.Weuse
the terms pseudofunctor,pseudonatural transformation, and pseudoadjunction rather
than homomorphism, strong natural transformation, and biadjunction, respectively.
Let us now review some aspects of 2-dimensional monad theory [10]. By a 2-
monad on a 2-category Cwe mean a 2-functor S:C→Cequipped with 2-natural
transformations m:S2→Sand e:1C→S, called the multiplication and unit of
the 2-monad, respectively, satisfying the usual axioms for a monad in a strict sense.
As usual, we often refer to a 2-monad by mentioning only its underlying 2-functor,
leaving implicit the rest of its data. Similar conventions will be used for other kinds
of structures considered in the rest of the paper.
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Relative pseudomonads, Kleisli bicategories, and… 2795
For a 2-category Cand 2-monad S:C→C, we write Ps-T-AlgC(or Ps-T-Alg
if no confusion arises) for the 2-category of pseudoalgebras, pseudomorphisms and
algebra 2-cells, and S-AlgC(or S-Alg) for the locally full sub-2-category of Ps-S-AlgC
spanned by strict algebras. Here, by a pseudoalgebra we mean an object A∈C, called
the underlying object of the algebra, equipped with a morphism a:SA →A, called
the structure map of the algebra, and invertible 2-cells
S2A
SA
SA
A,
Sa
a
mAa
¯a
ASA
A,
1A
eA
a
˜a
called the associativity and unit 2-cells of the algebra, subject to two coherence
axioms [59]. We have a strict algebra when the associativity and unit 2-cells are
identities, in which case (as in analogous cases below) the coherence conditions are
satisfied trivially. For pseudoalgebras Aand B(and in particular for strict algebras),
a pseudomorphism from Ato Bconsists of a morphism f:A→Band an invertible
2-cell
SA
A
SB
B,
Sf
f
ab
¯
f(2.1)
required to satisfy two coherence axioms [10,59]. For pseudomorphisms f,g:A→
B,analgebra 2-cell between them is a 2-cell α:f→gthat satisfies one coherence
axiom [10]. We have a forgetful 2-functor U:Ps-S-Alg →Cwith a left pseudoadjoint
F:C→Ps-S-Alg, defined by mapping an object X∈Cto the free algebra on X,
which is the strict algebra having SX as its underlying object and mX:S2X→SX
as its structure map. The components of the unit of the pseudoadjunction are the
components of the unit of the 2-monad.
In our applications, we will consider several 2-monads on CAT (restricting to Cat
in an evident way), for which we invite the readers to consult [10,41,43]. Among them,
the 2-monads for (strict) monoidal categories, symmetric (strict) monoidal categories,
categories with finite limits, categories with finite products, and categories with a
terminal object.
Bicategories and pseudomonads
For a bicategory C, we write the associativity and unit isomorphisms as natural families
of invertible 2-cells
(hg)f∼
=h(gf),1Yf∼
=f,f∼
=f1X,(2.2)
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2796 M. Fiore et al.
which we leave unnamed. By the coherence theorem for bicategories [45](which
also follows from the bicategorical Yoneda lemma [60], see [26]), every bicategory
is biequivalent to a 2-category. In virtue of this, we shall often treat bicategories as if
they were 2-categories.
Example 2.1 Fundamental to our applications is the bicategory Prof of profunctors [8,
42,60]. Its objects are small categories; and for small categories Xand Y, the hom-
category Prof[X,Y]is defined to be CAT[Yop×X,Set]. The composite of profunctors
F:X→Yand G:Y→Zis given by the profunctor G◦F:X→Zdefined by the
coend formula
(G◦F)(z,x)=def y∈Y
G(z,y)×F(y,x). (2.3)
For a small category X, the identity profunctor IdX:X→Xis defined by letting
IdX(x,y)=def X[x,y].(2.4)
It remains to prove that these definitions give rise to a bicategory. In the literature, it
is often suggested that this can be proved by direct calculations, left to the readers. In
Sect. 3, instead, we give a more conceptual proof by describing Prof as the Kleisli
bicategory associated to the relative pseudomonad of presheaves.
Apseudomonad on a bicategory Cis given by a pseudofunctor T:C→C, pseudo-
natural transformations n:T2→Tand i:1C→T, called the multiplication and
unit of the pseudomonad, respectively, and invertible modifications α,ρ, and λ, called
the associativity,right unit, and left unit, respectively, of T, fitting in the diagrams
T3T2
T2T,
nT n
Tn
n
α
TT2
T
T
1
iT
n
1
Ti
λρ
(2.5)
and subject to two coherence conditions [39]. The notions of a strict algebra and
pseudoalgebra, of strict morphism and pseudomorphism, and of algebra 2-cell make
sense also for pseudomonads, giving rise to bicategories Ps-T-Alg and T-Alg. When
Cis a 2-category, these are again 2-categories.
Every pseudomonad has also an associated Kleisli bicategory [14], which can be
defined in complete analogy with the one-dimensional case; but we do not spell this
out since we will give an alternative account of the Kleisli construction in Sect. 3.
Importantly, in constrast with the situation for algebras discussed above, the Kleisli
construction for a pseudomonad T:C→Cproduces only a bicategory even when
Cis a 2-category, with the associativity and unit isomorphisms of Tused to give the
associativity and unit isomorphisms of the Kleisli bicategory (see also Theorem 4.1
below).
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Relative pseudomonads, Kleisli bicategories, and… 2797
3 Relative pseudomonads
In ordinary category theory, the notion of a monad has an equivalent alternative presen-
tation, via the notion of a Kleisli triple [47], which is particularly convenient to define
Kleisli categories. The notion of a Kleisli triple admits a natural generalization, given
by the notion of a relative monad [1], which is obtained by allowing the underlying
mapping on objects of the Kleisli triple to be defined relative to a fixed functor (see [1]
for details). Similarly, in 2-dimensional category theory, the notion of a pseudomonad
can be rephrased equivalently as the notion of a no-iteration pseudomonad [53], which
is the 2-dimensional analogue of the notion of a Kleisli triple. Here, we introduce rel-
ative pseudomonads, which generalize no-iteration pseudomonads in the same way as
relative monads generalize Kleisli triples, i.e. by allowing the mapping on objects that
is part of a no-iteration pseudomonad to be defined relatively to a fixed pseudofunctor
between bicategories. From now until the end of this section, we consider a fixed
pseudofunctor between bicategories J:C→D.
Definition 3.1 Arelative pseudomonad T over J:C→Dconsists of
•an object TX ∈D, for every X∈C,
•a family of functors (−)∗
X,Y:D[JX,TY]→D[TX,TY]for X,Y∈C,
•a family of morphisms iX:JX →TX in Dfor X∈C,
•a natural family of invertible 2-cells μg,f:(g∗f)∗→g∗f∗,for f:JX →TY,
g:JY →TZ,
•a natural family of invertible 2-cells ηf:f→f∗iX,for f:JX →TY in D,
•a family of invertible 2-cells θX:iX∗→1TX,forX∈C,
such that the following conditions hold:
•for every f:JX →TY,g:JY →TZ,h:JZ →TV, the diagram
((h∗g)∗f)∗
μh∗g,f
(μh,gf)∗
((h∗g∗)f)∗
∼
=
(h∗g)∗f∗
μh,gf∗
(h∗(g∗f))∗
μh,g∗f
(h∗g∗)f∗
∼
=
h∗(g∗f)∗
h∗μg,f
h∗(g∗f∗)
(3.1)
commutes, and
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2798 M. Fiore et al.
•for every f:JX →TY, the diagram
f∗(η f)∗
∼
=
(f∗iX)∗μf,iXf∗iX∗
f∗θX
f∗1TX
(3.2)
commutes.
We introduce some terminology. For a relative pseudomonad Tover J:C→Das
in Definition 3.1, we refer to the family of morphisms iX:JX →TX,forX∈C,as
the unit of T, and to the family of 2-cells μ,η, and θas the associativity,right unit,
and left unit of T, respectively. Finally, we refer to the axioms in (3.1) and (3.2)as
the associativity and unit axioms for T, respectively. Note that in order to simplify the
notation we have omitted the subscripts on the functors (−)∗
X,Yand we will henceforth
continue to do so. We furthermore adopt the convention of writing Xrather than iX
in a subscript of μand θ. So, for example, we have
μf,X:(f∗iX)∗→f∗iX∗,
ηX:iX→iX∗iX.
We also omit the detailed definition of some 2-cells in diagrams, labelling arrows only
with the main 2-cell involved in its definition, and omitting subscripts. In all such
cases, the precise definition of the 2-cell can be easily deduced from its domain and
codomain.
We wish to make precise in what sense relative pseudomonads are a generalization
of no-iteration pseudomonads [53, Definition 2.1]. This will be useful for relating
relative pseudomonads and ordinary pseudomonads.
Lemma 3.2 Let T be a relative pseudomonad over J :C→D.
(i) For every f :JX →T Y and g :JY →T Z, the diagram
g∗f
ηg∗f
g∗ηf
(g∗f)∗iX
μf,giX
g∗f∗iX
commutes.
(ii) For every f :JX →T Y , t he diagram
(iY∗f)∗μY,f
(θYf)∗
iY∗f∗
θYf∗
f∗
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Relative pseudomonads, Kleisli bicategories, and… 2799
commutes.
(iii) For every X ∈C, the diagram
iX
ηX
1
iX∗iX
θXiX
iX
commutes.
Proof The proof is a modified version of the proof of the redundancy of three axioms
in the original definition of a monoidal category [33] (see also [32]), which has a
version also for pseudomonads [48, Proposition 8.1].
Proposition 3.3 A no-iteration pseudomonad is the same thing as a relative pseu-
domonad over the identity.
Proof The two notions involve exactly the same data, except for the direction of the
invertible 2-cells. Then, using the numbering of axioms for a no-iteration pseudomonad
in [53, Definition 2.1], the equivalence between axioms for a relative pseudomonad
and those for a no-iteration pseudomonad are given as follows:
Relative pseudomonads No-iteration pseudomonads
Naturality of μ⇔Axioms 6 and 7
Naturality of η⇔Axiom 4
Associativity axiom ⇔Axiom 8
Unit axiom ⇔Axiom 2
Lemma 3.2, part (i) ⇔Axiom 5
Lemma 3.2, part (ii) ⇔Axiom 3
Lemma 3.2, part (iii) ⇔Axiom 1.
Note that it follows that Axioms 1, 3 and 5 for a no-iteration pseudomonad in [53,
Definition 2.1] are redundant, in that they can be derived from the others.
The next remarks use Proposition 3.3 and the analysis of the relationship between
ordinary pseudomonads and no-iteration pseudomonads in [53] to show how a pseu-
domonad can be regarded as a relative pseudomonad over the identity 1C:C→C
and, conversely, how every relative pseudomonad over the identity determines a pseu-
domonad.
Remark 3.4 (From pseudomonads to relative pseudomonads) The combination of [53,
Theorem 6.1] and Proposition 3.3 shows that every pseudomonad T:C→Cwith data
as in Sect. 2induces a relative pseudomonad over the identity 1C:C→C. Explicitly,
for X∈C, we already have TX ∈Cand a morphism iX:X→TX as part of the
pseudomonad structure. For a morphism f:X→TY, we define f∗:TX →TY
by letting f∗=def nYT(f). The three families of invertible 2-cells μ,η,θfor a
pseudomonad are then obtained in an evident way. For example, for f:X→TY,we
let ηf:f→f∗iXbe the composite 2-cell
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2800 M. Fiore et al.
fλnYiTY f∼
=nYT(f)iX.
where the unnnamed isomorphism 2-cell is a pseudonaturality of i.
Remark 3.5 (From relative pseudomonad over the identity to pseudomonads) The
combination of Proposition 3.3 and [53, Theorem 3.6] shows that every relative
pseudomonad over an identity pseudofunctor induces a pseudomonad. The explicit
definitions are a bit involved, and therefore checking the coherence diagrams directly is
not straightforward, but we shall outline a more conceptual account of the construction
of a pseudomonad from a relative pseudomonad in Remark 4.5.
We introduce a generalization of the notion of pseudoadjunction between bicate-
gories [12,60], extending to the 2-categorical setting the notion of a relative adjunction
considered in [63] and [1, Section 2.2].
Definition 3.6 Let G:E→Dbe a pseudofunctor. A relative left pseudoadjoint F
to Gover J:C→D, denoted
E
G
C
J
F
D,
consists of
•an object FX ∈E, for every object X∈C;
•a family of morphisms iX:JX →GFX,forX∈C;
•a family of adjoint equivalences
D[JX,GA]
(−)
⊥E[FX,A],
G(−)iX
(3.3)
for X∈C,A∈E.
For a relative pseudoadjunction as in Definition 3.6, the components of the unit and
counit of the adjoint equivalences in (3.3) will be written
ηf:f→G(f)iX,ε
u:(G(u)iX)→u,
respectively, where f:JX →GA and u:FX →A. Note that a relative pseudoad-
junction over the identity 1C:C→Cis equivalent to a pseudoadjunction in the usual
sense [35,60]. We now establish a relative variant of a standard fact that a pseudoad-
junction of bicategories gives rise to a pseudomonad [35,60], namely that a relative
pseudoadjunction determines a relative pseudomonad. The next lemma will be useful
for proving this.
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Relative pseudomonads, Kleisli bicategories, and… 2801
Lemma 3.7 Let
E
G
C
J
F
D
be a relative pseudoadjunction. Then there is an essentially unique way of extending
the function mapping X ∈Cto F X ∈Eto a pseudofunctor F :C→Eso that the
maps iX:JX →GFX, for X ∈C, become the 1-cell components of a pseudonatural
transformation i :J⇒GF.
Proof For a morphism f:X→Yin C, we define F(f)=def (iYJ(f)). The unit of
the adjoint equivalence in (3.3) gives us invertible 2-cells
ψf:iYJ(f)→GF(f)iX(3.4)
for f:X→Y.For f:X→Y,g:Y→Z, we need an invertible 2-cell
φg,f:F(gf)→F(g)F(f). By the definition, we have
F(gf)=(iZJ(gf))∼
=(iZJ(g)J(f)),F(g)F(f)=GF(g)F(f)iX.
So we can define φg,fusing the composite
(iZJ(g)J(f))ψ−1
GF(g)iYJ(f)ψGF(g)GF(f)iX∼
=GF(g)F(f)iX,
where the unnamed isomorphism is given by the pseudofunctoriality of G.ForX∈C,
we need an invertible 2-cell φX:F(1X)→1FX. By definition, F(1X)=(iX)=
(G(1FX)iX), and so we define φXto be ε1FX :(G(1FX)iX)→1FX. With these
definitions, the pseudonaturality 2-cells for i:J⇒GF are the 2-cells in (3.4). The
proof of the coherence conditions is routine (cf. [35,60]).
Theorem 3.8 Let
E
G
C
F
J
D
be a relative pseudoadjunction. Then the function sending X ∈Cto G F (X)∈D
admits the structure of a relative pseudomonad over J .
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2802 M. Fiore et al.
Proof For X∈C, define TX =def GFX. The relative pseudoadjunction gives mor-
phisms iX:X→TX,forX∈C, and adjoint equivalences
D[JX,GFY]
(−)
⊥E[FX,FY]
G(−)iX
(3.5)
for X,Y∈C. We then define (−)∗:D[JX,TY]→D[TX,TY]by letting
f∗=def G(f). It now remains to define the families of invertible 2-cells μ,ηand θ.
For μg,f:(g∗f)∗→g∗f∗, observe that
(g∗f)∗=G(G(g)f),g∗f∗=G(g)G(f),
and so we define μg,fto be the composite
GG(g)fηGG(g)G(f)iX∼
=G(G(gf)iX)εG(gf)
∼
=G(g)G(f).
The 2-cells ηf:f→f∗iXare given by the units of the adjunction (3.5), which satisfy
the required naturality condition. For θX:iX∗→1TX, we recall that iX∗=G(iX),
and so we define θXto be the composite 2-cell
G(iX)ηG(G(1FX)iX)εG(1FX)∼
=1GFX .
It remains to establish the coherence conditions. While it is possible to show this
directly, it is more illuminating to argue in terms of universal properties. Simply
restating the adjunction in (3.5), we observe that, given f:JX →GA in Dand
u:FX →Ain E, for every 2-cell φ:f→G(u)iX, there is a unique 2-cell ψ:f→
u, the adjoint transpose, such that the diagram
fηf
φ
G(f)iX
G(ψ) iX
G(u)iX
commutes. Accordingly, we can characterize μg,fand θXas follows. There are 2-cells
˜κg,f:G(G(g)f)→G(gf), ˜κX:G(iX)→G(1FX)
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Relative pseudomonads, Kleisli bicategories, and… 2803
being the image under Gof the unique 2-cells (G(g)f)→gfand iX→1FX
such that the diagrams
G(g)fη
η
G(G(g)f)iX
˜κg,fiX
G(g)G(f)iX∼
=G(gf)iX
iX
η
∼
=
G(iX)iX
˜κXiX
1GFX iX∼
=G(1FX)iX
commute. The 2-cells μg,fand θXthen arise by composing these 2-cells with pseudo-
functoriality 2-cells of G. The coherence diagrams follow readily, and we give details
in the 2-categorical case, where the characterizing diagrams for μg,fand θXreduce
to the diagrams
iX
η
1
iX∗iX
θXiX
iX,
g∗fη
g∗ηf
(g∗f)∗iX
μg,fiX
g∗f∗iX.
For the associativity condition in (3.1), we have commuting diagrams
(h∗g)∗fη
(h∗g)∗η
μh,gf
(h∗g)∗f)∗iX
μh∗g,fiX,
(h∗g)∗f∗iX
μh,gf∗iX
h∗g∗f
h∗g∗η
h∗g∗f∗iX
(h∗g)∗fη
μg,hf
(h∗g)∗f∗iX
(μh,gf)∗iX
h∗g∗fη
h∗η
h∗g∗η
(h∗g∗f)∗iX
μh,g∗fiX
h∗(g∗f)∗iX
h∗μf,giX
h∗g∗f∗iX.
Note that the triangles in these diagrams commute by part (i) of Lemma 3.2. Since
both of the composites on the right-hand side of the diagrams lie in the image of G,we
deduce by universality that they are equal, as required. For the unit condition in (3.2)
we have a commuting diagram
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2804 M. Fiore et al.
fηf
ηf
f∗iX
ηf∗iX
f∗iX
ηf∗iX
f∗ηiX
1
(f∗iX)∗iX
μf,iXiX
f∗iX∗iX
f∗θXiX
f∗iX.
Here, the triangles commute by part (i) and (iii) of Lemma 3.2. Again, since the
composite of (f∗θX)(μf,iX)(ηf∗)lies in the image of Gwe deduce by universality
that it equals the identity, as required.
Using Theorem 3.8, we can introduce our fundamental example of a relative pseu-
domonad, given by the presheaf construction.1
Example 3.9 There is a relative pseudoadjunction of the form
COC
U
Cat J
P
CAT ,
where COC is the 2-category of locally small cocomplete categories, cocontinuous
functors, and natural transformations, and U:COC →CAT is the evident forgetful
functor. The category P(X)=def [Xop,Set]of presheaves over a small category Xis
the colimit completion of Xin the sense that, for every locally small cocomplete A,
composition with the Yoneda embedding yX:X→P(X)induces an equivalence of
categories
CAT[X,A]COC[P(X), A].
U(−)yX
Thus Pprovides a relative left pseudoadjoint to U. By Theorem 3.8 we obtain a relative
pseudomonad over the inclusion J:Cat →CAT. For a functor F:X→P(Y),
F∗:P(X)→P(Y)is the left Kan extension of Falong the Yoneda embedding,
defined by the coend formula
F∗(p)(y)=def x∈X
F(x)(y)×p(x). (3.6)
1The possibility of introducing a notion of relative pseudomonad to include the presheaf construction as
an example was mentioned in [1, Example 2.7].
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Relative pseudomonads, Kleisli bicategories, and… 2805
The invertible 2-cells ηFfit into the diagram
XyX
F
ηF
⇒
P(X)
F∗
P(Y).
The 2-cells μF,Gand θXare uniquely determined by the universal property of left Kan
extensions.
There is an analogous relative pseudomonad arising from the relative pseudoad-
junction
FIL
U
Cat J
D
CAT ,
where FIL is the 2-category of locally small categories with filtered colimits, functors
preserving such colimits, and all natural transformations [2]. Here, for X∈Cat,we
define D(X)∈CAT to be the full subcategory of P(X)spanned by small filtered
colimits of representables.
4 Kleisli bicategories
We introduce the Kleisli bicategory of a relative pseudomonad, extending to the
2-dimensional setting the definition of the Kleisli category of a relative monad [1,
Section 2.3].
Theorem 4.1 Let T be a relative pseudomonad over J :C→D. Then there is a
bicategory Kl(T), called the Kleisli bicategory of T , having the objects of Cas objects,
and hom-categories given by Kl(T)[X,Y]=
def D[JX,TY],forX,Y∈C.
Proof We begin by defining composition in Kl(T).Let f:JX →TY and g:JY →
TZ. We define g◦f:JX →TZ as the composite in D
JX fTY g∗
TZ.
This obviously extends to 2-cells, so as to obtain the required composition functors. For
X∈C, the identity morphism on Xin Kl(T)is iX:JX →TX. For the associativity
isomorphisms, let f:JX →TY,g:JY →TZ and h:JZ →TV. Since
(h◦g)◦f=(h∗g)∗f,h◦(g◦f)=h∗(g∗f),
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2806 M. Fiore et al.
we define the associativity isomorphism αh,g,f:(h◦g)◦f→h◦(g◦f)to be the
composite 2-cell
(h∗g)∗fμh,gf(h∗g∗)f∼
=h∗(g∗f).
For the right and left unit, let f:JX →TY. Since f◦iX=f∗iX, we define
ρf:f◦iX→fto be the 2-cell ηf:f→f∗iX. Since iY◦f=iY∗f, we define
λf:iY◦f→fto be the composite 2-cell
iY∗fθYf1TY f∼
=f.
We now need to show that these natural isomorphisms satisfy the required coherence
conditions. We give the proof making explicit the bicategoricalstructure of Cand D.We
only need to show that the associativity, left unit, and right unit isomorphisms satisfy
the coherence conditions for a bicategory. The coherence axiom for associativity is
obtained via the following diagram:
((k∗h)∗g)∗f
μf(μ g)∗f
(k∗h∗)g)∗f
∼
=
((k∗h)∗g∗)f
(μ g∗)f
∼
=
(k∗(h∗g))∗f
μf
((k∗h∗)g∗)f
∼
=∼
=
(k∗h)∗(g∗f)
μ(g∗f)
(k∗(h∗g)∗)f
∼
=
(k∗μ) f(k∗(h∗g∗)) f
∼
=
(k∗h∗)(g∗f)
k∗((h∗g)∗f)k∗(μ f)k∗((h∗g∗)f)∼
=k∗(h∗(g∗f)) ,
where, starting from the rhombus on the right-hand side and proceeding clockwise,
we use naturality of the associativity in D, coherence of associativity in D, naturality
of the associativity in Dagain, and finally the associativity coherence axiom for a
relative pseudomonad in (3.1). The coherence axiom for the units is obtained via the
following diagram:
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Relative pseudomonads, Kleisli bicategories, and… 2807
g∗fηg∗f
∼
=
1
(g∗iY)∗fμg,Yf(g∗iY∗)f∼
=
(g∗θY)f
g∗(iY∗f)
g∗(θYf)
(g∗1TY)f∼
=g∗(1TY f)
∼
=
g∗f,
where, starting from the triangle on the top left-hand side, we use the coherence axiom
for units of the relative pseudomonad in (3.2), naturality of the associativity in D, and
the coherence axiom for units of D.
Note that, as mentioned in Sect. 2for ordinary pseudomonads, Kl(T)is only a
bicategory even if Cand Dare 2-categories.
Example 4.2 It is straightforward to identify the bicategory of profunctors of Exam-
ple 2.1 with the Kleisli bicategory associated to the relative pseudomonad of presheaves
Pof Example 3.9. First of all, both bicategories have small categories as objects. Sec-
ondly, for small categories Xand Ywe have
Prof[X,Y]=[Yop ×X,Set],Kl(P)[X,Y]=CAT[X,P(Y)].
Thus, we have a canonical isomorphism of hom-categories
τ:Prof[X,Y]→Kl(P)[X,Y],
given by exponential adjoint transposition. Furthermore, these isomorphisms are com-
patible with composition and identities. For composition, it suffices to observe that,
for profunctors F:X→Yand G:Y→Z, there is a canonical natural isomorphism
τ(G◦F)∼
=(τ G)◦(τ F),
where the composition on the left-hand side is that of Prof, as defined in (2.3), while
the composition on the right is the one of Kl(P), which is given by the functorial
composite of τ(F):X→P(Y)and (τ G)∗:P(Y)→P(Z), the latter being defined
by the formula for left Kan extensions in (3.6). For identities, simply note that, for
a small category X, the adjoint transpose of the identity profunctor on X, as defined
in (2.4), is exactly the Yoneda embedding, which is the identity on Xin Kl(P).
In one-dimensional category theory, every monad determines two universal adjunc-
tions relating the base category with the category of Eilenberg-Moore algebras and
the Kleisli category for the monad. For pseudomonads, the construction of the bicat-
egory of pseudoalgebras is well-known and it has been considered for no-iteration
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2808 M. Fiore et al.
pseudomonads in [53, Section 4], but we do not need its counterpart for relative pseu-
domonads here. We focus instead on the counterpart of the Kleisli adjunction, which
has not been considered for no-iteration pseudomonads. The first step is the following
lemma.
Lemma 4.3 Let T be a relative pseudomonad over J :C→D. Then the function
sending X ∈Kl(T)to T X ∈Dadmits the structure of a pseudofunctor G T:Kl(T)→
D.
Proof For X,Y∈Kl(T), we define the functor GT
X,Y:Kl(T)[X,Y]→D[GTX,
GTY]to be
(−)∗:D[JX,TY]−→D[TX,TY].
By inspection of the definitions, we can define the pseudofunctoriality 2-cells of GT
to be exactly some of the 2-cells that are part of the data of a relative pseudomonad,
namely
μg,f:GT(g◦f)→GT(g)GT(f), θX:GT(iX)→1GTX.
In order to have a pseudofunctor, we need to verify that the following three coherence
diagrams commute:
GT(h◦g)◦f
GT(μ f,g,h) μh◦g,f
GTh◦(g◦f)
μh,g◦f
GT(h◦g)GT(f)
μh,gGT(f)
GT(h)GT(g◦f)
GT(h)μ
g,f
GT(h)GT(g)GT(f),
GT(f)GT(ρ f)
1GT(f)
GT(f◦iX)μf,XGT(f)GT(1X)
GT(f)θX
GT(f),
GT(iY◦f)μY,f
GT(λ f)
GT(iY)GT(f)
θYGT(f)
GT(f).
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Relative pseudomonads, Kleisli bicategories, and… 2809
The first and second diagrams follow at once from the coherence conditions in (3.1)
and (3.2) that are part of the definition of a relative pseudomonad. The third is part (ii)
of Lemma 3.2.
By analogy with the one-dimensional case, we expect that the pseudofunctor
GT:Kl(T)→Dhas some form of left pseudoadjoint. The next result makes this
precise.
Theorem 4.4 Let T :C→Dbe a relative pseudomonad over J :C→D. Then
GT:Kl(T)→Dhas a relative left pseudoadjoint over J :C→D,
Kl(T)
GT
C
J
FT
D.
Proof For X∈C, we define FTX=def X. Then we have GTFTX=TX,sothe
relative pseudomonad provides a morphism iX:JX →GTFTX. For these to act like
the components of the unit of a relative pseudoadjunction one needs to show that the
functor
Kl(T)[FTX,Y]GT(−)iXD[JX,GTY]
is an adjoint equivalence. Indeed, Kl(T)[FTX,Y]=D[JX,TY]=D[JX,GTY]
and the functor GT(−)iXis naturally isomorphic to the identity. This is because,
for f:JX →TY,wehaveGT(f)◦iX=f∗iXand there is an invertible 2-
cell ηf:f→f∗iX, suitably natural.
Remark 4.5 Theorem 4.4 allows us to give a more conceptual account of the con-
struction of the pseudomonad associated to a relative pseudomonad over the identity
1C:C→Cin Remark 3.5. Given a relative pseudomonad Tover 1C:C→C,we
can construct a pseudoadjunction between Cand Kl(T)as in Theorem 4.4. Then,
the pseudomonad associated to this pseudoadjunction is exactly the pseudomonad
described in Remark 3.5. So we have established again the coherence conditions for
a pseudomonad, in a clean (albeit indirect) way.
Let us also note that if we start with a relative pseudomonad Tover J:C→
D, form the associated Kleisli relative pseudoadjunction (as in Theorem 4.4), and
take the induced relative pseudomonad (as in Theorem 3.8), we then retrieve the
original relative pseudomonad. The only issues arise at the 2-cell level and we leave
the verifications to the interested readers.
In the other direction, suppose that we start with a relative pseudoadjunction
E
G
C
J
F
D,
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2810 M. Fiore et al.
form the induced relative pseudomonad T=GF over J:C→D(as in Theorem 3.8),
and then take the induced relative pseudoadjunction
Kl(T)
GT
C
J
FT
D,
as in Theorem 4.4. We expect a comparison and indeed we have a pseudofunctor
C:Kl(T)→Edefined on objects by letting C(X)=def FX,for X∈C. On hom-
categories, for X,Y∈C, we define
CX,Y:D[JX,TY]→E[FX,FY]
by letting C(f)=def f,whereweusedthatTY =GF(Y).
One can compose adjunctions between categories and, similarly, pseudoadjunctions
between bicategories. It does not make sense to compose relative pseudoadjunctions,
but one can form the composite of a relative pseudoadjunction and a pseudoadjunction.
Proposition 4.6 Let
E
G
C
J
F
D,
E
F
⊥E
G
be a relative pseudoadjunction and a pseudoadjunction, respectively. Then there is a
relative pseudoadjunction of the form
E
GG
C
J
FF
D.
Proof The construction is evident and thus omitted.
We conclude this section by extending some results on no-iteration pseudomonads
to relative pseudomonads. In the next proposition, part (i) generalizes [53, Proposi-
tion 3.1], part (ii) generalizes [53, Proposition 3.2], while part (iii) does not seem to
have been made explicit for no-iteration pseudomonad yet.
Proposition 4.7 Let T be a relative pseudomonad over J :C→D.
(i) The function T :C→Dadmits the structure of a pseudofunctor.
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Relative pseudomonads, Kleisli bicategories, and… 2811
(ii) The family of morphisms i X:JX →TX, forX ∈C, admits the structure of a
pseudonatural transformation i :J→T.
(iii) The family of functions (−)∗:D[JX,TY]→D[TX,TY]for X,Y∈C, admits
the structure of a pseudonatural transformation.
Proof Parts (i) and (ii) follow from Lemma 3.7 and Theorem 4.4 via Remark 4.5,
but we also give explicit proofs. We begin from part (i). For f:X→Y, we define
T(f):TX →TY by letting Tf =def (iYJ(f))∗. We then define the pseudofunc-
toriality 2-cells. First, we need invertible 2-cells τg,f:T(gf)→T(g)T(f),for
f:X→Yand g:Y→Z. By definition, we have
T(gf)=(iZJ(g)J(f))∗,T(g)T(f)=(iZJ(g))∗(iYJ(f))∗.
We then define τg,fas the composite 2-cell
(iZJ(g)J(f))∗η((iZJ(g))∗iYJ(f))∗μ(iZJ(g))∗(iYJ(f))∗
Secondly, we need invertible 2-cells τX:T(1X)→1TX for X∈C. But since T(1X)=
iX∗by definition, we let τX=def θX, the component of the left unit of the relative
pseudomonad.
One should check the three coherence laws for a pseudofunctor. The coherence law
for τg,finvolves a pasting of the associativity condition in (3.1), part (i) of Lemma 3.2
and all the naturality conditions for the families 2-cells of a relative pseudomonad.
One of the coherence laws for τXcomes from the unit condition in (3.2), while the
other is from part (ii) of Lemma 3.2.
For part (ii), the required pseudonaturality 2-cell for f:X→Yfits into the diagram
JX
iX
J(f)
⇓¯
if
JY
iY
TX T(f)TY
Since T(f)=(iYJ(f))∗, we can simply let
¯
if=def ηiYJ(f).(4.1)
We should check two coherence conditions for pseudonatural transformations. The
composition condition involves a pasting of a naturality of ηto a diagram coming
from part (i) of Lemma 3.2. The identity condition is just part (iii) of Lemma 3.2 .
Finally, for part (iii), to see the pseudonaturality in X,takeu:X→X, observe
that f∗T(u)=f∗(iXu)∗and note the 2-cell
(fu)∗η(f∗iXu)∗μf∗(iXu)∗.
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2812 M. Fiore et al.
For the pseudonaturality in Y,takev:Y→Y, observe that (T(v) f)∗=((iYv)∗f)∗
and T(v) f∗=(iYv)∗f∗, and note the 2-cell
(iYv)∗f∗μ(iYv)∗f∗.
There are coherence conditions to check, but they are straightforward.
5 Lax idempotent relative pseudomonads
We isolate a special class of relative pseudomonads, which appears to be the appro-
priate generalization to our setting of the notion of a lax idempotent 2-monad, or
Kock–Zöberlein 2-monad, or KZ-doctrine [38,65]. An extensive analysis of these 2-
monads, with useful equivalent formulations, was given in the course of a study of
general property-like 2-monads in [37]. For pseudomonads, the more general notion of
lax idempotent pseudomonad on a bicategory was introduced in [60], with yet another
characterisation of the notion, and studied further in [48,52].
In order to state the definition of a lax idempotent relative pseudomonad, it is
convenient to use the notion of a left extension in a bicategory [40, §2.2.], which we
now recall. We consider a fixed morphism i:X→Xin a bicategory C. By definition,
aleft extension of a morphism f:X→Yalong iconsists of a morphism f:X→Y
and a 2-cell η:f→fisuch that composition with ηinduces a bijection between
2-cells f→gi and 2-cells f→gfor every morphism g:X→Y. In this case, one
says that ηexhibits fas the left extension of falong i.
Let us fix again a pseudofunctor between bicategories J:C→D.
Definition 5.1 Alax idempotent relative pseudomonad over Jis a relative pseu-
domonad Tover J:C→Dsuch that the following conditions hold:
•for all f:JX →TY, the 2-cell ηf:f→f∗iXexhibits f∗:TX →TY as a
left extension of falong iX,
•for all f:JX →TY,g:JY →TZ, the diagram
g∗f
ηg∗f
g∗ηf
(g∗f)∗iX
μf,giX
g∗f∗iX
commutes,
•for all X∈C, the diagram
iX
ηiX
1
iX∗iX
θXiX
iX
commutes.
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Relative pseudomonads, Kleisli bicategories, and… 2813
Our next goal is to give alternative characterizations of lax idempotent relative
pseudomonads, which we will use to discuss further the relative pseudomonad of
presheaves. For this, we formulate the relative version of the notion of a left Kan
pseudomonad introduced in [52, Definition 3.1].
Definition 5.2 Arelative left Kan pseudomonad over J:C→Dconsists of:
•an object TX ∈D, for every X∈C,
•a morphism iX:JX →TX in D, for every X∈C,
•a morphism f∗:TX →TY, for every f:JX →TY,
•an invertible 2-cell ηf:f→f∗iXwhich exhibits f∗as the left extension of f
along iX, for every f:JX →TY,
such that the following conditions hold:
•the 2-cell g∗ηf:g∗f→g∗f∗iXexhibits g∗f∗as the left extension of g∗falong
iX, for all f:JX →TX,g:JY →TZ,
•the identity 2-cell 1:iX→iXexhibits 1TX as a left extension of iXalong iX,for
all X∈C.
Let us now assume that we have an object TX ∈Dfor every X∈C, a morphism
iX:JX →TX in Dfor every X∈C, a morphism f∗:TX →TY for every
f:JX →TY and an invertible 2-cell ηf:f→f∗iXexhibiting f∗as the left
extension of falong iXfor every f:JX →TY. Note that this gives us all the data
for a relative left Kan pseudomonad, but does not require all its axioms. Below, we
refer to this data simply as T:C→D. It is evident that, for all X,Y∈Cwe have an
adjunction
D[JX,TY]
(−)∗
⊥D[TX,TY],
(−)iX
(5.1)
whose unit has components the invertible 2-cells ηf:f→f∗iX,for f:JX →TY,
and whose counit has components 2-cells that will be written εu:(ui
X)∗→u,for
u:TX →TY. We can then state our characterizations of lax idempotent relative
pseudomonads as follows.
Theorem 5.3 The following conditions are equivalent:
(i) T :C→Dadmits the structure of a lax idempotent relative pseudomonad over
J:C→D.
(ii) For every f :JX →T Y , the 2-cell εf∗:(f∗iX)∗→f∗is invertible and there
are isomorphisms
μf,g:(g∗f)∗→g∗f∗,θ
X:(iX)∗→1TX .
(iii) The bicategory Ewith objects of the form T X, for X ∈C, and hom-categories
given by defining E[TX,TY]to be the full subcategory of D[TX,TY]spanned
by the morphisms u :TX →T Y such that u ∼
=f∗,forsome f:JX →TY in
D, is a sub-bicategory of D.
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2814 M. Fiore et al.
(iv) There exists a sub-bicategory Eof Dsuch that f ∗:TX →T Y is in Efor all
f:JX →TY inDand εu:(uiX)∗→u is invertible for all u :TX →TY in
E.
(v) T :C→Dis a relative left Kan pseudomonad over J :C→D.
Proof To prove that (i) implies (ii), observe that the axioms for a lax idempotent
relative pseudomonad imply that the following diagram commutes:
f∗iX
ηf∗iX
f∗ηiX
1
(f∗iX)∗iX
μiX,fiX
f∗i∗
XiX
f∗θXiX
f∗iX.
By one of the triangular laws for the adjunction in (5.1), εf∗is the composite
(f∗iX)∗μf,iXf∗(iX)∗f∗θXf∗
and it is therefore invertible. For (ii) implies (iii), the given isomorphisms show that E
as defined is closed under composition and contains identities. The implication from
(iii) to (iv) is immediate. For the implication from (iv) to (v), observe that the following
diagram commutes:
g∗f
ηg∗f
g∗ηf
(g∗f)∗iX
(g∗ηf)∗iX
g∗f∗iX
ηg∗f∗iX
1
(g∗f∗iX)∗iX
εg∗f∗iX
g∗f∗iX.
This shows that g∗ηfis the composite of ηg∗f(which exhibits (g∗f)∗as an extension
of g∗falong iX) with an invertible 2-cell. It then follows that g∗ηfexhibits g∗f∗as
the left extension of g∗falong iX, as required. Similarly, we have
iX
1
ηiXiX∗iX
ε1TX
iX,
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Relative pseudomonads, Kleisli bicategories, and… 2815
which implies that the identity 1 :iX→iXexhibits 1TX as a left extension of iXalong
iX.
Finally, we show that (v) implies (i). We begin by defining the remaining parts
of the data for a relative pseudomonad, namely the families of invertible 2-cells
μf,g:(g∗f)∗→g∗f∗and θX:iX∗→1TX. Using the universal property of
ηg∗f, we define μf,gto be the unique 2-cell such that
g∗f
ηg∗f
g∗ηf
(g∗f)∗iX
μf,giX
(g∗f)∗iX.
Similarly, using the universal property of ηiX, we define θXto be the unique 2-cell
such that
iX
ηX
1
iX∗iX
θXiX
1TX iX.
It then remains only to check the coherence conditions of Definition 3.1. There are two
ways of doing this. The first is by a diagram-chasing arguments using the universal
properties defining μf,gand θX. The second is to express μf,gand θXin terms of
the unit and counit of the adjunction. Taking that approach, first observe that the
diagram
(h∗ui
X)∗(h∗ηuiX)∗
(h∗(ui
X)∗iX)∗εh∗(uiX)∗
(h∗εuiX)∗
h∗(ui
X)∗
h∗εu
(h∗ui
X)∗
εh∗uh∗u
(5.2)
commutes by a triangle identity and the naturality of ε. The associativity coherence
condition is then given by the diagram
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2816 M. Fiore et al.
((h
∗
g)
∗
f)
∗
((h
∗
g)
∗
f
∗
i)
∗
(h
∗
g)
∗
f
∗
(h
∗
g
∗
i)
∗
f
∗
h
∗
g
∗
f
∗
h
∗
(g
∗
f
∗
i)
∗
h
∗
(g
∗
f)
∗
(h
∗
(g
∗
f)
∗
i)
∗
(h
∗
g
∗
f)
∗
((h
∗
g
∗
i)
∗
f)
∗
((h
∗
g
∗
i)
∗
f
∗
i)
∗
(h
∗
g
∗
f
∗
i)
∗
(h
∗
(g
∗
f
∗
i)
∗
i)
∗
((h
∗
g)
∗
η
f
)
∗
ε
(h
∗
g)
∗
f
∗
(h
∗
η
g
)
∗
f
∗
ε
h
∗
g
∗f
∗
h
∗
ε
g
∗
f
∗
h
∗
(g
∗
η
f
)
∗
ε
h
∗
(g
∗
f)
∗
(h
∗
η
g
∗
f
)
∗
(ε
h
∗
g
∗f)
∗
((h
∗
η
g
)
∗
f)
∗
(ε
h
∗
g
∗f
∗
i)
∗
(h
∗
η
g
∗
f
∗
i
)
((h
∗
g
∗
i)
∗
η
f
)
∗
((h
∗
η
g
)
∗
f
∗
i)
∗
ε
(h
∗
g
∗
i)
∗
f
∗
ε
h
∗
g
∗
f
∗
ε
h
∗
(g
∗
f
∗
i)
∗
(h
∗
(g
∗
η
f
)
∗
i)
(h
∗
g
∗
η
f
)
∗
where, starting from the top in a clockwise direction, we use interchange, two natu-
ralities of ε, the diagram in (5.2), a naturality of ε, a naturality of η, and finally an
interchange again. Finally, the unit condition is given by the following diagram:
f∗ηf∗
ηf∗
1
(f∗iX)∗
(f∗ηX)∗
1
(f∗iX)∗
εf∗
(f∗(iX)∗iX)∗
(f∗εiX)∗
εf∗iX∗
f∗f∗iX∗
f∗ε1TX
where we have two uses of the triangle identities and a naturality
Example 5.4 We can apply Theorem 5.3 to show that the relative pseudomonad of
presheaves of Example 3.9 is lax idempotent. For this, observe that for X,Y∈Cat,
there is an adjunction of the form
CAT[X,P(Y)]
(−)∗
⊥CAT[P(X), P(Y)]
(−)yX
(5.3)
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Relative pseudomonads, Kleisli bicategories, and… 2817
in which the components of the unit are natural isomorphisms. The left adjoints in (5.3)
factor through COC, the sub-2-category of cocomplete locally small categories and
cocontinuous functors, and if U:P(X)→P(Y)is cocontinuous, then εUis an
isomorphism. Part (iii) of Theorem 5.3 then applies. A similar example arises by
considering the Ind-completion, which is also lax idempotent. This is because the
corresponding left adjoint factors through FIL the sub-2-category of Ind-complete
categories and functors preserving filtered colimits; and if U:D(X)→D(Y)pre-
serves filtered colimits then εUis an isomorphism.
Remark 5.5 Theorem 5.3 and [52, Theorems 4.1 and 4.2] imply that a pseudomonad T
on a bicategory Cis lax idempotent in the usual sense if and only if it is lax idempotent
as a relative pseudomonad over the identity 1C:C→Cin the sense of Definition 5.1.
6 Liftings, extensions, and compositions
We now discuss a general method to extend a 2-monad to the Kleisli bicategory of
a relative pseudomonad, which we will apply in Sect. 7to extend several 2-monads
from the 2-category Cat of small categories and functors to the bicategory Prof of
small categories and profunctors.
Let us begin by introducing the setting in which we will work. We fix a pseudo-
functor between 2-categories J:C→D, a relative pseudomonad Tover Jwith data
as in Definition 3.1 and a 2-monad S:D→Dwith data as in Sect. 2. We assume that
the 2-monad Srestricts along J. Explicitly, this means that we have a dotted functor
CJ
S
D
S
C
J
D,
and that, for X∈C, the components of the multiplication and the unit, written
mX:S2X→SX and eX:X→SX, respectively, are in C. This implies that the pseud-
ofunctor J:C→Dcan be lifted to pseudofunctors J:Ps-S-AlgC→Ps-S-AlgDand
J:S-AlgC→S-AlgD(the definition of these 2-categories is recalled in Sect. 2),
making the following diagram commute:
S-AlgC
JS-AlgD
Ps-S-AlgC
JPs-S-AlgD
C
J
D,
(6.1)
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2818 M. Fiore et al.
where the vertical arrows in the top square are inclusions and those in the bottom
square are forgetful 2-functors. We shall deal with two types of liftings, one involving
only strict algebras (Definition 6.1) and another one involving both strict algebras
and pseudoalgebras (Definition 6.2). We begin by defining the simpler type of lifting,
involving only strict algebras.
Definition 6.1 Alifting of T to strict algebras for S , denoted
S-AlgC
¯
T
U
S-AlgD
U
C
T
D,
consists of
•a strict algebra structure on TA, for every A∈S-AlgC,
•a pseudomorphism structure on f∗:TA →TB, for every pseudomorphism
f:JA→TB,
•a pseudomorphism structure on iA:JA→TA, for every A∈S-AlgC,
such that
•μf,g:(g∗f)∗→g∗f∗is an algebra 2-cell for every pair of pseudomorphisms
f:JA→TBand g:JB →TC,
•ηf:f→f∗iAis an algebra 2-cell for every pseudomorphism f:JA →TB,
•θA:iA∗→1TA is an algebra 2-cell for A∈S-AlgC.
Note that a lifting of Tto strict algebras gives immediately a relative pseudomonad
¯
Tover the pseudofunctor J:S-AlgC→S-AlgDsuch that applying the forgetful
2-functors to the data of ¯
Treturns the corresponding data of T. We shall give several
examples of liftings of the relative monad of presheaves to strict algebras for some
2-monads in Sect. 7. However, it is not useful to work with liftings to categories of
strict algebras for other 2-monads, since typically for a strict algebra Athere is no
evident structure of strict algebra structure on TA. In order to address this situation,
we introduce the following definition.
Definition 6.2 Alifting of T to pseudoalgebras for S , denoted
S-AlgC
¯
TPs-S-AlgD
C
T
D
consists of the following data:
•a pseudoalgebra structure on TA, for every A∈S-AlgC,
•a pseudomorphism structure on f∗:TA →TB, for every pseudomorphism
f:JA→TB,
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Relative pseudomonads, Kleisli bicategories, and… 2819
•a pseudomorphism structure on iA:JA→TA, for every A∈S-AlgC,
such that
•μf,g:(g∗f)∗→g∗f∗is an algebra 2-cell for every pair of pseudomorphisms
f:JA→TBand g:JB →TC,
•ηf:f→f∗iAis an algebra 2-cell for every pseudomorphism f:JA →TB,
•θA:iA∗→1TA is an algebra 2-cell for every A∈S-AlgC.
Similarly to what happened for liftings to strict algebras, a lifting of Tto
pseudoalgebras gives a relative pseudomonad ¯
T, but now over the pseudofunc-
tor J:S-AlgC→Ps-S-AlgD, again suitably related to ¯
Tvia the appropriate forgetful
2-functors. Note here that for the inclusion J:Cat →CAT, the corresponding inclu-
sion J:S-AlgCat →Ps-S-AlgCAT is not merely about size distinction, but involves
both strict algebras and pseudoalgebras. Indeed, the notion of a relative pseudomonad
was designed to encompass these situations as well.
Our next goal is to show how a lifting of a relative pseudomonad Tgives rise
to a pseudomonad on the Kleisli bicategory of T. In the one-dimensional situation,
such a step commonly involves passing via a distributive law [6]. In our setting,
where we are dealing with both coherence and size issues, such an approach would
be rather complicated, as one would have to adapt the theory of pseudo-distributive
laws [34,49,50] to relative pseudomonads. However, it is possible to take a more direct
approach.
Theorem 6.3 Assume that T has a lifting to either strict algebras or pseudoalgebras
for S. Then S has an extension to a pseudomonad ˜
S:Kl(T)→Kl(T)on the Kleisli
bicategory of T .
Proof We only deal with the case of a lifting to pseudoalgebras, since the case of
lifting to strict algebras is completely analogous. First, we consider the relative pseu-
domonad ¯
Tover J:S-AlgC→Ps-S-AlgDand its Kleisli bicategory Kl(¯
T).The
objects of Kl(¯
T)are strict algebras with underlying object in C, and its hom-categories
are given by
Kl(¯
T)[A,B]=Ps-S-AlgD[JA,TB].
Secondly, we observe that there is a forgetful pseudofunctor U:Kl(¯
T)→Kl(T),
defined on objects by sending a strict algebra to its underlying object. To define the
action on hom-categories, let A,B∈S-AlgC. Then, the required functor is determined
by the diagram
Kl(¯
T)[A,B]UA,BKl(T)[A,B]
Ps-S-Alg[JA,TB]UA,B
D[JA,TB].
We claim that Uhas a left pseudoadjoint. The action of the left pseudoadjoint on
objects is defined by sending Xto SX, the free pseudoalgebra on X(which is in
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2820 M. Fiore et al.
fact a strict algebra since Sis a 2-monad). Next, for X∈C, we define morphisms
˜eX:X→SX in Kl(T)as the composite
JX eXSJX =JSX iSX TSX
in D. We wish to show that these are suitably universal. For this, let us observe that
the diagram
Kl(T)[X,A]Kl(¯
T)[SX,A]
U(−)◦˜eX
Ps-S-AlgD[JSX,TA]
D[JX,TA]Ps-S-AlgD[SJX,TA]
U(−)eX
(6.2)
commutes up to natural isomorphism, since if f:SJX →TAis a pseudomorphism,
then
f◦˜eX=f∗˜eX=f∗iSX eX∼
=fe
X.
Since the horizontal arrow at the bottom of (6.2) is an equivalence, we have the desired
universality of the morphism ˜eX. Now that we have a pseudoadjunction
Kl(T)
F
⊥Kl(¯
T),
U
we obtain the desired extension ˜
Sas the pseudomonad associated to this pseudoad-
junction via Theorem 3.8.
We conclude this section by showing how to compose a relative pseudomonad and
a 2-monad.
Theorem 6.4 Assume that T admits a lifting to pseudoalgebras of S. Then the function
sending X ∈Cto T S(X)∈Dadmits the structure of a relative pseudomonad over
J:C→D.
Proof First, recall that, by Theorem 4.4, we have a relative pseudoadjunction
Kl(T)
GT
C
FT
J
D.
(6.3)
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Relative pseudomonads, Kleisli bicategories, and… 2821
Secondly, let us consider the pseudomonad ˜
S:Kl(T)→Kl(T)constructed in the
proof of Theorem 6.3 and its associated Kleisli bicategory Kl(˜
S). Applying again
Theorem 4.4, this time in the case of an ordinary pseudomonad, we have a pseudoad-
junction
Kl(T)
F˜
S
⊥Kl(˜
S).
G˜
S
(6.4)
By Proposition 4.6, we can then compose the pseudoadjunctions in (6.3) and the
relative pseudoadjunction in (6.4) so as to obtain a new relative pseudoadjunction
Kl(˜
S)
GTG˜
S
C
F˜
SFT
J
D.
By Theorem 3.8 we then obtain a relative pseudomonad over J:C→D. Unfold-
ing the definitions, one readily checks that the underlying function of this relative
pseudomonad sends X∈Cto TS(X)∈D, as required.
7 Substitution monoidal structures
We apply our results to obtain a homogeneous method for extending several 2-monads
from the 2-category Cat of small categories and functors to the bicategory Prof of
small categories and profunctors, encompassing all the examples considered in the
theory of variable binding [24,55,61], concurrency [13], species of structures [23],
models of the differential λ-calculus [21], and operads [25].
The simplest examples of liftings for the relative pseudomonad for presheaves are
with respect to 2-monads on CAT whose strict algebras are locally small categories
equipped with suitable classes of limits. These 2-monads are co-lax, as discussed
in [37]. The specific examples of 2-monads that we consider here are those for cat-
egories with terminal object, categories with chosen finite products (by which we
mean categories with chosen terminal object and binary products) and categories with
chosen finite limits (by which we mean categories with chosen terminal object and
pullbacks). Each of these 2-monads is flexible in the sense of [9,10] and restricts along
the inclusion J:Cat →CAT to a 2-monad on the 2-category Cat of small categories,
so as to determine a situation as in (6.1). We speak of small (or locally small) strict
algebras to indicate small (or locally small) categories equipped with a strict algebra
structure.
We make some preliminary observations about the pseudomorphisms (cf. (2.1)) in
the cases under consideration. Since limits are determined up to a unique isomorphism,
the pseudomorphisms are exactly the functors that preserve the specified limits in the
usual, up to isomorphism, sense: the coherence conditions for a pseudomorphism
are automatic [37]. Similarly, one sees directly that any 2-cell between functors that
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2822 M. Fiore et al.
preserve the relevant limits is an algebra 2-cell. In the terminology of [37], these 2-
monads Sare fully property-like. It follows in particular that S-AlgCAT[A,B]can be
regarded as a full subcategory of CAT[A,B]. All this is in fact an abstract consequence
of the fact that the 2-monads in question are all co-lax. That fact is evident and the
general theory appears in [37].
Theorem 7.1 Let S :CAT →CAT be the 2-monad for categories with terminal
object, or categories with finite products, or categories with finite limits. Then the
relative pseudomonad of presheaves P :Cat →CAT has a lifting to strict S -algebras,
¯
P:S-Alg(Cat)→S-Alg(CAT).
Proof Let us begin by observing that we have a choice of limits in Set, so for any
X∈Cat,P(X)has chosen limits defined pointwise. Thus, there is a strict S-algebra
structure on P(X). Furthermore, the Yoneda embedding yX:X→P(X)preserves
those limits. Hence, if A∈S-AlgCat is a small strict algebra, then yA:A→P(A)
is a pseudomorphism of S-algebras in an evident fashion. Composition with yAthus
gives us a functor
S-AlgCAT[A,P(B)]S-AlgCAT [P(A), P(B)].
(−)yA
Now, suppose that F:A→P(B)is a pseudomorphism, that is to say, Fpreserves
the relevant limits. Then the left Kan extension F∗:P(A)→P(B)also preserves
these limits. This is critical, and for the separate classes of limits needs to be proved
on a case by case basis. The case when Sis the 2-monad for a terminal object is
simple. If Fpreserves the terminal, then so does F∗yA(being naturally isomorphic
to F). But the Yoneda embedding yA:A→P(A)preserves the terminal object,
and hence so does F∗. The case when Sis the 2-monad for finite products can be
seen as a corollary of the results in [31] (see also Theorem 7.3), but we provide a
direct argument. Suppose that Fand hence F∗yApreserves finite products. As the
Yoneda embedding yA:A→P(A)preserves finite products, F∗preserves finite
products of representables. But the objects of P(A)are colimits of representables.
Since F∗and products with objects (are left adjoints and so) preserve colimits, it
follows that F∗preserves finite products. Finally, the case when Sis the monad for
finite limits is similar, though in this case the result is standard. If Ahas finite limits
and F:A→P(B)preserves finite limits, then Fis flat [46, §VII.10, Corollary 3]
and hence F∗preserves finite limits.
Thus, in each case, F∗is a pseudomorphism of strict S-algebras; and, as we observed
above, any 2-cell between pseudomorphisms will be an algebra 2-cell. Hence, the left
Kan extension gives us a functor
S-AlgCAT[A,P(B)](−)∗
S-AlgCAT[P(A), P(B)].
Now we exploit the fact that the relative pseudomonad for presheaves is lax idem-
potent (see Example 5.4). So we have an adjunction
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Relative pseudomonads, Kleisli bicategories, and… 2823
CAT[A,P(B)]
(−)∗
⊥CAT[P(A), P(B)].
(−)yA
(7.1)
We observed that S-Alg[A,P(B)]and S-Alg[P(A), P(B)]are full subcategories of
CAT[A,P(B)]and CAT[P(A), P(B)], and it is clear from the above discussion that
this adjunction restricts to an adjunction
S-AlgCAT[A,P(B)]
(−)∗
⊥S-AlgCAT[P(A), P(B)].
(−)yA
In view of Theorem 5.3, the claim is proved.
Remark 7.2 With the experience of these examples of liftings, it is easy to give exam-
ples of 2-monads which do not lift as above.
•Consider the 2-monad for a category with zero object (i.e. an object which is both
terminal and initial). No category of presheaves of sets over a non-empty category
has a zero object. So the 2-monad cannot lift. The same applies to the monad for
direct sums or biproducts (in the terminology of [44]).
•Consider the 2-monad for a category with initial object. Given a category Awith
initial object, while the presheaf category P(A)does indeed have an initial object,
the Yoneda embedding does not preserve it. Hence the 2-monad cannot lift.
•Consider the 2-monad for a category with equalisers. Given a category Awith
equalisers, the presheaf category P(A)also has equalisers, and the Yoneda embed-
ding yA:A→P(A)preserves them. But now suppose that Ahas equalisers and
that F:A→Set preserves them. It does not follow that F∗:P(A)→Set pre-
serves equalisers. For a counterexample one can obviously just take Ato be the
fork (i.e. the generic equaliser). Then for example take F:A→Set mapping the
parallel pair to the identity and twist on 2 with equaliser 0. Because of this failure
it follows that the 2-monad cannot lift.
Next, we consider 2-monads associated with various notions of monoidal category.
To start with, we consider 2-monads which are flexible in the sense of [9,10] and we
have again a situation as in (6.1).
Theorem 7.3 Let S :CAT →CAT be the 2-monad for monoidal categories, or
symmetric monoidal categories, or monoidal categories in which the unit is a terminal
object, or symmetric monoidal categories in which the unit is a terminal object. The
relative pseudomonad of presheaves P :Cat →CAT has a lifting to strict S -algebras,
¯
P:S-AlgCat →S-AlgCAT .
Proof The base case is that of a monoidal category. We discuss that case and derive the
others. We use the analysis in [31] of the univeral property of Day’s convolution tensor
product [18]. We write Mon (respectively, MON) for the 2-category of small (respec-
tively, locally small) monoidal categories, strong monoidal functors and monoidal
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2824 M. Fiore et al.
natural transformations. For cocomplete categories A,B, a functor F:A×B→C
is separately cocontinuous if for every a∈A,b∈Bboth F(a,−):B→Cand
F(−,b):A→Care cocontinous. We write COC[A,B;C]for the category of
such functors and natural transformations between them. A cocomplete category A
equipped with a monoidal structure is monoidally cocomplete if the tensor product
is separately cocontinuous. We then have a straightforward 2-category MONCOC
of monoidally cocomplete locally small categories, strong monoidal cocontinuous
functors, and monoidal transformations.
In [18] Day showed how for any small monoidal category A, the category P(A)
of presheaves on Acan be equipped with a monoidal structure, called the convolution
tensor product, which makes P(A)into a biclosed monoidally cocomplete category,
defined by letting
(F1ˆ
⊗F2)(a)=def a1,a2∈A
F1(a1)×F2(a2)×A[a,a1⊗a2]
for F1,F2∈P(A)and a∈A. Furthermore, the Yoneda embedding yA:A→P(A)
has then the structure of a strong monoidal functor. For A∈Mon and B∈MONCOC,
we have the adjoint equivalence obtained in [31]
MON[A,B]
(−)∗
⊥MONCOC[P(A), B],
(−)yA
(7.2)
as required. In particular, for any A,B∈Mon, we have an adjoint equivalence
MON[A,P(B)]
(−)∗
⊥MONCOC[P(A), P(B)].
(−)yA
In our terminology, the adjoint equivalences in (7.2) amount to saying that we have a
relative pseudoadjunction
MONCOC
Mon
P
MON .
This provides exactly a lifting of the relative pseudomonad P:Cat →CAT to a
relative pseudomonad ¯
P:S-AlgCat →S-AlgCAT. All these considerations extend to
symmetric monoidal categories, again by the results in [18,31]. For the 2-monads for
monoidal categories with the condition that the unit is terminal, the lift follows from
the above, observing that the unit of the convolution monoidal structure is the Yoneda
embedding of the unit on the base category and so it remains a terminal object.
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Relative pseudomonads, Kleisli bicategories, and… 2825
Our final group of examples of a lifting involve 2-monads on CAT which are not
flexible. In this case, we have a lifting to pseudoalgebras in the sense of Definition 6.2.
Theorem 7.4 Let S :Cat →Cat be the 2-monad for either strict monoidal cate-
gories, or symmetric strict monoidal categories, or strict monoidal category in which
the unit is terminal, or symmetric strict monoidal categories in which the unit is
terminal. Then the relative pseudomonad P :Cat →CAT has a lifting to pseudo- S-
algebras,
¯
P:S-AlgCat →Ps-S-AlgCAT .
Proof There is a direct and an indirect approach to this. Directly, one follows through
the arguments of the previous section making the necessary adjustments. Indirectly,
observe that in each case S, the flexible 2-monad associated to S, is the 2-monad whose
strict algebras are categories with unbiased structure (in the sense of [43]) as in the list in
Theorem 7.3.NowS-Alg is a full sub-2-category of S-Alg ∼
=Ps-S-Alg. So the lifting
of the relative pseudomonad of presheaves P:Cat →CAT to ¯
P:S-Alg(Cat)→
S-Alg(CAT)restricts to S-Alg(Cat)→S-Alg(Cat).
Corollary 7.5 All the 2-monads on Cat listed in Theorems 7.1,7.3, and 7.4 admit an
extension to pseudomonads on Prof.
Proof Immediate consequence of Theorems 6.3 and 7.1,7.3, and 7.4.
For each of the monads S:Cat →Cat above, one can consider the Kleisli bicat-
egory associated to the pseudomonad ˜
S:Prof →Prof determined by Corollary 7.5.
The composition functors of these Kleisli bicategories can be understood as general-
izations of various kinds of substitution monoidal structures [22–25,27,36,57], among
those giving rise to the notions of a many-sorted Lawvere theory and of a coloured
operad. We conclude the paper by illustrating this idea in the case of coloured operads.
Example 7.6 As an illustration of the theory developed here, we revisit the construc-
tion of the bicategory of generalized species of structures of [23] and relate more
precisely its composition with the substitution monoidal structure for coloured oper-
ads [4](seealso[19]). For this, let us begin by recalling the definition of the 2-monad
S:Cat →Cat for symmetric strict monoidal categories. Let X∈Cat.Forn∈N,
define Sn(X)to be the category having as objects n-tuples ¯x=(x1,...,xn)of objects
xi∈Xand as morphisms (σ, ¯
f):¯x→¯xgiven by pairs consisting of a permutation
σ∈nand an n-tuple of morphisms fi:xi→x
σ(i). We then let
S(X)=def
n∈N
Sn(X).
The category S(X)is equipped with a strict symmetric monoidal structure: the tensor
product, written ¯x⊕¯x, is given by concatenation of sequences, and the unit is given
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2826 M. Fiore et al.
by the empty sequence; the symmetry is given by a permutation of identity maps.
This definition can be extended easily to obtain a 2-functor S:Cat →Cat.The
multiplication of the monad is given by taking a sequence of sequences and forgetting
the bracketing, while the unit has components eX:X→S(X)mapping x∈Xto the
singleton sequence (x)∈S(X). By the theory developed above, and Corollary 7.5 in
particular, we obtain a pseudomonad
˜
S:Prof →Prof .(7.3)
For our purposes, it is convenient to describe explicitly the relative pseudomonad
associated to ˜
S. Its action on objects is the function mapping X∈Cat to S(X)∈Cat.
The component of the unit for X∈Cat is the profunctor ˜eX:X→S(X)corresponding
to the functor
XeXS(X)yS(X)PS(X).
The extension functors of the relative pseudomonad have the form
(−):Prof[X,S(Y)]→Prof [S(X), S(Y)],
where X,Yare small categories. For a functor F:X→PS(Y), we can define
the functor F:S(X)→PS(Y)recalling that, since S(Y)has a symmetric strict
monoidal structure, PS(Y)has an unbiased (in the sense of [43]) symmetric monoidal
structure. Hence, by the universal property of S(X), we have an essentially unique F
fitting into a diagram of the following form:
XeX
F
∼
=
S(X)
F
PS(Y).
For brevity, we omit the description of the invertible natural transformations
ηF:F⇒F◦˜eX,μ
F,G:(G◦F)⇒G◦F,κ
X:(˜eX)⇒IdX.
The Kleisli bicategory of ˜
Sis the bicategory S-Prof of S-profunctors defined in [25],
which has small categories as objects and hom-categories defined by
S-Prof[X,Y]=Prof [X,S(Y)]=CAT[S(Y)op ×X,Set].
Indeed, one can readily check that composition and identity morphisms of S-Prof ,as
defined in [25], coincide with those given by instantiating the general definition of a
Kleisli bicategory. Following [25], we write CatSym for the bicategory of categorical
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Relative pseudomonads, Kleisli bicategories, and… 2827
symmetric sequences, which is defined as the opposite of S-Prof . Thus, the objects
of CatSym are small categories and its hom-categories are given by
CatSym[X,Y]=
def S-Prof[Y,X]=CAT[S(X)op ×Y,Set].
We write F[¯x;y]for the result of applying F:S(X)op ×Y→Set to (¯x,y)∈
S(X)op ×Y. Given categorical symmetric sequences F:X→Yand G:Y→Z,
i.e. functors F:S(X)op ×Y→Set and G:S(Y)op ×Z→Set, their composite
G◦F:X→Zin CatSym is given by considering Fand Gas S-profunctors in
the opposite direction, taking their composition in S-Prof using the definition of
composition in a Kleisli bicategory, and then regarding the result as a categorical
symmetric sequence from Xto Z. Explicitly, one obtains that
(G◦F)( ¯x;z)=def
m∈N(y1,...,ym)∈Sm(Y)
G[y1,...,ym;z]×
¯x1∈S(X)
···¯xm∈S(X)
F[¯x1;y1]× ···× F[¯xm;ym]×S(X)[¯x,¯x1⊕···⊕ ¯xm].
(7.4)
Happily, this formula yields the definition of the substitution monoidal structure for
coloured operads given in [3] by considering the special case where Xand Yare
discrete and coincide with a fixed set of colours of the coloured operads under con-
sideration.
The bicategory CatSym can be seen to be equivalent to the bicategory of generalized
species of structures Esp previously introduced in [23]. To see this, let us recall the
definition of Esp. For this, observe that the duality pseudofunctor (−)⊥:Prof →
Prof defined by X⊥=def Xop allows us to turn this pseudomonad in (7.3)intoa
pseudocomonad. The bicategory Esp is then defined as the co-Kleisli bicategory of
this pseudocomonad. More explicitly, its objects are small categories and its hom-
categories are given by
Esp[X,Y]=Prof[S(X), Y]=CAT[Yop ×S(X), Set].
The bicategory Esp is then equivalent to CatSym, via the pseudofunctor that sends X
to Xop. Indeed,
CatSym[X,Y]=CAT[S(Xop)×Y,Set]∼
=[Y×S(X)op,Set]∼
=Esp[Xop,Yop ].
Furthermore, the composition operation of categorical symmetric sequences defined
in (7.4) corresponds exactly to composition of generalized species of structures defined
via co-Kleisli composition given in [23].
Acknowledgements Nicola Gambino acknowledges gratefully the support of EPSRC, under Grant EP/M0
1729X/1, and of the US Air Force Research Laboratory, under Agreement Number FA8655-13-1-3038. We
are grateful to the anonymous referee for insightful comments, which led in particular to a simplification
of the material in Sect. 5.
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2828 M. Fiore et al.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-
tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
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