ALGEBRAIC WEAK FACTORIZATION SYSTEMS
IN DOUBLE CATEGORIES
by
PATRICK M. SCHULTZ
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2014
DISSERTATION APPROVAL PAGE
Student: Patrick M. Schultz
Title: Algebraic Weak Factorization Systems in Double Categories
This dissertation has been accepted and approved in partial fulfillment of
the requirements for the Doctor of Philosophy degree in the Department of
Mathematics by:
Dr. Daniel Dugger Chair
Dr. Boris Botvinnik Core Member
Dr. Dev Sinha Core Member
Dr. Arkady Vaintrob Core Member
Dr. Peter Lambert Institutional Representative
and
Kimberly Andrews Espy Vice President for Research & Innovation
Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2014
ii
© 2014 Patrick M. Schultz
iii
DISSERTATION ABSTRACT
Patrick M. Schultz
Doctor of Philosophy
Department of Mathematics
June 2014
Title: Algebraic Weak Factorization Systems in Double Categories
We present a generalized framework for the theory of algebraic weak
factorization systems, building on work by Richard Garner and Emily Riehl.
We define cyclic 2-fold double categories and bimonads (or bialgebras) and
lax/colax bimonad morphisms inside cyclic 2-fold double categories. After
constructing a cyclic 2-fold double category FF(D) of functorial factorization
systems in any sufficiently nice 2-category D, we show that bimonads and
lax/colax bimonad morphsims in FF(Cat) agree with previous definitions of
algebraic weak factorization systems and lax/colax morphisms. We provide a
proof of one of the core technical theorems from previous work on algebraic
weak factorization systems in our generalized framework. Finally, we show that
this framework can be further generalized to cyclic 2-fold double multicategories,
incorporating Quillen functors of several variables.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Patrick M. Schultz
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene
University of California, Santa Cruz
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2014, University of Oregon
Master of Science, Mathematics, 2009, University of California at Santa Cruz
Bachelor of Science, Computer Science, 2005, University of California at Santa
Cruz
AREAS OF SPECIAL INTEREST:
Category Theory
Categorical methods in Homotopy Theory
Categorical Logic and Type Theory
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, Department of Mathematics, University of
Oregon, Eugene, 2009–2014
Teaching Assistant, Department of Mathematics, University of California,
Santa Cruz, 2005–2009
Teaching Assistant, COSMOS, Santa Cruz, California, 2003–2009
v
ACKNOWLEDGEMENTS
I would like to thank my advisor, Dan Dugger, for his patience and support
in allowing me to follow my interests. I would also like to thank Emily Riehl and
Mike Shulman for helpful conversations and feedback on drafts of this thesis.
vi
To Lizzie, for putting up with me through far too many years of grad school.
vii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. WEAK FACTORIZATION SYSTEMS . . . . . . . . . . . . . . . . . . . . . 9
2.1. Arrow Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2. Functorial Factorizations . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3. Algebraic Weak Factorization Systems . . . . . . . . . . . . . . . 13
III. DOUBLE CATEGORIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1. Review of Double Categories . . . . . . . . . . . . . . . . . . . . . 21
3.2. Arrow Objects in a Double Category . . . . . . . . . . . . . . . . 25
3.3. Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4. Double Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
IV. 2-FOLD DOUBLE CATEGORIES . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1. 2-Fold Double Categories . . . . . . . . . . . . . . . . . . . . . . . 36
4.2. Monads in 2-Fold Double Categories . . . . . . . . . . . . . . . . 38
V. CYCLIC 2-FOLD DOUBLE CATEGORIES . . . . . . . . . . . . . . . . . . 42
viii
Chapter Page
VI. FUNCTORIAL FACTORIZATIONS . . . . . . . . . . . . . . . . . . . . . . 48
VII. ALGEBRAIC WEAK FACTORIZATION SYSTEMS . . . . . . . . . . . . 61
VIII. R-ALG AND L-COALG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
IX. COMPOSITION OF L-COALGEBRAS . . . . . . . . . . . . . . . . . . . . 71
X. A UNIVERSAL PROPERTY FOR THE PUSHOUT PRODUCT . . . . . 88
10.1. Review of Cyclic Double Multicategories . . . . . . . . . . . . . 89
10.2. The Universal Property . . . . . . . . . . . . . . . . . . . . . . . . . 92
10.3. Arrow Objects in Cyclic Double Multicategories . . . . . . . . . 97
XI. CYCLIC 2-FOLD DOUBLE MULTICATEGORIES . . . . . . . . . . . . . 101
11.1. Multimorphisms of Bimonads . . . . . . . . . . . . . . . . . . . . 104
11.2. Functorial Factorizations . . . . . . . . . . . . . . . . . . . . . . . . 106
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
ix
CHAPTER I
INTRODUCTION
The theory of model categories has a long history, and has proven to
be indispensable to several recent advances in mathematics, such as higher
category theory, so-called spectral algebraic geometry, even finding applications
in computer science and the foundations of mathematics with homotopy type
theory.
In the modern treatment, a model category is defined to consist of two weak
factorization systems on a category C (e.g. [MP12]). A weak factorization system
is a structure which consists of two classes of morphisms of C , call them L and
R, such that solutions to certain lifting problems involving one morphism from
each class always exist, plus an axiom that every morphism of C factors as a
morphism from L followed by a morphism from R. In the past 20 or so years,
most authors have added the requirement that this factorization can be chosen in
a natural/functorial way.
Taking this one step further, in [GT06] the category theorists Marco Grandis
and Walter Tholen proposed a strengthening of weak factorization systems which
they called natural weak factorization systems, today most often referred to as
algebraic weak factorization systems, or awfs for short. An awfs strengthens the
structure in a way which provides a canonical choice of solution to every lifting
problem, in such a way that these choices are coherent or natural in a precise
sense. The structure of an awfs consists of a monad and a comonad on the
category of arrows satisfying some axioms, and the categories of algebras and
1
coalgebras for these respectively provide an algebraic analogue of the right and
left classes of maps of the factorization system.
It at first seems as though this extra structure is too strict, and that examples
would be hard to find. But in [Gar07] and [Gar09], the category theorist
Richard Garner provided a modification of Quillen’s small object argument
which generates algebraic weak factorization systems, and which furthermore has
much nicer convergence properties than Quillen’s original construction, and often
generates a smaller and easier to understand factorization. Best of all, Garner’s
small object argument operates under almost identical assumptions as Quillen’s,
so that in practice any cofibrantly generated weak factorization system can be
strengthened to an algebraic one.
In her Ph.D. thesis, [Rie11] and [Rie13], Emily Riehl began the project of
developing a full-fledged theory of algebraic model structures, built out of two awfs
analogously to an ordinary model structure. Since then, she and her collaborators
have continued to develop and find applications of this theory, e.g. [CGR12],
[BR13], and [BMR13]. Of particular interest for us, she gives the first definition of
algebraic Quillen functors.
If we define a lax functor of weak factorization systems to be a functor
between categories each equipped with a wfs which takes morphisms in the
right class of the first to morphisms in the right class of the second, then a
right Quillen functor between model categories is simply a functor which is a
lax functor with respect to both weak factorization systems making up the model
structures. Likewise, a colax functor of wfs preserves the left classes, and a left
Quillen functor is colax with respect to both wfs. It is a basic fact from model
2
category theory that given an adjunction between weak factorization systems, the
left adjoint is colax if and only if the right adjoint is lax.
An algebraic version of Quillen functors should continue to have this
property, as the definition Riehl gives does, but making this precise requires
some pieces of classical category theory: the mates correspondence, and double
categories. The mates correspondence is a natural bijection between natural
transformations involving two pairs of adjoint functors which generalizes the
hom-set bijection of an adjunction. The naturality of the mates correspondence is
best formulated using double categories, and for this reason double categories
play a central role in this thesis. Double categories are a kind of two-
dimensional categorical structure, similar to a 2-category but with separate classes
of vertical and horizontal morphisms, and with square shaped 2-cells which
can be composed both vertically and horizontally. Double categories were first
defined by Ehresmann in the ‘60’s and then largely ignored, but have recently
enjoyed a resurgence of interest, see e.g. [Shu08], [DPP10], [FGK10].
In [Gar07] and [Gar09], Garner proves as a technical tool that algebraic
weak factorization systems can be seen as bialgebras in a category of functorial
factorizations, supporting the intuition that an awfs is given by a functorial
factorization equipped with (co)algebraic structure. The category of functorial
factorizations he constructs is not a symmetric or braided monoidal category,
but a so-called two-fold monoidal category, which is a generalization of braided
monoidal category having two compatible monoidal structures, and in which the
definition of bialgebra still makes sense.
We find this a very nice conceptual way of understanding algebraic weak
factorization systems, but it has the shortcoming of being unable to say anything
3
about functors between awfs on different categories. It is one of our primary
goals of this thesis to extend this awfs-as-bialgebras perspective to include the
(co)lax morphisms of awfs defined in [Rie11]. To do this, we have had to
find a common generalization of double categories, used to formalize the mates
correspondence and the duality relating lax and colax morphisms, and two-fold
monoidal categories, in which the notion of bialgebra makes sense. We call this
common generalization a two-fold double category.
We show that a kind of bialgebra can be defined in any two-fold double
category, which we call bimonads, and that the natural generalization of bialgebra
morphism bifurcates into lax and colax morphisms of bimonads. One main result
of this thesis is that there is a two-fold double category of functorial factorizations
(in any 2-category), and that bimonads and (co)lax morphisms of bimonads in this
two-fold double category correspond precisely to awfs and (co)lax morphisms of
awfs.
In the second part of her thesis, published as [Rie13], Riehl develops a
theory of monoidal algebraic model categories, ultimately based on an algebraic
strengthening of the notion of two-variable Quillen adjunction. Classically, a 2-
variable Quillen adjunction is a functor of two variables with both adjoints (one
in each variable), such that the induced pushout-product of two maps in the left
classes is again in the left class. The primary motivation for this definition is to be
able to define monoidal model categories, in which the tensor product is part of a
2-variable Quillen adjunction. To give an algebraic version of this definition, Riehl
had to extend the mates correspondence to multivariable adjunctions, which she
does with her coauthors in [CGR12]. The mates correspondence for multivariable
adjunctions is most easily understood in terms of cyclic double multicategories,
4
a kind of structure defined in [CGR12] which generalizes double categories to
allow for morphisms with multiple inputs, with a cyclic action which formalizes
the mates correspondence.
In order to incorporate multivariable morphisms into the bialgebraic view of
awfs, we have developed a common generalization of two-fold double categories
and cyclic double multicategories. Another main result of this thesis is that the
pushout product—central to the definition of Quillen bifunctor, and hence to
monoidal model categories, simplicial model categories, etc.—satisfies a universal
property in the framework of cyclic double multicategories. The author is
particularly pleased with this result, as the need for the pushout product in
the axioms of monoidal model categories and simplicial model categories had
always seemed slightly mysterious and ad hoc. This universal property provides
a conceptual explanation: the pushout product defines the universal way of lifting
a multivariable adjunction to arrow categories. This also allows us to define
multivariable morphisms of bimonads in a cyclic two-fold double multicategory,
generalizing the multivariable morphisms of awfs given in [Rie13].
A primary motivation for this work was to develop the theory of awfs at
a high level of generality. In particular, all of the constructions and theorems
of this thesis work just as well in any 2-category satisfying minor completeness
conditions as they do in the 2-category of categories. For example, in [BMR13]
the authors make use of enriched algebraic weak factorization systems, in which
stronger enriched lifting properties are required. (Note that this is different than
enriched model categories in the sense of, e.g., simplicial model categories.) This
thesis provides a framework in which the core theory of awfs can be developed in
great generality, saving the effort of reproving results for enriched awfs and any
5
other variations of awfs yet to be considered, and it makes a start of proving the
most important results in this greater generality.
Overview
In chapter II, we review the definitions of algebraic weak factorization system
and morphisms of algebraic weak factorization systems, trying to lead up to the
(abstract) definitions in a natural way. Then in chapter III we review the definition
of double category, as well recording some constructions which will be needed
later on. Of these, the definitions of arrow objects in a double category and of
fully-faithful lax double functors are (to the best of our knowledge) original.
In chapter IV we introduce a definition of two-fold double categories.
Generalizing bialgebras in a two-fold monoidal category, we define bimonads
and (co)lax morphisms of bimonads in a two-fold double category.
In [CGR12], the authors show that the mates correspondence can be
conveniently expressed as the existence of a cyclic action on a double category
of adjunctions. In chapter V, we show how to generalize the cyclic action as
in [CGR12] to the two-fold double categories defined in chapter IV, defining what
we call a cyclic two-fold double category. We show that a cyclic action interacts
well with bimonads in a two-fold double category, extending to a cyclic action on
the category of bimonads. This cyclic action is the abstract form of the fact that an
algebraic Quillen adjunction can be specified either by a lax stucture on the right
adjoint or a colax structure on the left adjoint.
In chapter VI, we begin the core work of this thesis, constructing a cyclic two-
fold double category of functorial factorizations in an arbitrary double category
which has all arrow objects. Then in chapter VII we show that given any 2-
6
category D with arrow objects, bimonads in the cyclic two-fold double category
of adjunctions in D are precisely algebraic weak factorization systems in D.
Garner proves in [Gar07] and [Gar09] that instead of specifying both the
monad and comonad halves of the awfs structure, it is equivalent to define the
comonad, plus a functorial composition on the category of coalgebras. This
generalizes the classical fact that the left (and right) class of maps is closed
under composition, but more importantly provides a convenient technical tool
for constructing algebraic weak factorization systems. Similarly, in [Rie11] Riehl
proves that an equivalent definition of colax morphism of awfs is a functor which
lifts to the categories of coalgebras for the comonads, and which also preserves
the composition of coalgebras. She uses both of these theorems repeatedly
throughout the paper.
In chapter VIII we lay the groundwork towards proving a generalization
of these theorems in the framework of cyclic two-fold double categories by
reviewing the standard universal property for Eilenberg-Mac Lane categories for
monads and comonads, first given in [Str72], and showing the particular form this
universal property takes in the special case of comonads arising from an awfs.
Then in chapter IX, we give the (surprisingly difficult and technical) proofs that
the results mentioned above about composition of coalgebras continue to hold at
our higher level of generality.
In chapter X we show that a natural generalization of the universal property
for arrow objects in a double category (defined in section 3.2.) to cyclic double
multicategories in fact uniquely characterizes the pushout/pullback product. This
allows us to abstract away the the pushout/pullback product, isolating precisely
the properties which are necessary to make the theory of multivariable Quillen
7
adjunctions work, and providing a conceptual explanation for the appearance
of pushout/pullback products in the definitions of monoidal model category,
simplicial model category, etc.
In chapter XI we define a common generalization of the cyclic two-fold
double categories of chapter V and the cyclic double multicategories of [CGR12],
which we call a cyclic two-fold double multicategory. We give a definition
of multivariable morphisms of bimonads, showing that this definition is stable
under the cyclic action. We then generalize our construction of a cyclic two-
fold double category of functorial factorizations from chapter VI to a cyclic two-
fold double multicategory, and show that multivariable morphisms of bimonads
recover the definition of multivariable adjunction of awfs given in [Rie13]. The fact
that these multivariable morphisms are stable under the cyclic action generalizes
the classical fact that if a functor which is part of a 2-variable adjunction preserves
the left classes, in that the pushout product of morphisms in the left classes is in
the left class, then each of the two adjoints satisfy similar properties involving a
mix of left and right classes.
8
CHAPTER II
WEAK FACTORIZATION SYSTEMS
We will begin by briefly reviewing the notions of functorial factorization,
weak factorization system, and algebraic weak factorization system.
2.1. Arrow Categories
Let C be a category. Its arrow category C 2 is the category whose objects are
arrows in C and whose morphisms are commutative squares. The arrow category
comes with two functors dom, cod∶C 2 → C , along with a natural transformation
κ∶dom⇒ cod. The component of κ at an object f of C 2 is simply f ∶dom f → cod f .
Moreover, C 2 satisfies a universal property: there is an equivalence of categories
Fun(2, Fun(X ,C )) ≃ Fun(X ,C 2) (2.1)
given by composition with κ. Here, 2 is the ordinal, i.e. the category with two
objects and a single non-identity arrow. In other words, C 2 is the cotensor of C
with the category 2 in the 2-category Cat.
We will make this universal property more explicit in the next lemma,
separating out the 1-dimensional and the 2-dimensional parts of (2.1):
Lemma 2.1. Let C be a category.
i) For any category X , pair of functors F, G∶X → C , and natural transformation
α∶ F ⇒ G, there is a unique functor αˆ∶X → C 2 such that dom αˆ = F, cod αˆ = G,
9
and
X C 2 C
αˆ
dom
cod
⇓κ = X C .F
G
⇓α (2.2)
ii) For any functors F, F′, G, G′∶X → C and a commutative square of natural
transformations
F F′
G G′,
γ
α β
φ
there is a unique natural transformation η∶ αˆ → βˆ such that dom η = γ and cod η =
φ, hence
X C
F
G
G′
⇓α
⇓φ = X C 2 C
αˆ
βˆ
dom
cod
⇓η ⇓κ = X C .
F
F′
G′
⇓γ
⇓β (2.3)
Definition 2.2. Let D be any 2-category. For any object A in D , the arrow object of
A, if it exists, is an object A2 satisfying the universal property (2.1). If every object
has an arrow object, i.e. if D has cotensors by 2, we will say D has arrow objects.
In practice, we will work with arrow objects in a 2-category using the two
parts of lemma 2.1.
Finally, we will record here a simple proposition which will be needed later.
Proposition 2.3. Any arrow object A2 has an internal category structure
A3 A2 Ac
dom
cod
i
10
where A3 is the pullback of the span
A2 A A2cod dom
Proof. Using the universal property, we define i and c by the equations dom i = id,
cod i = id, κi = idid, dom c = dom p1, cod c = cod p2, and κc = κp2 ○ κp1, where p1
and p2 are the projections of the pullback.
2.2. Functorial Factorizations
Definition 2.4. A functorial factorization on a category C consists of a functor E
and two natural transformations η and e which factor κ, as in
C2 C
dom
cod
⇓κ = C2 C.
dom
E
cod
⇓η
⇓e
This determines for any arrow f in C a factorization f = e f ○ η f . The
factorization is natural, meaning that for any morphism (u, v)∶ f ⇒ g in C 2 (i.e.
commutative square in C ), the two squares in
⋅ ⋅
⋅ ⋅
⋅ ⋅
u
η f ηg
E(u,v)
e f eg
v
commute.
A functorial factorization also determines two functors L, R∶C2 → C2 such
that dom L = dom, cod R = cod, cod L = dom R = E, κL = η, and κR = e, by the
11
universal property of C2. The components of the factorization of f can then also
be referred to as L f and R f , now thought of as objects in C 2. There are also two
canonical natural transformations, η⃗∶ id ⇒ R and e⃗∶ L ⇒ id, determined by the
commuting squares
dom E
cod cod
η
κ e
id
and
dom dom
E cod
id
η κ
e
respectively. These make L and R into (co)pointed endofunctors of C 2.
An algebra for the pointed endofunctor R is an object f in C 2 equipped with a
morphism t⃗∶R f ⇒ f , such that t⃗ ○ η⃗ f = id f . Similarly, a coalgebra for the copointed
endofunctor L is an f equipped with a morphism s⃗∶ f ⇒ L f , such that e⃗ f ○ s⃗ = id f .
Lemma 2.5. Let f ∶X → Y be a morphism in C . An R-algebra structure on f ∈ C 2 is
precisely a choice of lift t in the square
X X
E f Y.
L f f
R f
t (2.4)
Dually, an L-coalgebra structure on f is precisely a choice of lift s in the square
X E f
Y Y.
L f
f R fs (2.5)
12
Moreover, a morphism (u, v)∶ f ⇒ g in C 2 is a morphism of R-algebras if it
commutes with the lifts t and t′, that is, if in the diagram
⋅ ⋅
⋅ ⋅
⋅ ⋅
u
L f Lg
E(u,v)
R f Rgt
v
t′
we have t′v = E(u, v)t.
2.3. Algebraic Weak Factorization Systems
To simplify the discussion of weak factorization systems, we will start by
introducing a notation. For any two morphisms l and r in C , write l ⧄ r to mean
that for every commutative square
⋅ ⋅
⋅ ⋅
u
l r
v
w (2.6)
there exists a lift w. In this case, we will say that l has the left lifting property with
respect to r, and that r has the right lifting property with respect to l. Similarly, for
two classes of morphisms L and R, we will say L⧄R if l ⧄ r for every l ∈ L and
r ∈R. Finally, we will write L⧄ for the class of morphisms having the right lifting
property with respect to every morphism of L, and ⧄R for the class of morphisms
having the left lifting property with respect to every morphism of R.
13
Definition 2.6. A functorial weak factorization system on a category C consists of a
functorial factorization on C and two classes L and R of morphisms in C , such
that
– for every morphism f in C , L f ∈ L and R f ∈R,
– L⧄ =R and ⧄R = L.
It a simple and standard proof that the lifting property condition can be
replaced by two simpler conditions:
Lemma 2.7. A functorial weak factorization system can equivalently be defined to be a
functorial factorization on C and two classes L and R of morphisms in C , such that
– for every morphism f in C , L f ∈ L and R f ∈R,
– L⧄R,
– L and R are both closed under retracts.
In fact, the functorial factorization by itself already determines the two classes
of morphisms, with L the class of morphisms admitting an L-coalgebra structure,
and R the class of morphisms admitting an R-algebra structure. The lifting
properties also follow directly from the functorial factorization, as the next lemma
shows.
Lemma 2.8. For any L-coalgebra (l, s) and any R-algebra (r, t), there is a canonical
choice of lift in the square (2.6). Any morphism of R-algebras (u1, v1)∶ (r, t)⇒ (r′, t′) and
any morphism of L-coalgebras (u2, v2)∶ (l′, s′) ⇒ (l, s) preserves these canonical choices
of lifts.
14
Proof. The construction is shown in the diagram
⋅ ⋅
⋅ ⋅
⋅ ⋅
u
Ll Lr
E(u,v)
Rl
t
Rrs
v
(2.7)
Commutativity of (2.6) follows immediately from (2.4) and (2.5).
That a morphism of R-algebras preserves these canonical lifts can be seen in
the diagram ⋅ ⋅ ⋅
⋅ ⋅ ⋅
⋅ ⋅ ⋅
u
Ll
u′
Lr Lr′
E(u,v)
Rl
E(u′,v′)t
Rr
t′
Rr′s
v v′
noting that u′tE(u, v)s = t′E(u′, v′)E(u, v)s = t′E(u′u, v′v)s.
This, together with the classical fact that the class of objects admitting a
(co)algebra structure for a (co)pointed endofunctor is closed under retracts, gives
a third equivalent definition of a functorial weak factorization system.
Lemma 2.9. A functorial weak factorization system can equivalently be defined to be a
functorial factorization on C such that
– for every morphism f in C , L f admits an L-coalgebra structure, and R f admits an
R-algebra structure.
An R-algebra structure on R f consists of a morphism µ⃗ f ∶R2 f → R f in C 2 such
that µ⃗ f ○ η⃗R f = idR f , while an L-coalgebra structure on L f consists of a morphism
δ⃗ f ∶ L f → L2 f such that e⃗L f ○ δ⃗ f = idL f . We might hope that it is possible to choose
15
these structures for all f in a natural way, such that they form the components of
natural transformations µ⃗∶R2 ⇒ R and δ⃗∶ L⇒ L2.
If we want these choices of lifts to be fully coherent, we should also ask
that for any R-algebra ( f , t), the lift constructed as in (2.7) for the square (2.4) is
equal to t, and similarly for L-coalgebras and (2.5). Lastly, we should ask that
the components µ⃗ f and δ⃗ f are (co)algebra morphisms. These conditions, plus one
more ensuring that there is an unambiguous notion of a morphism with both
L-algebra and R-coalgebra structures, lead to the definition of an algebraic weak
factorization system, first given in [GT06] (there called natural weak factorization
systems), and further refined in [Gar07] and [Gar09].
Definition 2.10. An algebraic weak factorization system on a category C consists of a
functorial factorization (L, e⃗, R, η⃗) together with natural transformations µ⃗∶R2 ⇒ R
and δ⃗∶ L⇒ L2, such that
– R = (R, η⃗, µ⃗) is a monad and L = (L, e⃗, δ⃗) a comonad on C 2, and
– the natural transformation ∆ = (δ,µ)∶ LR ⇒ RL determined by the equation
eL ○ δ = µ ○ ηR (= idE) as in lemma 2.1 is a distributive law, which in this case
reduces to the single condition δ ○ µ = µL ○ E∆ ○ δR.
Just as we saw that a functorial factorization already determines the left and
right classes of morphisms, there is a condition we can place on a functor F
between categories equipped with functorial factorizations which implies that F
preserves the left class.
Definition 2.11. Let C andD be categories equipped with functorial factorizations
(E1, η1, e1) and (E2, η2, e2) respectively. A colax morphism of functorial factorizations
16
is a pair (F,φ) where F∶C → D is a functor and φ is a natural transformation
C2 C
D2 D
E1
Fˆ F
E2
⇓φ
such that
C2 C
D2 D
E1
Fˆ F
E2
cod
⇓φ
⇓e2
= C2 C
D2 D
E1
cod
Fˆ F
cod
⇓e1
⇓id
and
C2 C
D2 D
dom
E1
Fˆ F
E2
⇓η1
⇓φ = C2 C
D2 D.
dom
Fˆ F
dom
E2
⇓id
⇓η2
In components, given a morphism f ∶X → Y in C, these two equations simply say
that the following diagram commutes:
FX E2(F f )
F(E1 f ) FY.
L2(F f )
F(L1 f ) R2(F f )φ f
F(R1 f )
Here, Fˆ∶C2 → D2 is the obvious lift of F to the arrow categories, sending an
object ( f ∶X → Y) in C2 to the object (F f ∶ FX → FY) in D2.
17
Proposition 2.12. Let (F,φ) be a colax morphism of functorial factorizations as above.
Then F preserves the left class of morphisms, i.e. if f ∶X → Y in C has an L1-coalgebra
structure, then F f has an L2-coalgebra structure.
Proof. Let f ∶X → Y be a morphism in the left class in C, with L1-coalgebra
structure given by the lift s in
X E f
Y Y.
L f
f R fs
Then F f has an L2-coalgebra structure given by φ f F(s), as shown by the
commutativity of the diagram
FX E2(F f )
F(E1 f )
FY FY
L2(F f )
F f
F(L1 f )
R2(F f )
φ f
F(R1 f )Fs
If F has a right adjoint G, then the natural transformation φ determines a
natural transformation φ′∶E1Gˆ → GˆE2 which ensures that G preserves the right
class of morphisms analogously to proposition 2.12, and making G what is called
a lax morphism of functorial factorizations. The relationship between φ and φ′ is
what is known as the mates correspondence (see e.g. [CGR12]), and we say that
φ′ is the mate of φ, and vice versa.
18
It turns out that the mates correspondence is best understood in the context
of double categories, which we review in the next section. The theory of mates
underlies the definition of adjunctions of awfs, and is the reason why double
categories are an essential part of our general framework.
19
CHAPTER III
DOUBLE CATEGORIES
Recall in definition 2.11, a colax morphism of functorial factorizations is a
pair (F,φ), where φ is a natural transformation
C2 C
D2 D
E1
Fˆ F
E2
⇓φ (3.1)
In [Rie11] it is proven that if F ⊣ G is an adjunction, then specifying a natural
transformation φ making F a colax morphism uniquely determines a natural
transformation θ making G a lax morphism. The transformation θ is called the
mate of φ, and is found by composing φ with the unit and counit of Fˆ and F
respectively.
This mates correspondence (see example 3.3) defines a bijection between
natural transformations of the form (3.1) and natural transformations of the form
C2 C
D2 D.
E1
Gˆ
E2
G⇓θ
The collection of square 2-cells, where the vertical 1-cells are required to be the
left adjoint of an adjunction, and the horizontal 1-cells are allowed to be arbitrary
functors, can be organized into a structure called a double category. Similarly, there
is a double category where the vertical 1-cells are required to be right adjoints
(with some subtlety regarding the direction vertical 1-cells and 2-cells point), and
20
the naturality of the mates correspondence can be expressed by saying these two
double categories are isomorphic.
Double categories are a fundamental structure for this thesis, primarily due to
the importance of the mates correspondence to the algebraic analogue of Quillen
functors: lax and colax morphisms of awfs.
In section 3.1., we begin by giving an overview of double categories. Then in
section 3.2. we give a generalization of definition 2.2, defining arrow objects in a
double category by means of a universal property. This is needed to be able to
define functorial factorizations and (co)lax morphisms of functorial factorizations
in general double categories, which we do in chapter VI.
We will ultimately want to define an algebraic weak factorization system to
be a sort of bialgebraic object in a (two-fold) double category. To prepare the way
for this definition, in section 3.3. we give (one possible version of) the definition
of monads in a double category.
3.1. Review of Double Categories
We first give the most concise definition of a double category, which we will
then break down into more concrete terms.
Definition 3.1. A (strict) double category is an internal category object in the (large)
category of categories.
So a double category D consists of a category D0 and a category D1, along
with functors s, t∶D1 → D0, i∶D0 → D1, and ⊗∶D1 ×D0 D1 → D1 satisfying the
usual axioms of a category. We will call the objects of D0 the 0-cells (or just
objects) of D, and the morphisms of D0 the vertical 1-cells. Thus D0 forms the
21
so-called vertical category of D. We will call the objects of D1 the horizontal 1-cells
of D, and the morphisms of D1 are the 2-cells.
A morphism φ∶X → Y in D1, where s(X) = C, t(X) = C′, s(Y) = D, t(Y) = D′,
s(φ) = f , and t(φ) = g will be drawn as
C C′
D D′
X
f g
Y
⇓φ (3.2)
where the tick-mark on the horizontal 1-cells serves as a further reminder that
the horizontal 1-cells are of a different nature than the vertical 1-cells. The
composition in D0 provides a vertical composition of vertical 1-cells and 2-
cells, while the composition functor ⊗∶D1 ×D0 D1 → D1 provides a horizontal
composition of horizontal 1-cells and 2-cells.
For any object C in D0, i(C) is the unit horizontal 1-cell
C C
IC
and acts as an identity with respect to the horizontal composition.
A 2-cell θ for which sθ = tθ = id will be called globular. We will sometimes
draw globular 2-cells as
C C′,X
Y
⇓θ
to save space and help readability of diagrams.
Example 3.2. For any 2-category D, there is an associated double category Sq(D)
of squares in D, in which the vertical and horizontal 1-cells are both just 1-cells in
22
D, and 2-cells
C C′
D D′
j
f g
k
⇓φ
are simply 2-cells φ∶ gj⇒ k f in D.
Example 3.3. Given any 2-category D, there is a double category LAdj(D). The
horizontal 1-cells are just 1-cells in D, while the vertical 1-cells are fully specified
adjunctions f ⊣ g (meaning the unit and counit have been chosen) pointing in the
direction of the left adjoint. 2-cells
C C′
D D′
j
( f⊣g) ( f ′⊣g′)
k
⇓φ
are natural transformations involving the left adjoints, φ∶ f ′ j⇒ k f .
Similarly there is a double category RAdj(D) where the vertical 1-cells still
point in the direction of the left adjoint, but the 2-cells are natural transformations
involving the right adjoints, φ∶ jg⇒ g′k.
The mates correspondence (see e.g. [CGR12]) gives an isomorphism of double
categories LAdj(D) ≅RAdj(D), which sends a 2-cell φ in LAdj(D) to the 2-cell
C C′ C′
D D D′
j
f f ′g
k
g′⇓e ⇓φ ⇓η
given by composing with the unit/counit of the adjunctions.
23
Example 3.4. Given any category M , there is a pseudo double category Span(M )
of spans in M . The vertical category of Span(M ) is just M , while horizontal
1-cells
C DX
are given by spans
C X D
j k
in M , and 2-cells
C D
C′ D′
X
f g
Y
⇓θ
are given by commutative diagrams
C X D
C′ Y D.′
f
j k
θ g
j′ k′
The horizontal composition of spans is given by pullback. It is because this
horizontal composition is only determined up to isomorphism that this example
is not a strict double category.
Definition 3.5. For any double category D, there is an associated 2-category
Hor(D), called the horizontal 2-category of D. The objects and 1-cells of Hor(D)
are the objects and horizontal 1-cells of D, while 2-cells φ∶X ⇒ Y in Hor(D) are
24
the globular 2-cells in D, i.e. those of the form
C D
C D
X
Y
⇓φ
Notice that Hor(Sq(D)) is isomorphic to D.
Definition 3.6. Given a double category D, define double categories Dvop and
Dhop, obtained by reversing the direction of the vertical and horizontal 1-cells
respectively, and changing the orientation of the 2-cells as appropriate. For
example, a 2-cell (3.2) in Dvop is a 2-cell
D D′
C C′
Y
f g
X
⇓φ
in D.
In terms of definition 3.1, Dvop is the double category obtained by replacing
the categories D0 and D1 with their opposites, while Dvop is the obtained by
swapping the horizontal source and target functors s and t.
3.2. Arrow Objects in a Double Category
In the following we will need an extension of the universal property (2.1) to
double categories. Fortunately, this is quite straightforward.
25
Let D be a double category. Given an object C of D, the arrow object C2, if it
exists, is an object together with a diagram
C2 C,
dom
cod
⇓κ
such that any 2-cell
A C
d1
d0
⇓α
uniquely factors through κ, as
A C2 C.αˆ
dom
cod
⇓κ
Given a vertical 1-cell F∶C → D in D, the lift to arrow objects Fˆ∶C2 → D2, if it
exists, is a vertical 1-cell Fˆ∶C2 → D2 together with 2-cells
C2 C
D2 D
dom
Fˆ F
dom
⇓γ1
C2 C
D2 D
cod
Fˆ F
cod
⇓γ0
satisfying
C2 C
D2 D
dom
Fˆ F
dom
cod
⇓γ1
⇓κ
= C2 C
D2 D,
dom
cod
Fˆ F
cod
⇓κ
⇓γ0
26
such that for any 2-cells
A C
d1
d0
⇓α B D
d′1
d′0
⇓α′
and
A C
B D
d1
G F
d′1
⇓λ1
A C
B D
d0
G F
d′0
⇓λ0
satisfying
A C
B D
d1
G F
d′1
d′0
⇓λ1
⇓α′
= A C
B D
d1
d0
G F
d′0
⇓α
⇓λ0
there is a unique 2-cell
A C2
B D2
αˆ
G Fˆ
αˆ′
⇓θ
such that the horizontal composition of θ with γ0 and γ1 is respectively equal to
λ0 and λ1.
Remark 3.7. Note that in most naturally occurring examples, the 2-cells γ0 and γ1
will be isomorphisms (or even identities). This just says that given any g∶X → Y
in C2, the lift Fˆ(g) will be some Fˆ(g)∶ FX → FY. We are not aware of any examples
where the γi are not isomorphisms, though when we generalize this universal
property to the multivariable setting, they will necessarily not be isomorphisms
(rather they will be projections out of a pullback).
27
Definition 3.8. A double category D has arrow objects if for every object C of D
there is an object C2 and 2-cell κ, and for every vertical 1-cell F there is a vertical
1-cell Fˆ and 2-cells γ0 and γ1, satisfying the universal properties given above.
The intuition that this is a generalization of lemma 2.1 is supported by the
following two propositions, the (easy) proofs of which are left to the reader.
Proposition 3.9. If the double category D has arrow objects, then so does Hor(D).
Proposition 3.10. If the 2-category D has arrow objects, then so does Sq(D).
Proof. A simple check. The 2-cells γ0 and γ1 will always be identities.
3.3. Monads
We will define a monad in a double category D to be a tuple (C, T, η,µ), in
which C is an object, T∶C → C is a horizontal 1-cell, and η and µ are 2-cells
C C
C C
idC
T
⇓η
C C C
C C
T T
T
⇓µ
satisfying the usual unit and associativity conditions. In other words, a monad in
D is simply a monad in the 2-category Hor(D). The non-identity vertical 1-cells
come into play in the morphisms of monads.
Given two monads (C, T, η,µ) and (D, S, η′,µ′), a monad morphism from
(C, T) to (D, S) consists of a pair ( f ,φ), where f is a vertical 1-cell C → D and φ
is a 2-cell
C C
D D
T
f f
S
⇓φ
28
which commutes with the unit and multiplication 2-cells in the sense of the two
equations
C C
C C
D D
idC
T
f f
S
⇓η
⇓φ
=
C C
D D
D D
idC
f f
idD
S
⇓id f
⇓η′
(3.3)
and
C C C
C C
D D
T T
T
f f
S
⇓µ
⇓φ
=
C C C
D D D
D D
T
f
T
f f
S S
S
⇓φ ⇓φ
⇓µ′
(3.4)
Definition 3.11. Given any double category D, we will write Mon(D) for the
category of monads inD, consisting of monads and monad morphisms as defined
above. The category Comon(D) of comonads in D is defined to be the category
Mon(Dop) of monads in Dop.
Example 3.12. The category Mon(Span(Set)) is precisely the category of small
categories. It is an easy and enlightening exercise to work this out for oneself.
Proposition 3.13. The categories of (co)monads and (co)lax morphisms in a 2-category
D can be given in terms of (co)monads in the double category of squares as follows:
Moncolax(D) = Mon(Sq(D))
Comoncolax(D) = Comon(Sq(D))
Monlax(D) = Mon(Sq(Dop))op
Comonlax(D) = Comon(Sq(Dop))op
29
where by Dop we mean the 2-category obtained by reversing the direction of all 1-cells (but
not 2-cells).
Proof. Immediate from the definitions. Those readers unfamiliar with (co)lax
morphisms of monads can take this as the definition.
3.4. Double Functors
The natural notion of functor between double categories is a straightforward
generalization of lax functors between monoidal categories. Recall that we are
using the symbol ⊗ to denote horizontal composition.
Definition 3.14. Let D and E be double categories. A lax double functor F∶D → E
consists of:
– Functors F0∶D0 → E0 and F1∶D1 → E1 such that sF1 = F0s and tF1 = F0t
– Natural transformations with globular components
F⊗∶ F1X⊗ F1Y → F1(X⊗Y) and FI ∶ IF0C → F1(IC),
which satisfy the usual coherence axioms for a lax monoidal functor.
A lax double functor F for which the components of FI and F⊗ are identities
will be called strict. For the intermediate notion where the components of FI and
F⊗ are (vertical) isomorphisms, we will simply refer to F as a double functor.
Proposition 3.15. A lax double functor F∶D → E induces a functor F∶Mon(D) →
Mon(E).
30
Proof. This works just like the case for monoidal categories. For instance, if X is a
monad in D, FX has the multiplication
C C C
C C
C C
FX FX
F(X⊗X)
FX
⇓F⊗
⇓Fµ
The fact that F takes monad morphisms to monad morphisms can easily be
checked using the naturality of FI and F⊗.
We will have need for a condition on a lax double functor which implies a
sort of converse to proposition 3.15. This condition is a slight strengthening of the
notion of fully-faithful functor which makes sense for lax functors.
Definition 3.16. A lax double functor F∶D → E is fully-faithful on 2-cells if, for
any fixed X1, X2, Y, f , g, the induced function from 2-cells in D of the shape
C0 C1 C2
D0 D1
f
X1 X2
g
Y
to 2-cells in E of the shape
FC0 FC1 FC2
FD0 FD1,
F f
FX1 FX2
Fg
FY
31
which takes a 2-cell θ to F(θ) ○ F⊗, is a bijection, and if similarly the induced
function from 2-cells of the shape
C C
D0 D1
IC
f g
Y
to
FC FC
FD0 FD1
IFC
F f Fg
FY
taking a 2-cell φ to F(φ) ○ FI is a bijection.
Remark 3.17. Definition 3.16 implies the function on 2-cells θ with a single
horizontal 1-cell in the domain is also bijective. We leave the details of the (simple)
proof to the reader.
Proposition 3.18. Let F∶D → E be a fully-faithful lax double functor. Given any
horizontal 1-cell X in D, a monoid structure on FX in E lifts uniquely to a monoid
structure on X such that the induced functor F∶Mon(D)→Mon(E) takes X to FX.
Similarly, a vertical 1-cell f ∶X → Y for which F f is a monoid morphism must also
be a monoid morphism.
In other words, the diagram of categories is a pullback:
Mon(D) Mon(E)
D E.
F
U U
F
Proof. Simply use the surjectivity of the fully-faithful functor to lift the unit and
multiplication 2-cells from FX to X, then use the injectivity to show that the
unitary and associativity equations on FX imply those on X.
32
CHAPTER IV
2-FOLD DOUBLE CATEGORIES
It is well known that the notion of bialgebra or bimonoid—an object with
both monoid and comonoid structures which are compatible in a certain sense—
makes sense not only in a symmetric monoidal category, but also in more general
braided monoidal categories. A bimonoid in a braided monoidal category C can
be defined to be a monoid in the category of comonoids in C , or equivalently as
a comonoid in the category of monoids in C . The braiding is necessary to ensure
that the monoidal structure in C lifts to a product in Mon(C ) and Comon(C ).
Less well known is the fact that the definition of bimonoid works just as well
in a more general context still: the so-called 2-fold monoidal categories. A 2-fold
monoidal category has two different monoidal structures, call them (⊗, I) and
(⊙,⊥), which are themselves compatible in certain sense. This compatibility can
be stated in a way analogous to the definition of bimonoid given in the previous
paragraph: a (strict) 2-fold monoidal category is a monoid object in the category
StrMonCatl of strict monoidal categories and lax functors, or equivalently a
monoid object in the category StrMonCatc of strict monoidal categories and colax
functors. Notice that monoid objects in the category of strict monoidal categories
and strong monoidal functors (in which the components of the lax structure are
isomorphisms) are precisely (strict) braided monoidal categories.
More concretely, the compatibility between the monoidal structures amounts
to the existence of maps
m∶⊥ ⊗ ⊥→⊥, c∶ I → I ⊙ I, j∶ I →⊥,
33
making (⊥, j, m) a ⊗-monoid and (I, j, c) a ⊙-comonoid, and a natural family of
maps
zA,B,C,D∶ (A⊙ B)⊗ (C⊙D)→ (A⊗C)⊙ (B⊗D)
satisfying some coherence axioms.
Example 4.1.
– Any braided monoidal category can be made into a 2-fold monoidal
category in which the two monoidal structures coincide.
– Any monoidal category (C ,⊗, I) with finite products has a 2-fold monoidal
structure with (⊙,⊥) given by the product and terminal object. Dually, a
monoidal category (C ,⊙,⊥) with finite coproducts has a 2-fold monoidal
structure with (⊗, I) given by the coproduct and initial object.
Because the ⊙-monoidal structure is lax monoidal with respect to the ⊗-
monoidal structure, it lifts to the category Mon⊗(C ) of ⊗-monoids in C . Dually,
the ⊗-monoidal structure lifts to the category Comon⊙(C ) of ⊙-comonoids in
C . Thus, we could define the category Bimon(C ) of bimonoids in C to be
either Comon⊙(Mon⊗(C )) or Mon⊗(Comon⊙(C )), and it turns out that these
are canonically isomorphic. In either case, a bimonoid is an object A with a ⊗-
monoid structure (η,µ) and a ⊙-comonoid structure (e, δ), such that the following
34
four diagrams commute:
I A
I ⊙ I A⊙ A,
η
c δ
η⊙η
A⊗ A A
⊥ ⊗ ⊥ ⊥,
µ
e⊗e e
m
A
I ⊥,
eη
j
A⊗ A A
(A⊙ A)⊗ (A⊙ A) (A⊗ A)⊙ (A⊗ A) A⊙ A.
µ
δ⊗δ δ
zA,A,A,A µ⊙µ
(4.1)
In [Gar07] and [Gar09], Garner proves that given any category C , there is a
2-fold monoidal category of functorial factorizations on C . Given two functorial
factorizations (E1, L1, R1) and (E2, L2, R2), the factorization E1 ⊗ E2 factors an
arrow f ∶X → Y as
X E2R1 f Y
L2R1 f ○L1 f R2R1 f (4.2)
while the factorization E1 ⊙ E2 factors f as
X E2L1 f Y.
L2L1 f R1 f ○R2L1 f (4.3)
Garner shows that bimonoids in this 2-fold monoidal category are equivalent
to algebraic weak factorization systems on C . In other words, a bialgebra
structure on a functorial factorization (E, η, e) is precisely a choice of monad and
comonad making E an awfs. However, as this structure only contains functorial
factorizations on a fixed category C , it can say nothing about morphisms between
factorization systems on different categories.
In order to address this shortcoming, we will generalize this 2-fold monoidal
category definition to double categories, where there are two different horizontal
35
compositions which are compatible in a way analogous to the two monoidal
structures in a 2-fold monoidal category. In chapter VI we will construct a 2-
fold double category of functorial factorizations, generalizing Garner’s 2-fold
monoidal category, and in chapter VII we will see that bimonads and bimonad
morphisms in this 2-fold double category are exactly awfs and colax morphisms
of awfs.
4.1. 2-Fold Double Categories
We will start with a concise formal definition, and then expand on the
definition more concretely.
Definition 4.2. A 2-fold double category D with vertical category Vert(D) = D0 is a
2-fold monoid object in the 2-category Cat/D0 of categories over D0.
Breaking this down, we have a categoryD1, a functor p∶D1 → D0, two functors⊗,⊙∶D1 ×D0 D1 → D1 commuting with p, and two functors I,⊥∶D0 → D1 which are
sections of p, such that ⊗, ⊙, I, and ⊥ satisfy all the axioms of a 2-fold monoidal
category. In particular, each fiber of p has a 2-fold monoidal structure.
A monoid object in Cat/D0 is equivalently a double category where the
source and target functors s, t∶D1 → D0 are equal, and with the vertical category
D0. Conversely, any double category D in which all horizontal 1-cells have equal
domain and codomain, and all 2-cells have equal vertical 1-cells as domain and
codomain, is equivalently a monoid object in Cat/D0. We will alternate between
these two descriptions as convenient.
Using this shift of perspective, D has two underlying double categories,
both with vertical category D0 and with source and target functors both equal to
p∶D1 → D0. The double category D⊗ has the rest of the double category structure
36
given by the functors I and ⊗, while the double category D⊙ uses the functors ⊥
and ⊙.
Using this double category interpretation, we will find it convenient to think
of a 2-fold double category as a double category with two different but interacting
horizontal compositions. Notice that from this perspective, all horizontal 1-cells
are endomorphisms.
Remark 4.3. It may seem somewhat ad hoc to force a 2-fold monoid object in a
slice of C into a double category mold, with the odd looking restriction to having
only endomorphisms in the horizontal direction. We will make essential use
of double functors from D⊙ and D⊗ to genuine double categories (without the
endomorphism restriction), and it is mostly for this reason that we have found
the double categorical perspective useful, if perhaps only psychologically.
We did give some thought to how one might define a 2-fold double category
with non-endomorphism horizontal 1-cells and 2-cells, and while it seems like
there might be a workable definition, it would require a very large increase in
complexity. As we are mostly interested in the monads and comonads in a 2-fold
double category, which are structures on endomorphism horizontal 1-cells, this
restriction was of no concern to this work.
Now let us explicitly look at the 2-fold monoidal structure from the double
categorical perspective. For any object C there are 2-cells
C C
C C
⊥C⊗⊥C
⊥C
⇓m
C C
C C
IC
IC⊙IC
⇓c
C C
C C
IC
⊥C
⇓j (4.4)
37
and for any four horizontal morphisms W, X, Y, Z∶C C there is a 2-cell
C C
C C.
(W⊙X)⊗(Y⊙Z)
(W⊗Y)⊙(X⊗Z)
⇓z (4.5)
These are natural in the sense that, for any vertical morphism f ∶C → D we have
an equality
C C
C C
D D
⊥C⊗⊥C
⊥C
f f
⊥D
⇓m
⇓⊥ f
=
C C
D D
D D
⊥C⊗⊥C
f f
⊥D⊗⊥D
⊥D
⇓⊥ f⊗⊥ f
⇓m
and similarly for c and j, and for any four 2-cells θ1, . . . , θ4 of the appropriate
form, we have an equality
C C
C C
D D
(W⊙X)⊗(Y⊙Z)
(W⊗Y)⊙(X⊗Z)
f f
(W′⊗Y′)⊙(X′⊗Z′)
⇓z
⇓(θ1⊗θ3)⊙(θ2⊗θ4)
=
C C
C C
D D
(W⊙X)⊗(Y⊙Z)
f f
(W′⊙X′)⊗(Y′⊙Z′)
(W′⊗Y′)⊙(X′⊗Z′)
⇓(θ1⊙θ2)⊗(θ3⊙θ4)
⇓z
4.2. Monads in 2-Fold Double Categories
Definition 4.4. A monad in a 2-fold double category D is a monad in D⊗; a
comonad in D is a comonad in D⊙. Furthermore, we define the categories
Mon(D) = Mon(D⊗) and Comon(D) = Comon(D⊙).
38
So a monad X and a comonad Y in D are given by 2-cells
C C
C C
IC
X
⇓η
C C
C C
X⊗X
X
⇓µ
C C
C C
Y
⊥C
⇓e
C C
C C.
Y
Y⊙Y
⇓δ
The categories Mon(D) and Comon(D) come naturally equipped with
functors to D0, defined on objects and morphisms simply by applying p to
the underlying 1-cells and 2-cells respectively. It turns out that the interaction
between the ⊗ and ⊙ compositions in the 2-fold double category structure is
precisely what is needed to lift ⊙ to Mon(D) and to lift ⊗ to Comon(D). In
this way, we can define double categories Mon(D) and Comon(D), both having
D0 as vertical category.
These lifted compositions are defined as follows: Given two monads
(C, X, η,µ) and (C, Y, η′,µ′) in D, the horizontal composition
C C C
(X,η,µ) (Y,η′,µ′)
is the monoid with underlying horizontal 1-cell X⊙Y and unit and multiplication
2-cells
C C
C C
C C
IC
IC⊙IC
X⊙Y
⇓c
⇓η⊙η′
C C
C C
C C.
(X⊙Y)⊗(X⊙Y)
(X⊗X)⊙(Y⊗Y)
X⊙Y
⇓z
⇓µ⊙µ′
The unit for this composition is IC, given the trivial monad structure with η = µ =
idIC .
39
Similarly, the horizontal composition of two 2-cells in Mon(D) is given by
the ⊙-product of the underlying 2-cells in D. The fact that this commutes with
the unit and multiplication defined above follows from the naturality of c and z.
In this same way, we can define the horizontal composition of two 1-cells
(X, e, δ) and (Y, e′, δ′) in Comon(D) to be a comonad with underlying horizontal
1-cell X⊗Y, with horizontal unit ⊥with the trivial comonad structure. This allows
us to define (ordinary) categories Mon(Comon(D)) and Comon(Mon(D)).
Furthermore, these two categories are equivalent, leading to the next definition.
Definition 4.5. A bimonad in a 2-fold double category D is a monad in the double
category Comon(D), or equivalently a comonad in Mon(D). We can define a
category of bimonads in D as
Bimon(D) ∶= Mon(Comon(D)) ≃ Comon(Mon(D))
Concretely, a bimonad in D is a tuple (X, η,µ, e, δ) where X is a horizontal
1-cell, (X, η,µ) is a monad and (X, e, δ) is a comonad as above, such that four
40
equations hold:
C C
C C
C C
IC
X
X⊙X
⇓η
⇓δ
=
C C
C C
C C
IC
IC⊙IC
X⊙X
⇓c
⇓η⊙η
C C
C C
C C
X⊗X
X
⊥C
⇓µ
⇓e
=
C C
C C
C C
X⊗X
⊥C⊗⊥C
⊥C
⇓e⊗e
⇓m
C C
C C
C C
IC
X
⊥C
⇓η
⇓e
= C C
C C
IC
⊥C
⇓j
C C
C C
C C
C C
X⊗X
(X⊙X)⊗(X⊙X)
(X⊗X)⊙(X⊗X)
X⊙X
⇓δ⊗δ
⇓z
⇓µ⊙µ
=
C C
C C
C C
X⊗X
X
X⊙X
⇓µ
⇓δ
(4.6)
A bimonoid morphism is simply a 2-cell which is simultaneously a monoid
morphism and a comonoid morphism.
41
CHAPTER V
CYCLIC 2-FOLD DOUBLE CATEGORIES
Recall the notion of a cyclic double category from [CGR12]. A cyclic double
category D is a double category with an extra involutive operation. On objects
and horizontal 1-cells X∶C C , this operation is written
C●1 C●2X●
and respects horizontal identities and composition. The involution takes any
vertical 1-cell f ∶C → D to some σ f ∶D● → C●, and any 2-cell
C1 C2
D1 D2
X
f g
Y
⇓θ to
D●1 D●2
C●1 C●2
Y●
σ f σg
X●
⇓σθ
respecting vertical identities and composition.
The next example is the fundamental example of a cyclic double category.
Example 5.1. Recall from example 3.3 the two double categories LAdj(D) and
RAdj(D). If the 2-category D has an involution (−)●∶Dco → D, such as Cat with
(−)op, then the double category LAdj(D) has a natural cyclic action: on vertical
1-cells σ( f ⊣ g) = (g● ⊣ f ●), and if φ is a 2-cell
C C′
D D′
j
( f⊣g) ( f ′⊣g′)
k
⇓φ
42
with mate θ then σφ = θ●:
D● D′●
C● C′●
k●
(g●⊣ f ●) (g′●⊣ f ′●)
j●
⇓θ●
This cyclic action encodes the naturality of the mates correspondence using only
a single double category, and is a convenient alternative to the isomorphism
LAdj(D) ≅ RAdj(D). This simplification will be even more important when
we need the multivariable mates correspondence in chapters X and XI.
For a clear summary of the mates correspondence and the cyclic action on
LAdj, see [CGR12] Section 1.
Proposition 5.2. Let D be a cyclic double category with arrow objects. For any object C,
(C●)2 = (C2)●, as witnessed by
(C2)● C●cod
●
dom●
⇓σκ
For any vertical 1-cell F, the lift to arrow objects of F● is (Fˆ)●, as witnessed by the 2-cells
(D2)● D●
(C2)● C●
cod●
(Fˆ)● F●
cod●
⇓σγ0
(D2)● D●
(C2)● C●
dom●
(Fˆ)● F●
dom●
⇓σγ1
Proof. It is a very simple matter to verify the universal properties of section 3.2.
43
We will generalize this to a cyclic action on a 2-fold double category. Suppose
that D is a 2-fold double category. A cyclic action, written as above, must satisfy
the following:
– For every object C,
IC● = (⊥C)● and ⊥C●= (IC)●.
– For every composable pair of horizontal 1-cells X, Y∶C C ,
(X⊗Y)● = X● ⊙Y● and (X⊙Y)● = X● ⊗Y●
– For every vertical 1-cell f ∶C → D, there are equalities
D● D●
C● C●
ID●
σ f σ f
IC●
⇓Iσ f = D● D●
C● C●
(⊥D)●
σ f σ f
(⊥C)●
⇓σ⊥ f
D● D●
C● C●
⊥D●
σ f σ f
⊥C●
⇓⊥σ f = D● D●
C● C●
(ID)●
σ f σ f
(IC)●
⇓σI f
– For every horizontally composable pair of 2-cells
C C C
D D D
X
f
Y
f f
X′ Y′
⇓θ ⇓φ
44
there are equalities
D● D●
C● C●
(X′⊗Y′)●
σ f σ f
(X⊗Y)●
⇓σ(θ⊗φ) = D● D●
C● C●
X′●⊙Y′●
σ f σ f
X●⊙Y●
⇓σ(θ)⊙σ(φ)
D● D●
C● C●
(X′⊙Y′)●
σ f σ f
(X⊙Y)●
⇓σ(θ⊙φ) = D● D●
C● C●
X′●⊗Y′●
σ f σ f
X●⊗Y●
⇓σ(θ)⊗σ(φ)
One nice consequence of this definition is that a cyclic action on a 2-fold
double categoryD induces a cyclic action on the category of bimonads Bimon(D).
Proposition 5.3. Suppose D is a cyclic 2-fold double category. Then the category
Bimon(D) of bimonads inD carries a natural cyclic action (contravariant isomorphism).
Proof. The involution (−)● gives an isomorphism of double categories D⊗ ≅Dop⊙ .
Therefore it also induces an isomorphism
Mon(D) =Mon(D⊗) ≅Mon(Dop⊙ ) ≅ Comon(D⊙)op = Comon(D)op
as well as an isomorphism
Bimon(D) = Comon(Mon(D)) ≅ Comon(Comon(D)op)
≅ Mon(Comon(D))op = Bimon(D)op.
45
In more concrete terms, the involution takes a bimonad (X, η,µ, e, δ) to
(X, η,µ, e, δ)● = (X●, e●, δ●, η●, δ●),
swapping the monad and comonad structures. This is again a bimonad, as the
top two equations of (4.6) are interchanged under the involution, while the bottom
two equations are self-dual.
The action of the involution on bimonad morphisms can be broken down as
in the following lemma.
Lemma 5.4. Let (X, η,µ, e, δ) and (Y, η′,µ′, e′, δ′) be bimonads in a cyclic 2-fold double
category D, and let φ be a 2-cell in D
C C
D D.
X
f f
Y
⇓φ
Then ( f ,φ) is a monad morphism X → Y if and only if (σ f ,φ●) is a comonad morphism
Y● → X●. Dually, φ is a comonad morphism X → Y if and only if φ● is a monad morphism
Y● → X●.
Proof. Simply notice that the involution takes equations (3.3) and (3.4) to the
equations defining a comonad morphism in D.
This immediately implies a useful characterization of bimonoid morphisms.
Corollary 5.5. Given bimonads (X, η,µ, e, δ) and (Y, η′,µ′, e′, δ′) in a cyclic 2-fold
double category D, a bimonad morphism X → Y consists of a pair ( f ,φ) as above, such
that:
46
– Either ( f ,φ) is a monad morphism or (σ f ,φ●) is a comonad morphism, and
– Either ( f ,φ) is a comonad morphism or (σ f ,φ●) is a monad morphism.
47
CHAPTER VI
FUNCTORIAL FACTORIZATIONS
With the preliminary work done of defining cyclic 2-fold double categories,
and bimonads and bimonad morphisms in cyclic 2-fold double categories, our
next goal is to define a cyclic 2-fold double category in which bimonads are
precisely algebraic weak factorization systems, and bimonad morphisms are colax
morphisms of awfs. In fact, we can do this in much greater generality, beginning
with any cyclic double category which has arrow objects. The case to keep in mind
for intuition, in which a bimonad corresponds to a regular awfs on a category, is
the cyclic double category LAdj from 5.1.
Let D be a cyclic double category, and assume it has arrow objects in the
sense of section 3.2.. In this chapter, we will define a 2-fold double category
FF(D) of functorial factorizations in D, as follows:
– The objects and vertical 1-cells are the same as in D.
– Horizontal 1-cells C C in FF(D) are tuples (E, η, e), where E∶C2 → C
is a horizontal 1-cell in D, and
C2 C
dom
E
⇓η C2 CE
cod
⇓e
are 2-cells in D such that
C2 C
dom
E
cod
⇓η
⇓e = C2 C.
dom
cod
⇓κ
48
By the universal property of C2, such a horizontal 1-cell in FF(D) also
determines horizontal 1-cells L, R∶C2 → C2 in D such that dom ○L = dom,
cod ○R = cod, cod ○L = dom ○R = E, κ ○ L = η, and κ ○ R = e, and 2-cells
C2 C2.
L
id
⇓e⃗ C2 C2.
id
R
⇓η⃗
such that dom ○e⃗ = iddom, cod ○e⃗ = e, dom ○η⃗ = η, and cod ○η⃗ = idcod.
– The horizontal composition (E1, η1, e1)⊗ (E2, η2, e2) of two horizontal 1-cells
C C C
(E1,η1,e1) (E2,η2,e2)
in FF(D) is a horizontal 1-cell (E1⊗2, η1⊗2, e1⊗2), where
E1⊗2 = C2 C2 CR1 E2
η1⊗2 = C2 C2 Cid
R1
dom
E2
⇓η⃗1 ⇓η2
e1⊗2 = C2 C2 CR1 E2
cod
⇓e2
which also determines that R1⊗2 = R2 ○ R1.
– The horizontal unit IC for ⊗ is (dom, id, κ).
49
– The second horizontal composition (E1, η1, e1)⊙ (E2, η2, e2) is a horizontal
1-cell (E1⊙2, η1⊙2, e1⊙2), where
E1⊙2 = C2 C2 CL1 E2
η1⊙2 = C2 C2 CL1 dom
E2
⇓η2
e1⊙2 = C2 C2 CL1
id
E2
cod
⇓e⃗1 ⇓e2
which also determines that L1⊙2 = L2 ○ L1.
– The horizontal unit ⊥C for ⊙ is (cod, κ, id).
– 2-cells
C C
D D
(E1,η1,e1)
F F
(E2,η2,e2)
⇓θ
in FF(D) are given by 2-cells
C2 C
D2 D
E1
Fˆ F
E2
⇓θ
50
in D such that
C2 C
D2 D
E1
Fˆ F
E2
cod
⇓θ
⇓e2
= C2 C
D2 D
E1
cod
Fˆ F
cod
⇓e1
⇓γ0 (6.1)
and
C2 C
D2 D
dom
Fˆ F
dom
E2
⇓γ1
⇓η2
= C2 C
D2 D
dom
E1
Fˆ F
E2
⇓η1
⇓θ (6.2)
This also determines unique 2-cells
C2 C2
D2 D2
R1
Fˆ Fˆ
R2
⇓θR and
C2 C2
D2 D2
L1
Fˆ Fˆ
L2
⇓θL
such that composing horizontally with γ0 or γ1 gives γ0, γ1, or θ as
appropriate. For instance:
C2 C2 C
D2 D2 D
R1
Fˆ
dom
Fˆ F
R2 dom
⇓θR ⇓γ1 = C
2 C
D2 D
E1
Fˆ F
E2
⇓θ
51
– Given a pair of composable 2-cells in FF(D) as in
C C C
D D D
(E1,η1,e1)
F F
(E2,η2,e2)
F
(E′1,η′1,e′1) (E′2,η′2,e′2)
⇓θ1 ⇓θ2
the composite θ1 ⊗ θ2 is given by
C2 C2 C
D2 D2 D
R1
Fˆ
E2
Fˆ F
R′1 E′2
⇓θR1 ⇓θ2
while the composite θ1 ⊙ θ2 is given by
C2 C2 C
D2 D2 D
L1
Fˆ
E2
Fˆ F
L′1 E′2
⇓θL1 ⇓θ2
52
It is an easy exercise to check that these definitions satisfy (6.1) and (6.2). To
illustrate, we will demonstrate that θ1 ⊗ θ2 satisfies (6.1):
C2 C
D2 D
E1⊗2
Fˆ F
E1′⊗2′
cod
⇓θ1⊗θ2
⇓e1′⊗2′
= C2 C2 C
D2 D2 D
R1
Fˆ
E2
Fˆ F
R′1
E′2
cod
⇓θR1 ⇓θ2
⇓e′2
= C2 C2 C
D2 D2 D
R1
Fˆ
E2
cod
Fˆ F
R′1 cod
⇓θR1
⇓e2
⇓γ0
= C2 C
D2 D
E1⊗2
cod
Fˆ F
cod
⇓e1⊗2
⇓γ0
Example 6.1. Functorial factorizations in the double category D = Sq(Cat) of
squares in the 2-category of categories are precisely functorial factorizations as
defined in section 2.2.. The two horizontal compositions are the factorizations (4.2)
and (4.3).
It is straightforward to check that ⊗ and ⊙ are each associative and unital.
It takes more work to provide the compatibility between ⊗ and ⊙, which is the
content of the proof of the next proposition.
Proposition 6.2. FF(D) has the structure of a 2-fold double category.
Proof. The primary structure of FF(D) was given in the first part of this section.
What is left is to provide the coherence data (4.4) and (4.5).
53
First, note that IC is initial in the sense that, given any vertical morphism
F∶C → D and any functorial factorization (E, η, e) on D, there is a unique 2-cell
C C
D D
IC
F F
(E,η,e)
⇓
given by
C2 C
D2 D.
dom
Fˆ F
dom
E
⇓γ1
⇓η
Similarly, ⊥C is terminal. Thus there is only one possible way to define the 2-cells
m, c, and j, and naturality and all other coherence equations follows immediately
from this uniqueness.
We still need to construct the 2-cell z, which will take some work. We begin
by defining 2-cells
C C
C C
E1⊙E2
E1
⇓pE1,E2 and
C C
C C.
E1
E1⊗E2
⇓iE1,E2
for any pair of functorial factorizations. The 2-cell p is given by the underlying
2-cell in D
C2 C2 C
L1
E2
cod
⇓e2
54
and i is given by
C2 C2 C.
R1
dom
E2
⇓η2
To illustrate the verification that these give well-defined 2-cells in FF(D), we will
show that i satisfies (6.1) (keep in mind that when F is an identity, γ0 and γ1 are
also identities):
C2 C2 C
L1
dom
E2
cod
⇓η2⇓e2 = C2 C2 CL1
dom
cod
⇓κ
= C2 C.dom
E1
⇓η1
Moreover, it is straightforward to check that i and p are natural families of
2-cells. Specifically, for any pair of 2-cells θ1 and θ2
C C
C C
D D
E1⊙E2
E1
F F
E′1
⇓pE1,E2
⇓θ1
=
C C
D D
D D
E1⊙E2
F F
E′1⊙E′2
E′1
⇓θ1⊙θ2
⇓pE′1,E′2
C C
C C
D D
E1
E1⊗E2
F F
E′1⊗E′2
⇓iE1,E2
⇓θ1⊗θ2
=
C C
D D
D D
E1
F F
E′1
E′1⊗E′2
⇓θ1
⇓iE′1,E′2
55
As with any 2-cell in FF(D), p and i induce 2-cells in D
C2 C2
R1⊙2
R1
⇓pR and C2 C2.
L1
L1⊗2
⇓iL
such that
C2 C2 C
R1⊙2
R1
dom⇓pR = C2 C2 CL1 E2
cod
⇓e2 (6.3)
C2 C2 C
L1
L1⊗2
cod⇓iL = C2 C2 CR1 dom
E2
⇓η2 (6.4)
Now suppose given three functorial factorizations E1, E2, E3 on an object C.
We define a 2-cell in D
C2
C2 C2
C2
L3R1⊙2
L1⊗3 R2
⇓w
such that
C2
C2 C2 C
C2
L3R1⊙2
L1⊗3
dom
R2
⇓w = C2 C2 CL1
L1⊗3
E2⇓iL (6.5)
C2
C2 C2 C
C2
L3R1⊙2
L1⊗3
cod
R2
⇓w = C2 C2 C.R1⊙2
R1
E3⇓pR (6.6)
56
Using the universal property for C2, it suffices to check that
C2 C2 C
L1
L1⊗3
E2
cod
⇓iL
⇓e2 = C2 C2 CR1⊙2
R1
dom
E3⇓pR
⇓η3
and a quick check using equations (6.3) and (6.4) shows that both are equal to
C2
C2 C
C2
E2
cod
L1
R1
dom
E3
⇓e2
⇓η3
where the inner diamond is the equality cod L1 = dom R1 = E1.
We also check that w is natural with respect to 2-cells in FF(D) in the
following sense: given three 2-cells θ1, θ2, and θ3, there is an equality
C2
C2 C2
C2
D2 D2
D2
L3R1⊙2
L1⊗3
Fˆ Fˆ
R2
Fˆ
L′1⊗3 R′2
⇓w
⇓(θ1⊗θ3)L ⇓θR2 =
C2
C2 C2
D2
D2 D2.
D2
L3
Fˆ
R1⊙2
Fˆ Fˆ
L′3R′1⊙2
L′1⊗3 R′2
⇓(θ1⊙θ2)R ⇓θL3
⇓w
57
To verify this equation, it suffices to check equality upon right composition with
γ0 and γ1. We will illustrate the γ1 case, making use of the naturality of i:
C2
C2 C2 C
C2
D2 D2 D
D2
L3R1⊙2
L1⊗3
Fˆ Fˆ
dom
F
R2
Fˆ
L′1⊗3
dom
R′2
⇓w
⇓(θ1⊗θ3)L ⇓θR2 ⇓γ1 =
C2 C2 C
D2 D2 D
L1
L1⊗3
Fˆ
E2
Fˆ F
L1′⊗3′ E′2
⇓iL
⇓(θ1⊗θ3)L ⇓θ2
= C
2 C2 C
D2 D2 D
L1
Fˆ
E2
Fˆ F
L′1
L′1⊗3
E′2
⇓θL1 ⇓θ2
⇓iL
=
C2
C2 C2 C
D2
D2 D2 D.
D2
L3
Fˆ
R1⊙2
Fˆ Fˆ
dom
F
L′3R′1⊙2
L′1⊗3
dom
R′2
⇓(θ1⊙θ2)R ⇓θL3
⇓w
⇓γ1
Finally, given four functorial factorizations E1, E2, E3, E4 on an object C, we
define the 2-cell
C C
C C
(1⊙2)⊗(3⊙4)
(1⊗3)⊙(2⊗4)
⇓z1,2,3,4
in FF(D), where (1 ⊙ 2) is shorthand for (E1, η1, e1) ⊙ (E2, η2, e2), to have the
underlying 2-cell in D
C2
C2 C2 C.
C2
L3R1⊙2
L1⊗3
E4
R2
⇓w
The naturality of z follows immediately from that of w, but we still need to check
that this satisfies equations (6.1) and (6.2). We will leave the details to the reader,
58
but note that (6.2) comes down to the verification of the equality
C2
C2 C2 C
C2
L3
id
R1⊙2
L1⊗3
dom
E4
R2
⇓η⃗1⊙2
⇓w ⇓η4 = C2 C2 C2 C,L1⊗3 id
R2
dom
E4
⇓η⃗2 ⇓η4
which follows from equation (6.5) and the fact that dom ○iL = iddom.
Lemma 6.3. There is a strict double functor R∶FF(D)⊗ →D whose behavior on 2-cells
is
C C
D D
(E1,η1,e1)
F F
(E2,η2,e2)
⇓θ ↦ C2 C2
D2 D2
R1
Fˆ Fˆ
R2
⇓θR
and a double functor L∶FF(D)⊙ →D whose behavior on 2-cells is
C C
D D
(E1,η1,e1)
F F
(E2,η2,e2)
⇓θ ↦ C2 C2
D2 D2
L1
Fˆ Fˆ
L2
⇓θL
Corollary 6.4. R and L respectively induce functors
Mon(FF(D))→Mon(D) and Comon(FF(D))→ Comon(D).
Up to this point, we have demonstrated that given any double category
D having arrow objects, there is a 2-fold double category FF(D) of functorial
factorizations in D. The last thing we want to say about this construction is that a
cyclic action onD lifts to one on FF(D), and hence also to one on Bimon(FF(D)).
59
The cyclic action on objects and vertical morphisms is given directly by that
on D. Given a horizontal 1-cell (E, η, e) on an object C, we define the 1-cell
(E, η, e)● on C● to be (E●, e●, η●). This also implies that the cyclic action swaps L
and R for any given functorial factorization.
A quick look at the definitions of the two horizontal compositions is now
enough to see that for any two functorial factorizations E1 and E2, we have
(E1 ⊗ E2)● = E●1 ⊙ E●2 and (E1 ⊙ E2)● = E●1 ⊗ E●2
Similarly, the cyclic action on 2-cells in FF(D) is given by the cyclic action
in D on the underlying 2-cell. This gives a valid 2-cell in FF(D) since the cyclic
action simply swaps the equations (6.1) and (6.2).
Definition 6.5. Let D be a 2-category with with an involution (−)●∶Dco → D
and with arrow objects, and recall the cyclic double category LAdj(D) from
example 5.1. We define the cyclic category
AWFS(D) = Bimon(FF(LAdj(D)))
In the next chapter, we will see that AWFS(Cat) is the category whose objects
are categories C together with an awfs, and whose morphisms are adjunctions
F ⊣ G, where F∶C → D is equipped with the structure of a colax morphism of
awfs. The cyclic action takes this morphism to Gop ⊣ Fop, and a colax morphism
structure on Gop∶Dop → C op, which is the same as a lax morphism structure on
G∶D → C . It is in this way that the cyclic action encodes the equivalence between
a colax structure on the left adjoint and a lax structure on the right adjoint.
60
CHAPTER VII
ALGEBRAIC WEAK FACTORIZATION SYSTEMS
For this chapter, let D = Sq(D) be the double category of squares in a 2-
category D. We will show that bimonoids in FF(D) are precisely algebraic weak
factorization systems, and more generally that the morphisms in Bimon(FF(D))
are given by (co)lax morphisms of algebraic weak factorization systems.
Suppose that E = (E, η, e) is a functorial factorization on a category C , and
consider a monoid structure on E. As IC is initial, the unit of the monoid is
forced, and is simply η. The multiplication is given by a natural transformation
µ∶ER⇒ E satisfying equations (6.1) and (6.2), which now take the form e ○ µ = eR
and µ ○ (η ⋅ η⃗) = η.
The unit axioms for the monoid give the equations µ○Eη⃗ = idE = µ○ηR, which
together imply the equation µ ○ (η ⋅ η⃗) = η above. And finally, the associativity
axiom gives the equation µ ○ Eµ⃗ = µ ○ µR, where we write µ⃗ = µR∶R2 → R for the
natural transformation induced by the 2-cell µ.
Proposition 7.1. A monoid structure on an object (E, η, e) in FF(D) is given by a
natural transformation µ∶ER⇒ E, satisfying equations
e ○ µ = eR µ ○ Eη⃗ = idE = µ ○ ηR µ ○ Eµ⃗ = µ ○ µR. (7.1)
This determines a monad R = (R, η⃗, µ⃗), such that dom µ⃗ = µ and cod µ⃗ = idcod.
61
Similarly, a comonoid structure on (E, η, e) is given by a natural transformation
δ∶E⇒ EL, satisfying equations
δ ○ η = ηL Ee⃗ ○ δ = idE = eL ○ δ Eδ⃗ ○ δ = δL ○ δ, (7.2)
which determines a comonad L = (L, e⃗, δ⃗), such that dom δ⃗ = iddom and cod δ⃗ = δ.
Hence a functorial factorization which has both a monoid structure and a
comonoid structure in FF(D) is precisely an algebraic weak factorization system,
missing only the second bullet of definition 2.10: the distributive law condition.
This is not surprising, as it is the only condition requiring a compatibility between
the monad and comonad structures. We will see that a bialgebra in FF(D) adds
precisely this compatibility.
Proposition 7.2. A bimonoid structure on a horizontal morphism (E, η, e)∶C → C
in FF(D) is precisely an algebraic weak factorization system on C with underlying
functorial factorization system (E, η, e).
Proof. We have already shown how the monoid an comonoid structures give rise
to the monad and comonad of the awfs. All that remains is to show that the
equations (4.6) amount to just the distributive law, i.e. the equation
C2
C2 C2 C
C2
L
E
R
L
E
R
E
⇓∆
⇓δ
⇓µ =
C2
C2 C.
C2
ER
E
L E
⇓µ
⇓δ (7.3)
62
First of all, notice that the first three equations of (4.6) follow trivially from
the initiality of IC and the terminality of ⊥C in FF(D), hence they do not impose
any further conditions.
The fourth equation here takes the form
C2 C2 C
C2 C2 C2 C
C2 C2 C2 C
C2 C2 C
R E
RE⊙E
L E
LE⊗E
R E
L E
⇓δR ⇓δ
⇓w ⇓idE
⇓µL ⇓µ
=
C2 C2 C
C2 C
C2 C2 C,
R E
E
L E
⇓µ
⇓δ
and so to prove (7.3), it suffices to show that
C2
C2 C2
C2
L
R
R E⊙E
LE⊗E
L
R
⇓δR
⇓w
⇓µL
= C2C2 C2.
C2
LR
L R
⇓∆
63
We can check this using the universal property of C2 by composing with dom and
cod. First, use (6.5) and (6.6) to check that
C2
C2 C2 C
C2
L
R
R E⊙E
LE⊗E
L
dom
R
⇓δR
⇓w
⇓µL
= C2 C2 C
E
L
LE⊗E
L
E
⇓δ
⇓iL
⇓µL
C2
C2 C2 C
C2
L
R
R E⊙E
LE⊗E
L
cod
R
⇓δR
⇓w
⇓µL
= C2 C2 C.
R
RE⊙E
R
E
E
⇓δR
⇓pR ⇓µ
Then use the definitions of i and p to check that µ ○ i = µ ○ ηR = idE and that
p ○ δ = eL ○ δ = idE, so that the first row above just equals δ, and the second row
equals µ. Since ∆ also (by definition) satisfies dom∆ = δ and cod∆ = µ, we are
done.
The appropriate notion of morphism between awfs, analogous to left and
right Quillen functors and Quillen adjunctions, is (to our knowledge) first given
in [Rie11].
Definition 7.3. Suppose that (E1, η1,µ1, e1, δ1) and (E2, η2,µ2, e2, δ2) are awfs on C
and D respectively.
– A lax morphism of awfs (G, ρ)∶E1 → E2 consists of a functor G∶C → D and
a natural transformation ρ∶E2Gˆ ⇒ GE1, such that (1, ρ)∶ L2Gˆ ⇒ GL1 is a lax
morphism of comonads and (ρ, 1)∶R2Gˆ⇒ GR1 is a lax morphism of monads.
64
– A colax morphism of awfs (F,λ)∶E1 → E2 consists of a functor F∶C → D and
a natural transformation λ∶ FE1 ⇒ E2Fˆ, such that (1,λ)∶ FL1 ⇒ L2Fˆ is a
colax morphism of comonads and (λ, 1)∶ FR1 ⇒ R2Fˆ is a colax morphism
of monads.
Notice that a lax morphism of awfs induces a lift of the functor Gˆ to a functor
R1-Alg → R2-Alg. In that sense, G “preserves the right class,” so is analogous to
a right Quillen functor. Similarly, a colax morphism of awfs induces a lift of Fˆ to
L1-Coalg→ L2-Coalg, so is analagous to a left Quillen functor.
Proposition 7.4. Morphisms in Bimon(FF(D)) are precisely the colax morphisms of
awfs.
Proof. As above, let (E1, η1,µ1, e1, δ1) and (E2, η2,µ2, e2, δ2) be awfs on C and D
respectively. A morphism of bimonoids is given by a 2-cell
C C
D D
(E1,η1,e1)
F F
(E2,η2,e2)
⇓λ
which commutes with the monoid and comonoid structures. It is straightforward
to check that this implies the natural transformations
C2 C2
D2 D2
L1
Fˆ Fˆ
L2
⇓λL
C2 C2
D2 D2
R1
Fˆ Fˆ
R2
⇓λR
are colax morphisms of comonads and monads respectively.
65
Now takeD to be LAdj(D) instead of Sq(D). All of the above works without
change, as the only difference is that the vertical 1-cells are now left adjoints
equipped with the unit and counit of the adjunction. By proposition 5.3, there
is a cyclic action on AWFS(D) = Bimon(FF(D)) induced by the cyclic action on
FF(D). This action is given on awfs by
(E, η,µ, e, δ)● = (E●, e●, δ●, η●,µ●)
swapping the monad and comonad structures. If F ⊣ G is an adjunction in D and
λ is a 2-cell in FF(D) as above, then σλ is a 2-cell
Dop Dop
Cop Cop
(E2,η2,e2)●
Gop Gop
(E1,η1,e1)●
⇓(σλ)●
given by a 2-cell in D
D2 D
C2 C.
E2
G Gˆ
E1
⇑σλ
If (F,λ) is a colax morphism of awfs, it is not hard to show that (G, (σλ)●) is a lax
morphism of awfs. In this way, the cyclic action allows us to capture both types
of morphism of awfs in the same structure.
66
CHAPTER VIII
R-ALG AND L-COALG
For this chapter, we will continue to let D = Sq(D) be the double category of
squares in a 2-category D with arrow objects.
A weak factorization system on a category C is defined by two classes of
morphisms, L and R. In an algebraic weak factorization system, these classes
of morphisms are replaced by categories L-Coalg and R-Alg equipped with
functors to C2. In this chapter, we will discuss the universal property satisfied
by these categories, allowing us to define analogous objects in other 2-categories,
and record several technical lemmas which we will need in chapter IX. We will
focus on comonads, but there are dual results for monads which we leave to the
reader.
Recall from [Str72] the following proposition.
Proposition 8.1. Let C be a category, and L = (L, e, δ) be a comonad on C. The category
of coalgebras L-Coalg has a universal property as follows:
– There is a forgetful functor U∶L-Coalg→ C together with a natural transformation
α∶U ⇒ LU, satisfying eU ○ α = idU and δU ○ α = Lα ○ α.
– (U,α) is universal among such pairs satisfying such equations. Given another such
pair (F, β), where F∶X → C, there exists a unique functor Fˆ∶X → L-Coalg such
that UFˆ = F and αFˆ = β.
Any colax morphism of comonads (F,φ)∶ (C, L1, e1, δ1) → (D, L2, e2, δ2) induces a
functor F˜∶L1-Coalg→ L2-Coalg such that U2F˜ = FU1 and α2F˜ = φU1 ○ Fα1.
67
A natural transformation
X L-Coalg
Fˆ1
Fˆ2
⇓θˆ
is uniquely determined by the functors F1 = UFˆ1 and F2 = UFˆ2 and natural
transformations β1 = αFˆ1 and β2 = αFˆ2, and the natural transformation θ = Uθˆ∶ F1 ⇒ F2,
satisfying Lθ ○ β1 = β2 ○ θ.
For the rest of this chapter, assume that D has EM-objects for comonads, i.e. for
every comonad L in D there is an object L-Coalg satisfying the universal property
above.
It is not too hard to use this universal property to construct the free/forgetful
adjunction:
Proposition 8.2. For any comonad L on an object C in D, the 1-cell U∶L-Coalg → C
has a right adjoint Lˆ with ULˆ = L and αLˆ = δ. The counit of this adjunction is simply the
counit of L, e∶ULˆ⇒ idC, while the unit is a 2-cell αˆ∶ idL-Coalg ⇒ LˆU satisfying Uαˆ = α.
Proof. By proposition 8.1, to prove the existence of the 1-cell Lˆ, it suffices to verify
the equations eL ○ δ = idL and δL ○ δ = Lδ ○ δ, which are simply two of the comonad
axioms.
Using the 2-dimensional part of proposition 8.1, the existence of the 2-cell αˆ
follows from the equation Lα ○ α = δU ○ α, which is the remaining comonad axiom.
We leave the verification of the triangle identities for the adjunction to the
reader.
68
As our interest is in (co)monads in FF(D), which induce (co)monads on
arrow objects, it will be useful to record the universal property that results from
the interaction of the EM-object and arrow object universal properties.
Consider a comonad in FF(D) on an object C, i.e. a functorial factorization
with half of the awfs structure. We can combine the universal properties of EM-
objects and arrow objects into a universal property for L-Coalg, where now L is
the comonad in D arising from the comonad in FF(D).
Lemma 8.3. Let (E, η, e, δ) be a comonad in FF(D) on an object C. There is a 2-cell
C2
L-Coalg C
C2
codU
U E
⇓α
satisfying equations
L-Coalg C2 C
C2
U
U
dom
cod
E
⇓α
⇓κ = L-Coalg C2 CU dom
E
⇓η (8.1)
C2
L-Coalg C2 C
codU
U
E
cod
⇓α
⇓e = X C2 CU cod (8.2)
C2
L-Coalg C2 C
C2
codU
U
E
L E
⇓α
⇓δ =
C2
L-Coalg C2 C.
C2
codU
U
U
E
L
⇓α
⇓α⃗ (8.3)
where α⃗ is the unique 2-cell such that dom α⃗ = iddom U and cod α⃗ = α, the existence of
which is implied by equation (8.1).
69
Given any object X, together with a morphism F∶X → C2 and a 2-cell β∶ cod F⇒ EF
satisfying equations
1. β ○ κF = ηF
2. eF ○ β = idcod F
3. δF ○ β = Eβ⃗ ○ β
where β⃗∶ F⇒ LF is the unique 2-cell such that dom β⃗ = iddom F and cod β⃗ = β; there is a
unique morphism Fˆ∶X → L-Coalg such that UFˆ = F and αFˆ = β.
Given any pair of morphisms Fˆ1, Fˆ2∶X → L-Coalg and a 2-cell θ⃗∶ F2 ⇒ F2 such that
Eθ⃗ ○ β1 = β2 ○ cod θ⃗
(where Fi = UFˆi and βi = codαFˆi as in the previous paragraph), there is a unique 2-cell
θˆ∶ Fˆ1 ⇒ Fˆ2 such that Uθˆ = θ⃗.
Proof. U is simply the U from proposition 8.1, while the 2-cell α there is the
2-cell α⃗ here. The equation e⃗U ○ α⃗ = idF implies that dom α⃗ = iddom U. With
that observation, the rest of the equations follow immediately from the universal
property of C2 and the equations eU ○ α = idU and δU ○ α = Lα ○ α from
proposition 8.1.
70
CHAPTER IX
COMPOSITION OF L-COALGEBRAS
In an algebraic weak factorization system, the categories L-Coalg and R-Alg
respectively play the roles of the left and right classes of morphisms of the weak
factorization system. In an ordinary weak factorization system, these two classes
of morphisms are closed under composition. In [Gar09], this is strengthened to a
composition functor
L-CoalgΠC L-Coalg→ L-Coalg.
Furthermore, in [Rie11] it is shown that colax morphisms of awfs preserve
this composition. Similarly, there is a composition functor on R-Alg which is
preserved by lax morphisms of awfs.
In this chapter, we will generalize these results to the setting of bimonads
in FF(Sq(D)). In fact we will prove the following more general theorem, from
which the desired results will follow as corollaries using proposition 3.15.
Theorem 9.1. Let D be a 2-category with arrow objects and with EM-objects for
comonads. There is a lax double functor
Coalg∶Comon(FF(Sq(D)))→ Span(D0)/(−)2
where D0 is the ordinary category underlying the (strict) 2-category D, which is the
identity on the vertical categories, and which takes a comonad (E, η, e, δ) in FF(Sq(D))
71
to the span
C L-Coalg C.dom U cod U
In [Gar09] it is further shown that given a functorial factorization with
only the comonad half of the awfs structure, a composition functor on L-Coalg
uniquely determines the monad half of the structure. The paper [Rie11] makes
much use of this fact, and also extends it to morphisms of awfs. In our framework,
these results will follow from proposition 3.18 and the theorem:
Theorem 9.2. The lax double functor Coalg is fully-faithful.
First we should explain the notation Span(D0)/(−)2 . There is a natural family
of monads in Span(D0), given for each object C by the span
C C2 Cdom cod
with multiplication given by the composition of the internal category structure
of C2 given in proposition 2.3. That this is a natural family means that for any
morphism f ∶C → D in D0 there is a morphism of spans
C C2 C
D D2 D.
f
dom cod
f 2 f
dom cod
That this morphism of spans commutes with the multiplications follows easily
from the universal property of arrow objects.
The double category Span(D0)/(−)2 has the same vertical category as
Span(D0)—namely D0—with horizontal 1-cells C → C given by spans S equipped
72
with a (globular) morphism S⇒ C2, i.e. a commuting diagram
S
C C
C2
u v
p
dom cod
and with 2-cells given by 2-cells in Span(D0)which commute with these structure
maps, i.e. by pairs ( f , θ) such that
C S C
D S′ D
D2
f
u v
θ f
u′ v′
p′
dom cod
=
S
C C2 C
D D2 D.
u vp
f
dom cod
f 2 f
dom cod
The composition of two horizontal 1-cells in Span(D0)/(−)2
S1 S2
C C C
C2 C2
u1 v1
p1
u2 v2
p2
dom cod
dom cod
is given by their horizontal composition in Span(D0), and the structure map to
C2 is given by the horizontal composition of the p1 and p2 composed with the
multiplication of C2, i.e.
S1∏C S2 C2∏C C2 = C3 C.2(p1,p2) c
73
The identity for the horizontal composition is
C
C C,
C2
id id
i
dom cod
where i∶C → C2 is the identity of the internal category structure on C2 from
proposition 2.3.
We will now prove a couple of simple lemmas to establish the existence
of certain 2-cells in D using the arrow object universal property. First, notice
that any comonad (E, η, e, δ) in FF(Sq(D)) gives rise to the horizontal 1-cell in
Span(D0)/(−)2
L-Coalg
C C.
C2
dom U cod U
U
dom cod
For each of the following lemmas, let (E1, η1, e1, δ1) and (E2, η2, e2, δ2) be two
comonads in FF(Sq(D)), both on the same object C, and let X1,2 be the pullback
X1,2
L1-Coalg L2-Coalg
C
P1 P2
cod U
1 dom
U2
with structure map U1,2∶X → C2 given by the composition
X1,2 C3 C.2
(U1,U2) c
74
Recall from proposition 2.3 that c by definition satisfies the three equations
dom c = dom P1, cod c = cod P2, and κc = κP2 ○ κP1. We also record for later
reference:
dom U1,2 = dom c(U1, U2) = dom P1(U1, U2) = dom U1P1 (9.1)
cod U1,2 = cod c(U1, U2) = cod P2(U1, U2) = cod U2P2 (9.2)
κU1,2 = κc(U1, U2) = (κP2 ○ κP1)(U1, U2) = κU2P2 ○ κU1P1 (9.3)
Lemma 9.3. There is a 2-cell
L1-Coalg
X1,2 C2
U1P1
U1,2
⇓ζ
such that dom ζ = id and
L1-Coalg
X1,2 C2 C
U1P1
U1,2
cod⇓ζ =
C2
X1,2 C
C2
codUP1
UP2
dom
cod
⇓κ
Proof. Equation (2.3) becomes
X1,2 C2 C
U1,2
dom
cod
⇓κ =
C2
X1,2 C
C2
dom
cod
UP1
UP2
dom
cod
⇓κ
⇓κ
which is just equation (9.3)
75
Lemma 9.4. There is a 2-cell
L2-Coalg
X1,2 C2
C2
U2P2
U1,2 R1
⇓ν
such that cod ν = id and
L2-Coalg
X1,2 C2 C
C2
U2P2
U1,2
dom
R1
⇓ν =
L2-Coalg C2
X1,2 L1-Coalg C2 C
C2
U2
domP2
P1
U1,2
U1
U1
cod
E1
⇓ζ ⇓α1
Proof. We just need to verify equation (2.3):
L2-Coalg C2
X1,2 L1-Coalg C2 C
C2
U2
domP2
P1
U1,2
U1
U1
cod
E1
cod
⇓ζ ⇓α1 ⇓e1
=
L2-Coalg C2
X1,2 L1-Coalg C2 C
U2
domP2
P1
U1,2
U1 cod
⇓ζ
= X1,2 L1-Coalg C2 CP2 U1 dom
cod
⇓κ
where the first equation follows from (8.2), and the second by reducing cod ζ
using lemma 9.3.
76
Proof of Theorem 9.1. For notational convenience, let G = Coalg be the lax double
functor we need to establish. The double categories Comon(FF(Sq(D))) and
Span(D0)/(−)2 both have D0 as vertical category, and G0 (the component of G on
vertical categories) is simply the identity.
From the statement of the theorem, G takes an object in Comon(FF(Sq(D)))
to the span and structure map
L-Coalg
C C.
C2
dom U cod U
U
dom cod
To define the behavior of G on 2-cells, consider a 2-cell in Comon(FF(Sq(D))):
C C
D D.
(E1,η1,e1,δ1)
F F
(E2,η2,e2,δ2)
⇓φ
By corollary 6.4, φ induces a colax morphism of comonads from L1 to L2, hence
by proposition 8.1 there is an induced morphism φ˜ between the EM-objects such
that U2φ˜ = F2U1. We can then define Gφ to be the morphism of spans
C C2 L1-Coalg C2 C
D D2 L2-Coalg D2 D.
F
dom
F2
U1
φ˜
U1 cod
F2 F
dom U2 U2 cod
That φ˜ commutes with the structure maps is simply the commutativity of the
square U2φ˜ = F2U1.
77
Next we must define the coherence data GI and G⊗. We will define GI to be
the morphism of spans
C
C C
LI-Coalg
id id
GI
dom U cod U
defined via lemma 8.3 by the equations UGI = i∶C → C2 and αIGI is the identity
on dom i = cod i. The conditions of the lemma are trivially satisfied.
We will similarly use lemma 8.3 to define G⊗. Let X1,2, U1,2, ζ, and ν be as
defined earlier in the section. G⊗ is a morphism of spans
X1,2
C C.
L1⊗2-Coalg
dom U1P1 cod U2P2
G⊗
dom U1⊗2 cod U1⊗2
We will define G⊗ to be the 1-cell such that U1⊗2G⊗ = U1,2 and
C2
X1,2 L1⊗2-Coalg C
C2
cod
G⊗
U1⊗2
U1⊗2 E1⊗2
⇓α1⊗2 = L2-Coalg C2X C
C2 C2
U2
U2
codP2
U1,2 R1 E2
⇓ν ⇓α2
In other words, in the notation of lemma 8.3 let F = U1,2 and β = E2ν ○ α2P2, and
define G⊗ = Fˆ.
We now need to check equations 1-3 of lemma 8.3 to verify that G⊗ is well
defined. We will check these equationally to save space, but the reader may want
78
to draw out the diagrams for themselves to follow along. For the first equation:
E2ν ○ α2P2 ○ κU1,2
= E2ν ○ α2P2 ○ κU2P2 ○ κU1P1 Eq. (9.3)
= E2ν ○ (α2 ○ κU2)P2 ○ κU1P1
= E2ν ○ η2U2P2 ○ κU1P1 Eq. (8.1)
= η2R1U1,2 ○dom ν ○ κU1P1 Interchange
= η2R1U1,2 ○ E1ζ ○ α1P1 ○ κU1P1 Def of ν
= η2R1U1,2 ○ E1ζ ○ (α1 ○ κU1)P1
= η2R1U1,2 ○ E1ζ ○ η1U1P1 Eq. (8.1)
= η1⊗2U1,2 ○dom ζ Interchange; Def of η1⊗2
= η1⊗2U1,2 dom ζ = id
and the second:
e1⊗2U1,2 ○ E2ν ○ α2P2
= e2R1U1,2 ○ E2ν ○ α2P2 Def of e1⊗2
= cod ν ○ (e2U2 ○ α2)P2 Interchange
= idcod U1,2 . Eq. (8.2); cod ν = id
The third equation is a bit trickier to prove. We will need to prove two
intermediate equations first, using the arrow object universal property.
Lemma.
iLU1,2 ○ L1ζ ○ α⃗1P1 = β⃗ ○ ζ (9.4)
79
Proof. We must show the 2-cells become equal upon composition with dom and
cod:
dom(iLU1,2 ○ L1ζ ○ α⃗1P1) = iddom U1,2 = dom(β⃗ ○ ζ)
and
cod(iLU1,2 ○ L1ζ ○ α⃗1P1)
= cod iLU1,2 ○ E1ζ ○ cod α⃗1P1
= η2R1U1,2 ○ E1ζ ○ α1P1 Def of iL, α⃗
= η2R1U1,2 ○dom ν Def of ν
= E2ν ○ η2U2P2 Interchange
= E2ν ○ (α2 ○ κU2)P2 Eq. (8.1)
= (E2ν ○ α2P2) ○ κU2P2
= cod β⃗ ○ cod ζ Def of β⃗, ζ
= cod(β⃗ ○ ζ).
Lemma.
R1β⃗ ○ ν = wU1,2 ○ L2δR1 U1,2 ○ L2ν ○ α⃗2P2 (9.5)
80
Proof. Again we must prove equality after composing with dom and cod:
dom(R1β⃗ ○ ν)
= E1β⃗ ○dom ν
= E1β⃗ ○ E1ζ ○ α1P1 Def of ν
= E1(β⃗ ○ ζ) ○ α1P1
= E1(iLU1,2 ○ L1ζ ○ α⃗1P1) ○ α1P1 Eq. (9.4)
= E1iLU1,2 ○ E1L1ζ ○ (E1α⃗1 ○ α1)P1
= E1iLU1,2 ○ E1L1ζ ○ (δ1U1 ○ α1)P1 Eq. (8.3)
= E1iLU1,2 ○ δ1U1,2 ○ E1ζ ○ α1P1 Interchange
= dom wU1,2 ○dom δR1 U1,2 ○dom ν ○dom α⃗2P2 Defs of w, δR, ν, α⃗= dom(wU1,2 ○ L2δR1 U1,2 ○ L2ν ○ α⃗2P2)
and
cod(R1β⃗ ○ ν)
= cod β⃗ ○ cod ν
= E2ν ○ α2P2 Defs of β⃗, ν
= E2(pR ○ δR1 )U1,2 ○ E2ν ○ α2P2 pR ○ δR = id= E2pRU1,2 ○ E2δR1 U1,2 ○ E2ν ○ α2P2= cod wU1,2 ○ cod L2δR1 U1,2 ○ cod L2ν ○ cod α⃗2P2 Defs of w, L, α⃗= cod(wU1,2 ○ L2δR1 U1,2 ○ L2ν ○ α⃗2P2)
81
Now we are prepared to prove the third equation of lemma 8.3 validating
our definition of G⊗:
δ1⊗2U1,2 ○ E2ν ○ α2P2
= (E2w ○ δ2R1⊙1 ○ E2δR1 )U1,2 ○ E2ν ○ α2P2 Def of δ1⊗2= E2(wU1,2 ○ L2δR1 U1,2 ○ L2ν) ○ (δ2U2 ○ α2)P2 Interchange= E2(wU1,2 ○ L2δR1 U1,2 ○ L2ν) ○ (E2α⃗2 ○ α2)P2 Eq. (8.3)= E2(wU1,2 ○ L2δR1 U1,2 ○ L2ν ○ α⃗2P2) ○ α2P2= E2(R1β⃗ ○ ν) ○ α2P2 Eq. (9.5)
= E1⊗2β⃗ ○ E2ν ○ α2P2 Def of E1⊗2
The verification that the definitions of GI and G⊗ form natural families, and
of the coherence axioms for a lax double functor, is tedious, but follows from what
we have presented here without requiring any new ideas or ingenuity.
Corollary 9.5. For any awfs (E, η,µ, e, δ) on an object C in D, the multiplication µ
induces a composition functor on L-Coalg, and the functor between EM-objects induced
by any colax morphism of awfs preserves this composition.
Proof. Any awfs (E, η,µ, e, δ) has an underlying object in Comon(FF(Sq(D))), by
simply forgetting µ. The lax double-functor Coalg takes this to a span
C L-Coalg C.dom U cod U
The multiplication µ provides this object in Comon(FF(Sq(D))) with a monad
structure, and lax double-functors preserve monads, so µ induces a monad
82
structure on this span. A multiplication on this span is a morphism pi:
X
C C
L-Coalg
dom
UP1
cod UP2
pi
dom U cod
U
where X is the pullback in the composite span
X
L-Coalg L-Coalg
C C C.
P1 P2
dom
U cod U dom
U cod U
The morphism pi is the composition structure that we want. If D = Cat is the
2-category of small categories, then an object ( f , g) in X is a pair of morphisms
in C equipped with coalgebra structures, such that cod f = dom g, and pi( f , g) is
a morphism equipped with a coalgebra structure, with dompi( f , g) = dom f and
codpi( f , g) = cod g.
Of course, what we really want is that the morphism underlying the
coalgebra pi( f , g) is the composition g ○ f . But this is simply the fact that pi defines
a 2-cell in Span(D0)/(−)2 , hence commutes with the structure maps to C2. Recall
that the structure map for the horizontal composite X is defined using c∶C3 → C2,
hence Upi( f , g) = c(U f , Ug).
Now we will continue on to the proof of theorem 9.2. The proof is
surprisingly difficult and tedious—we will outline the main steps but leave many
of the routine verifications to the reader.
83
Proof of Theorem 9.2. The bijectivity of Coalg acting on 2-cells with domain I is
simple to check, since I is initial in Comon(FF(Sq(D))), and from lemma 8.3 it is
easy to see that there is a unique morphism !∶C → L-Coalg satisfying U! = i, with
α! = ηi.
Now let (Ei, ηi, ei, δi), i ∈ {1, 2, 3}, be three comonads in FF(Sq(D)) on
horizontal 1-cells E1, E2∶C2 → C and E3∶D2 → D, and let F∶C → D be a morphism.
Given a morphism X1,2 → L3-Coalg such that U3θ = F2U1,2, we need to prove the
unique existence of a 2-cell
C C C
D D
(E1,η1,e1,δ1)
F
(E2,η2,e2,δ2)
F
(E3,η3,e3,δ3)
⇓φ
such that φ˜G⊗ = θ.
Outline of proof:
– Define a morphism Lˇ1⊗2∶C2 → X1,2 such that
P1 Lˇ1⊗2 = Lˆ1 and P2 Lˇ1⊗2 = Lˆ2R1.
Show that U1,2 Lˇ1⊗2 = L1⊗2.
– Define a 2-cell
ψ∶U3θ Lˇ1⊗2 ⇒ F2
by ψ = F2e⃗1⊗2, noting that U3θ Lˇ1⊗2 = F2L1⊗2.
84
– Let the 2-cell ψ′ = Lˆ3ψ ○ αˆ3θ Lˇ1⊗2,
C2 X1,2 L3-Coalg
C2 D2
Lˇ1⊗2 θ
F2
Lˆ3⇓ψ′
be the mate of ψ under the adjunction U3 ⊣ Lˆ3.
– Define the desired 2-cell φ to be the codomain component of ψ′:
φ = cod U3ψ′ = E3ψ ○ α3θ Lˇ1⊗2∶ FE2R1 → E3F2
– First we must verify that φ defines a valid 2-cell in FF(Sq(D)) by checking
equations (6.1) and (6.2). Equation (6.1) is simple to show directly, while (6.2)
follows from the well definedness of U3ψ′. In fact, we have
φL = U3ψ′∶ F2L1⊗2 ⇒ L3F2
– Next we must verify that φ defines a valid 2-cell in Comon(FF(Sq(D))),
which means showing that it commutes with the comultiplication 2-cells:
C2 C2 C
C2 C2 C
D2 D2 D
R1 E2
L1⊗2
F2
E2R1
F2 F
L3 E3
⇓δ1⊗2
⇓φL ⇓φ
=
C2 C2 C
D2 D
D D2 D.
R1
F2
E2
F
E3
L3 E3
⇓φ
⇓δ3
85
To do this, first verify the existence of a 2-cell δˇ1⊗2∶ Lˇ1⊗2 ⇒ Lˇ1⊗2L1⊗2 satisfying
P1δˇ1⊗2 = Lˆ1iL ○ δˆ1 and P2δˇ1⊗2 = Lˆ2w ○ Lˆ2L2δR1 ○ δˆ2R1 (9.6)
where δˆi is the unique 2-cell with Ui δˆi = δ⃗i. Show that U1,2δˇ1⊗2 = δ⃗1⊗2.
Define
τ1 = Lˆ3φL ○ψ′L1⊗2 ○ θδˇ1⊗2 and τ2 = δˆ3F2 ○ψ′
and check that cod U3τ1 = E3φL ○φL1⊗2 ○ Fδ1⊗2 and cod U3τ2 = δ3F2 ○φ. Hence
to prove (9.6) it suffices to show τ1 = τ2. To do this, show that the mates of
each are equal to φL.
– We have defined a 2-cell φ in Comon(FF(Sq(D))), now we need to show
that the lax functor Coalg takes this φ to the 2-cell θ we began with, i.e. that
θ = φ˜G⊗. It is easy to see that
U3φ˜G⊗ = F2U1⊗2G⊗ = F2U1,2 = U3θ,
so it only remains to show that α3φ˜G⊗ = α3θ.
Begin by verifying the existence of a 2-cell ρ∶ idX1,2 ⇒ Lˇ1⊗2U1,2 such that
P1ρ = Lˆ1ζ ○ αˆ1P1 and P2ρ = Lˆ2ν ○ αˆ2P2,
and show that U1,2ρ = α⃗1⊗2G⊗.
Finally, show that
ψU1,2 ○ F2α⃗1⊗2G⊗ = idF2U1,2 ,
86
and use this to show that α⃗3φ˜G⊗ = α⃗3θ. Thus we have shown the existence of
the 2-cell φ such that φ˜G⊗ = θ, and the uniqueness follows by a very similar
computation.
Combining this with proposition 3.15 immediately implies:
Corollary 9.6. Suppose (E, η, e, δ) is a comonad in FF(Sq(D)). A composition on
L-Coalg is equivalent to completing E to an awfs.
87
CHAPTER X
A UNIVERSAL PROPERTY FOR THE PUSHOUT PRODUCT
In this chapter we will begin the work of incorporating adjunctions of several
variables into the framework given so far. These are essential to making precise
the definitions of monoidal model category and of a model category enriched in
a monoidal model category.
Recall that a monoidal category M is called biclosed if the tensor product has
adjoints in each variable, i.e. if there are functors homl, homr∶M op ×M →M and
isomorphisms
M (A⊗ B, C) ≅M (B, homl(A, C)) ≅M (A, homr(B, C))
natural in all three variables. If M has a model structure, then one of the
requirements for M to be a monoidal model category is that the three bifunctors
⊗, homl, and homr form a Quillen adjunction of two variables. There are three
equivalent conditions for this:
1. Given any cofibrations i∶A → B and j∶ J → K, the map i⊗ˆj defined by the
pushout
A⊗ J A⊗K
B⊗ J A⊗K∐
A⊗JB⊗ J
B⊗K
A⊗j
i⊗J
i⊗K
B⊗j
i⊗ˆj
is a cofibration (which is trivial if either i or j is).
88
2. Given any cofibration i∶A → B and fibration f ∶X → Y, the map
ˆhoml(i, f )∶homl(B, X)→ homl(A, X) ⨉
homl(A,Y)homl(B, Y)
is a fibration (which is trivial if either i or f is).
3. Given any cofibration j∶ J → K and fibration f ∶X → Y, the map
ˆhomr(j, f )∶homy(K, X)→ homr(J, X) ⨉
homr(J,Y)homr(K, Y)
is a fibration (which is trivial if either i or f is).
Proving the equivalence of these three conditions is a routine but tedious exercise
in adjunctions. Another exercise in adjunctions shows that ⊗ˆ, ˆhoml, and ˆhomr in
fact make up an adjunction of two variables on the arrow category M 2.
In this chapter, we will give a universal property satisfied by the functors ⊗ˆ,
ˆhoml, and ˆhomr which will trivialize these kinds of routine adjunction arguments,
as well as making precise the clear symmetry involved. Then in chapter XI we will
make use of this universal property in order to show that the algebraic analogue
of Quillen adjunctions of two variables (defined in [Rie13]) can be recovered as
multivariable morphisms of bimonads in a precise sense.
10.1. Review of Cyclic Double Multicategories
Just as we needed the mates correspondence to define adjunctions of
algebraic weak factorization systems, we will need an extension of the mates
correspondence to multivariable adjunctions in order to define multivariable
adjunctions of awfs. Fortunately, both of these tasks have been done in [Rie13]
89
and [CGR12]. We will review the necessary material from those papers in this
section.
A multicategory is a structure like a category, but where morphisms are
allowed to have a list of objects as their domain, which we write as
f ∶ (X1, . . . , Xn)→ Y
sometimes dropping the parenthesis when they are not needed for readability.
The composition takes a composable configuration f ○ (g1, . . . , gn), where f is as
above and gi is a morphism with codomain Xi, and produces a morphism with
codomain Y and with domain the concatenation of all the domains of the gi.
Example 10.1. Any monoidal category has an underlying multicategory, in which
the multimorphisms (X1, . . . , Xn)→ Y are simply defined to be unary morphisms
X1 ⊗ ⋯ ⊗ Xn → Y. In fact, monoidal categories can be defined to be the
multicategories having a certain representability property.
A cyclic multicategory involves both a duality on the objects, and a cyclic
action on the morphisms taking a morphism f ∶ (X1, . . . , Xn)→ X0 to a morphism
σ f ∶ (X●0, X1, . . . , Xn−1)→ X●n
In other words, the cyclic action cyclically permutes the objects in the domain
and codomain, applying the duality whenever an object moves from domain to
codomain or vice versa. There are some axioms governing the interplay between
the cyclic action and the composition. We will refer to [CGR12] for complete
details on the material of this section.
90
The canonical example of a cyclic multicategory is MAdjl, whose objects are
categories, and morphisms are multivariable mutual left adjoints. For example,
consider an adjunction of two variables
K ×M N⊗ K op ×N Mhoml M op ×N Khomr
N (k⊗m, n) ≅M (m, homl(k, n)) ≅K (k, homr(m, n))
Here ⊗ is adjoint on the left, while the homs are adjoint on the right, but we
can arrange for all three to be left adjoints as follows:
K ×M N⊗ N op ×K M ophomopl M ×N op K ophomopr
N (k⊗m, n) ≅M op(homl(k, n), m) ≅K op(homr(m, n), k).
Written as three mutual left adjoints, and swapping the order of the inputs to
homl, we expose the cyclical symmetry. There is similarly a cyclic multicategory
MAdjr whose morphisms are mutual right adjoints.
Cyclic double multicategories, first introduced in [CGR12], are like the cyclic
double categories defined in V but where the vertical category is enlarged to a
vertical cyclic multicategory. For complete details, see [CGR12], but the following
example should make the idea clear:
Example 10.2. Generalizing example 5.1, there is a cyclic double multicategory
MAdjl whose objects are categories, whose vertical cyclic multicategory is
91
MAdjl, and whose 2-cells
M1,M2 N1,N2
M0 N0
G0
u,v
F0
w
⇓φ
are natural transformations φ∶ F0(u, v) ⇒ wG0. The cyclic action permutes 2-cells
φ to 2-cells
M2,M
op
0 N2,N
op
0
M
op
1 N
op
1
v,wop
G1 F1
uop
⇓σφ
M
op
0 ,M1 N
op
0 ,N1
M
op
2 N
op
2
wop,u
G2 F2
vop
⇓σ2φ
where (G0, G1, G2) and (F0, F1, F2) are each systems of mutual left adjoints, and
where σφ and σ2φ are the two mates of φ.
The details of the mates correspondence are significantly more complicated
in the multivariable case. The advantage of working with cyclic double
multicategories is that these details are not important: the properties of the mates
correspondence that are needed in practice are captured by the cyclic action.
10.2. The Universal Property
Define a cyclic double multicategory J as follows. The objects are Ai, Bi, for
i ∈ {0, 1, 2}, and their duals. The horizontal 1-cells are di0, di1∶Bi → Ai. The vertical
1-cells are Fi∶ (Ai−1, Ai+1) → A●i and Gi∶ (Bi−1, Bi+1) → B●i , which form two orbits
under the cyclic action.
92
There are two types of 2-cells. There are
Bi Ai
Bi Ai
di1
id id
di0
⇓αi
for each i. We will often draw these 2-cells globularly.
There are also 2-cells
Bi+1, Bi−1 Ai+1, Ai−1
B●i A●i
di+1ki+1 ,di−1ki−1
Gi Fi
di●ki
⇓λiki+1,ki−1,ki
for all choices of (k0, k1, k2) ∈ {0, 1}3 except (0, 0, 0).
Notice that there is at most one element of every hom-set, so all compositions
and cyclic actions are uniquely defined. From now on, we will omit indices
whenever doing so is unambiguous.
Remark 10.3. The cyclic double multicategory J is generated under composition
by the αi and the λiki+1,ki−1,ki with exactly one of k0, k1, k2 equal to 1. These nine
λ generators are further generated under the cyclic action by only three, though
there are many choices of which three. These generators satisfy the relations
B1, B2 A1, A2
B●0 A●0
d1,d0
G0 F0
d●0
d●1
⇓λ
⇓α●
= B1, B2 A1, A2
B●0 A●0
d1,d0
d0,d0
G0 F0
d●1
⇓α,id
⇓λ
93
B1, B2 A1, A2
B●0 A●0
d0,d1
G0 F0
d●0
d●1
⇓λ
⇓α●
= B1, B2 A1, A2
B●0 A●0
d0,d1
d0,d0
G0 F0
d●1
⇓id,α
⇓λ
B1, B2 A1, A2
B●0 A●0
d1,d1
d1,d0
G0 F0
d●0
⇓id,α
⇓λ = B1, B2 A1, A2
B●0 A●0
d1,d1
d0,d1
G0 F0
d●0
⇓α,id
⇓λ
and their reflections under the cyclic action.
Example 10.4. Let MAdjr be the cyclic double multicategory of categories,
functors, and multivariable right adjunctions. If A0, A1, and A2 have the
necessary pushouts and pullbacks, then any multivariable right adjunction
F0∶A1 ×A2 → A0 extends to a functor F̂∶ J→MAdjr as follows.
– Bi is sent to A 2i , the arrow category of Ai.
– The d1 are sent to the domain functors dom∶A 2i → Ai and the d0 are sent to
the codomain functors cod∶A 2i → Ai.
– The α are sent to the canonical natural transformations dom⇒ cod.
94
– The Gi are sent to functors Fˆi. Given two morphisms f ∶A → B ∈ A1 and
g∶X → Y ∈ A2, Fˆ0( f , g) is defined as in the diagram
F0(B, Y)
F0(A, Y) ⨉
F0(A,X)F0(B, X) F0(B, X)
F0(A, Y) F0(A, X)
F0(1,g)
F0( f ,1)
Fˆ0( f ,g)
p2
p1
F0( f ,1)
F0(1,g)
(10.1)
It is a standard fact that the Fˆi form a two-variable adjunction between the
arrow categories.
– Looking at diagram (10.1),
(λ01,0,0) f ,g = p1∶ cod Fˆ0( f , g)→ F0(dom f , cod g)(λ00,1,0) f ,g = p2∶ cod Fˆ0( f , g)→ F0(cod f , dom g)(λ00,0,1) f ,g = id∶dom Fˆ0( f , g)→ F0(cod f , cod g).
The three relations (1)-(3) then correspond precisely to the commutativity of
the three regions in diagram (10.1).
Let I be the sub-category of J consisting of just the 1-cells Fi. Let CDMCat
denote the 2-category of cyclic double multicategories, functors, and horizontal
transformations.
Theorem 10.5. Fix a functor F∶ I→MAdjr, whose image is a two-variable mutual right
adjunction (F0, F1, F2) between categories A0,A1,A2 which have the necessary pushouts
95
and pullbacks. Then the functor Fˆ∶ J →MAdjr constructed in example 10.4 is terminal
in the category CDMCatF(J,MAdjr) of functors on J restricting to F on I.
Proof. Concretely, the theorem says that given the data of a functor J →MAdjr,
determining 2-variable adjunctions Fi and Gi and the rest of the structure spelled
out in remark 10.3, there is a unique 2-cell
B1,B2 A 21 ,A
2
2
B●0 A ●20
H1,H2
G0 Fˆ0
H●3
⇓θ
where Fˆi is the pullback product defined in (10.1), such that
B1,B2 A 21 ,A
2
2 A1,A2
B●0 A ●20 A ●0
H1,H2
G0
cod,cod
Fˆ0 F0
H●3 dom●
⇓θ ⇓id = B1,B2 A1,A2
B●0 A ●0
d0,d0
G0 F0
d●1
⇓λ (10.2)
B1,B2 A 21 ,A
2
2 A1,A2
B●0 A ●20 A ●0
H1,H2
G0
dom,cod
Fˆ0 F0
H●3 cod●
⇓θ ⇓p1 = B1,B2 A1,A2
B●0 A ●0
d1,d0
G0 F0
d●0
⇓λ (10.3)
B1,B2 A 21 ,A
2
2 A1,A2
B●0 A ●20 A ●0
H1,H2
G0
cod,dom
Fˆ0 F0
H●3 cod●
⇓θ ⇓p2 = B1,B2 A1,A2
B●0 A ●0
d0,d1
G0 F0
d●0
⇓λ (10.4)
96
Fix objects B1 ∈ B1, B2 ∈ B2. The Hi are the functors sending Bi to Hi(Bi) =
αBi ∶ d1Bi → d0Bi. The component of θ at (B1, B2) is a square
d1G0(B1, B2) F0(d0B1, d0B2)
d0G0(B1, B2) F0(d1B1, d0B2) ∏
F0(d1B1,d1B2)F0(d0B1, d1B2)
The top arrow is uniquely determined by equation (10.2), while the components
of the bottom arrow are uniquely determined by equations (10.3) and (10.4).
10.3. Arrow Objects in Cyclic Double Multicategories
Now let M be any cyclic double multicategory. We will take theorem 10.5 as
our definition of what it means for a general cyclic double multicategory to have
arrow objects. For future reference, we will spell this out more concretely.
Given an object C of M, an arrow object C2 is an object together with
a globular 2-cell κ∶dom ⇒ cod satisfying the same universal property as
in section 3.2. (this only involves the horizontal 2-category, so carries over
unchanged).
Given a vertical 1-cell F∶ (C1, C2) → C●0, the lift to arrow objects Fˆ is a vertical
1-cell Fˆ together with 2-cells
C21 , C
2
2 C1, C2
C●20 C●0
cod,cod
Fˆ F
dom●
⇓γ0
C21 , C
2
2 C1, C2
C●20 C●0
dom,cod
Fˆ F
cod●
⇓γ1
C21 , C
2
2 C1, C2
C●20 C●0
cod,dom
Fˆ F
cod●
⇓γ2
97
satisfying the equations
C21 , C
2
2 C1, C2
C●20 C●0
dom,cod
G0 F0
cod●
dom●
⇓γ1
⇓κ●
= C21 , C22 C1, C2
C●20 C●0
dom,cod
cod,cod
G0 F0
dom●
⇓κ,id
⇓γ0 (10.5)
C21 , C
2
2 C1, C2
C●20 C●0
cod,dom
G0 F0
cod●
dom●
⇓γ2
⇓κ●
= C21 , C22 C1, C2
C●20 C●0
cod,dom
cod,cod
G0 F0
dom●
⇓id,κ
⇓γ0 (10.6)
C21 , C
2
2 C1, C2
C●20 C●0
dom,dom
dom,cod
G0 F0
cod●
⇓id,κ
⇓γ1 = C
2
1 , C
2
2 C1, C2
C●20 C●0
dom,dom
cod,dom
G0 F0
cod●
⇓κ,id
⇓γ2 (10.7)
and which is universal, meaning that given any objects X0, X1, X2, horizontal 1-
cells di,0, di,1∶Xi → Ci, a vertical 1-cell G∶X1, X2 → X●0, globular 2-cells αi∶ di,1 ⇒ di,0,
and 2-cells
X1, X2 C1, C2
X●0 C●0
d1,0,d2,0
G F
d●0,1
⇓λ0
X1, X2 C1, C2
X●0 C●0
d1,1,d2,0
G F
d●0,0
⇓λ0
X1, X2 C1, C2
X●0 C●0
d1,0,d2,1
G F
d●0,0
⇓λ2
98
satisfying the three equations analogous to (10.5)–(10.7), there exists a unique 2-
cell
X1, X2 C21 , C
2
2
X●0 C●20
αˆ1,αˆ2
G Fˆ
αˆ●0
⇓θ
(where αˆi is the 1-cell determined by αi by the universal property of the arrow
object Ci) such that
X1, X2 C21 , C
2
2 C1, C2
X●0 C●20 C●0
αˆ1,αˆ2
G Fˆ F
αˆ●0
⇓θ ⇓γi = X1, X2 C1, C2
X●0 C●0
G F⇓λi
for each i ∈ {0, 1, 2}.
Similarly, we define the lift of a vertical 1-cell F∶ (C1, . . . , Cn) → C●0 to arrow
objects to be a vertical 1-cell Fˆ together with (n + 1) 2-cells γi satisfying (n + 1)
equations analogous to (10.5)–(10.7) and which is universal in the analogous way.
Definition 10.6. Let M be a double multicategory. We say M has arrow objects
if for every object C there is an arrow object C2, and if for every vertical 1-cell
F∶ (C1, . . . , Cn)→ C●0 there is a lift to arrow objects Fˆ.
We have given the universal property of arrow objects and lifts of vertical 1-
cells in ordinary double multicategories, but it is clear from the cyclical symmetry
of the construction that a cyclic action respects arrow objects. Specifically, for any
object C, (C2)● = (C●)2, and σ(Fˆ) = σ̂F for any vertical 1-cell F, with σ(γi) = γi+1.
Example 10.7. Let EAdj be the restriction of MAdj to finitely complete and
cocomplete categories (note that the functors are not required to preserve these
99
limits or colimits). Finitely complete and cocomplete categories are closed
under the formation of opposite categories, so EAdj is again a cyclic double
multicategory. Then theorem 10.5 shows that EAdj has arrow objects.
100
CHAPTER XI
CYCLIC 2-FOLD DOUBLE MULTICATEGORIES
In this last chapter we will complete our goal of defining a common
generalization of the cyclic double multicategories of [CGR12] and the cyclic
2-fold double categories introduced in chapter V, showing that there is a
natural notion of multivariable morphisms of bimonads in such structures, and
constructing a cyclic 2-fold double multicategory of functorial factorizations in
which the multivariable bimonad morphisms recover the multivariable (co)lax
morphisms of awfs defined in [Rie13].
A cyclic two-fold double multicategory M consists of the same underlying
data as a cyclic double multicategory, i.e. a vertical multicategory, horizontal
1-cells, and 2-cells of the form
C1, . . . , Cn C1, . . . , Cn
C●0 C●0
X1,...,Xn
F F
X●0
⇓θ
which compose vertically in the same way as in a cyclic double multicategory,
and where as in a two-fold double multicategory the horizontal 1-cells are
endomorphisms. There are two composition structures on the horizontal 1-cells,
(I,⊗) and (⊥,⊙), such that for any object C, (IC)● =⊥C● and (⊥C)● = IC● , and such
that for any composable pair of horizontal 1-cells X, and Y, (X ⊗Y)● = X● ⊙Y●
and (X⊙Y)● = X● ⊗Y●.
101
Perhaps surprisingly, given two composable 2-cells
C1, . . . , Cn C1, . . . , Cn C1, . . . , Cn
C●0 C●0 C●0
X1,...,Xn
F F
Y1,...,Yn
F
X●0 Y●0
⇓θ ⇓φ
there are (n + 1) different horizontal compositions:
C1, . . . , Cn C1, . . . , Cn
C●0 C●0
X1⊙Y1,...,Xi⊗Yi,...,Xn⊙Yn
F F
(X0⊙Y0)●
⇓θ⊗iφ
for i ∈ {1, . . . , n}, and
C1, . . . , Cn C1, . . . , Cn
C●0 C●0.
X1⊙Y1,...,Xn⊙Yn
F F
(X0⊗Y0)●
⇓θ⊗0φ
In all cases, there is exactly one ⊗ in the ith position, and the rest of the horizontal
compositions are ⊙. Notice that this pattern only holds when using the convention
of dualizing everything in the codomain. Similarly, given any vertical n-ary 1-cell
F, there are (n + 1) unit 2-cells:
C1, . . . , Cn C1, . . . , Cn
C●0 C●0
⊥C1 ,...,ICi ,...,⊥Cn
F F
⊥●C0
⇓IiF
102
for i ∈ {1, . . . , n}, and
C1, . . . , Cn C1, . . . , Cn
C●0 C●0
⊥C1 ,...,⊥Cn
F F
I●C0
⇓I0F
The horizontal compositions and units must respect the cyclic action, such that
the equations hold:
σ(θ ⊗i φ) = (σθ)⊗i+1 (σφ) σ(IiF) = I(i+1)σF
We require the existence of the families of globular coherence 2-cells m, c, j, z,
satisfying the same conditions as in a cyclic 2-fold double category. Notably, we
only require naturality of z with respect to unary 2-cells. It is unclear whether
there is any sensible compatibility between z and multivariable 2-cells that could
be asked for, but such a compatibility is not needed for our purposes.
Remark 11.1. The generalization from cyclic double categories to cyclic 2-fold
double categories can be thought of as relaxing the condition that (X● ⊗Y●)● =
X ⊗Y, and the similar condition on 2-cells. We add the notation X ⊙Y for the
left hand side, and the axioms for a cyclic 2-fold double category add coherence
conditions relating X⊙Y and X⊗Y, which would be trivial if they were equal.
Similarly, we might imagine discovering the structure of a cyclic 2-fold double
multicategory by dropping the requirement that σ(σ−1θ ⊗ σ−1φ) = θ ⊗ φ for θ and
φ two n-ary 2-cells. Thus we get a different horizontal composition σi(σ−iθ⊗σ−iφ)
for each i ∈ {0, . . . , n}, which we abbreviate as ⊗i.
103
11.1. Multimorphisms of Bimonads
The definition 4.5 of bimonads in a 2-fold double category uses only globular
2-cells, so works unchanged in a cyclic 2-fold double multicategory M. However,
using the multicategory structure of M we will now be able to expand the
category of bimonads in M to a multicategory Bimon(M), and the cyclic
structure of M will lift to Bimon(M), making it a cyclic multicategory.
Definition 11.2. LetM be a cyclic 2-fold double multicategory, let (Xi, ηi,µi, ei, δi),
where i ∈ {0, 1, 2}, be bimonads in M, and let F and φ be as in the diagram
C1, C2 C1, C2
C0 C0.
X1,X2
F F
X●0
⇓φ
Say that (F,φ) is a 0-colax morphism of bimonads if the following two equations
are satisfied:
C1, C2 C1, C2
C0 C0
X1,X2
F F
X●0
I●
⇓φ
⇓η●0
= C1, C2 C1, C2
C0 C0
X1,X2
⊥,⊥
F F
I●
⇓e1,e2
⇓I0F
C1, C2 C1, C2
C0 C0
X1,X2
F F
X●0
(X0⊗X0)●
⇓φ
⇓µ●0
= C1, C2 C1, C2
C0 C0.
X1,X2
X1⊙X1,X2⊙X2
F F
(X0⊗X0)●
⇓δ1,δ2
⇓φ⊗0φ
104
Likewise, (F,φ) is a 1-colax morphism of bimonads if the two equations
C1, C2 C1, C2
C0 C0
I,X2
X1,X2
F F
X●0
⇓η1,id
⇓φ = C1, C2 C1, C2
C0 C0
I,X2
I,⊥
F F⊥●
X●0
⇓id,e2
⇓I1F
⇓e●0
C1, C2 C1, C2
C0 C0
X1⊗X1,X2
X1,X2
F F
X●0
⇓µ1,id
⇓φ = C1, C2 C1, C2
C0 C0
X1⊗X1,X2
X1⊗X1,X2⊙X2
F F(X0⊙X0)●
X●0
⇓id,δ2
⇓φ⊗1φ
⇓δ●0
hold, and (F,φ) is 2-colax if the analogous two equations hold. We will call (F,φ)
a 2-variable colax morphism of bimonads if it is i-colax for all i ∈ {0, 1, 2}.
The definition of colax multimorphisms with arity n should be clear from the
n = 2 case.
It is straightforward to see that multimorphisms of bimonads compose
multicategorically, so we have the multicategory Bimon(M) of bimonads in M.
Furthermore, the definition of colax multimorphism is clearly symmetric with
respect to the cyclic action, so that Bimon(M) inherits a cyclic action.
Definition 11.3. Let M be a cyclic 2-fold double multicategory. The cyclic
multicategory Bimon(M) has as objects bimonads in M, and as morphisms has
colax multimorphisms of bimonads.
105
11.2. Functorial Factorizations
In this section, given a cyclic double multicategory M, we will construct a
cyclic 2-fold double multicategory FF(M) of functorial factorizations in M.
The objects and vertical multicategory of FF(M) are those of M. The
horizontal 1-cells of FF(M) are functorial factorizations inM. As with bimonads,
the definition of functorial factorization given in chapter VI involves only globular
2-cells, so no modification is necessary to define functorial factorizations in M.
Also as with bimonads, we will give an explicit definition of 2-ary 2-cell and
let the reader extend the (easy) pattern to n-ary 2-cells for arbitrary n.
Definition 11.4. Let (Ei, ηi, ei), i ∈ {0, 1, 2}, be functorial factorizations in M on
objects Ci. A 2-ary 2-cell in FF(M)
C1, C2 C1, C2
C●0 C●0
E1,E2
F F
E●0
⇓θ
is given by a 2-cell
C21 , C
2
2 C1, C2
C●20 C●0
E1,E2
Fˆ F
E0
⇓θ
106
in M satisfying three equations:
C21 , C
2
2 C1, C2
C●20 C●0
E1,E2
Fˆ F
E●0
dom●
⇓θ
⇓η●0
= C21 , C22 C1, C2
C●20 C●0
E1,E2
cod,cod
Fˆ F
dom●
⇓e1,e2
⇓γ0 (11.1)
C21 , C
2
2 C1, C2
C●20 C●0
dom,E2
E1,E2
Fˆ F
E●0
⇓η1,id
⇓θ = C
2
1 , C
2
2 C1, C2
C●20 C●0
dom,E2
dom,cod
Fˆ F
cod●
E●0
⇓id,e2
⇓γ1
⇓e●0
(11.2)
C21 , C
2
2 C1, C2
C●20 C●0
E1,dom
E1,E2
Fˆ F
E●0
⇓id,η2
⇓θ = C
2
1 , C
2
2 C1, C2
C●20 C●0
E1,dom
cod,dom
Fˆ F
cod●
E●0
⇓e1,id
⇓γ2
⇓e●0
(11.3)
The cyclic action on a 2-cell in FF(M) is simply given by the cyclic action on
the underlying 2-cell in M. This is well defined since the definition of 2-cell in
FF(M) is clearly stable under the cyclic action in M.
107
Proposition 11.5. Continuing the notation of the previous definition, the 2-cell θ induces
2-cells in M
C21 , C
2
2 C
2
1 , C
2
2
C●20 C●20
L1,L2
Fˆ Fˆ
R●0
⇓θˆ0
C21 , C
2
2 C
2
1 , C
2
2
C●20 C●20
R1,L2
Fˆ Fˆ
L●0
⇓θˆ1
C21 , C
2
2 C
2
1 , C
2
2
C●20 C●20
L1,R2
Fˆ Fˆ
L●0
⇓θˆ2
satisfying
C21 , C
2
2 C
2
1 , C
2
2 C1, C2
C●20 C●20 C●0
L1,R2
Fˆ
cod,cod
Fˆ F
L●0 dom●
⇓θˆ2 ⇓γ0 = C
2
1 , C
2
2 C1, C2
C●20 C●0
E1,cod
cod,cod
Fˆ F
dom●
⇓e1,id
⇓γ0
C21 , C
2
2 C
2
1 , C
2
2 C1, C2
C●20 C●20 C●0
L1,R2
Fˆ
dom,cod
Fˆ F
L●0 cod●
⇓θˆ2 ⇓γ1 = C
2
1 , C
2
2 C1, C2
C●20 C●0
dom,cod
Fˆ F
cod●
E●2
⇓γ1
⇓e●0
C21 , C
2
2 C
2
1 , C
2
2 C1, C2
C●20 C●20 C●0
L1,R2
Fˆ
cod,dom
Fˆ F
L●0 cod●
⇓θˆ2 ⇓γ2 = C
2
1 , C
2
2 C1, C2
C●20 C●0
E1,E2
Fˆ F
E●0
⇓θ
and a similar three equations for each of θˆ0 and θˆ1.
In general, an n-ary 2-cell θ in FF(M) induces 2-cells θˆi in M, i ∈ {0, . . . , n}.
Proof. We will verify the existence of θˆ2. The pattern extending to all other cases
should be evident.
108
By using the universal property for arrow objects in a cyclic double
multicategory, we only need to check the three equations obtained by composing
each side of the equations (10.5)–(10.7) with θˆ2. Equation (10.5) remains
unchanged after composition with θˆ2, equation (10.6) becomes (11.1), and
equation (10.7) turns into (11.2).
Note that equation (11.3) proves that θˆ2 respects the units/counits of L0, L1,
R2, i.e. that the unit condition for a 2-colax morphism of bimonads holds, so that
all three equations (11.1)–(11.3) go into establishing θˆi for each i.
To finish the construction of the cyclic 2-fold double multicategory FF(M),
we still need to define the horizontal composites and units for n-ary 2-cells. Given
2-cells
C21 , C
2
2 C1, C2
C●20 C●0
E1,E2
Fˆ F
E●0
⇓θ and
C21 , C
2
2 C1, C2
C●20 C●0
E′1,E′2
Fˆ F
E′0●
⇓φ
in M underlying 2-cells in FF(M) (i.e. satisfying equations (11.1)–(11.3)), define
C21 , C
2
2 C1, C2
C●20 C●0
E1⊙1′ ,E2⊗2′
Fˆ F
E●0⊙0′
⇓θ⊗2φ by
C21 , C
2
2 C
2
1 , C
2
2 C1, C2
C●20 C●20 C●0
L1,R2
Fˆ
E′1,E′2
Fˆ F
L●0 E′0●
⇓θˆ2 ⇓φ
and likewise for the other horizontal composites. Checking that this composite
2-cell satisfies equations (11.1)–(11.3) is easy but notationally cumbersome, so we
109
will verify that θ ⊗2 φ satisfies (11.2) to convey the idea:
C21 , C
2
2 C1, C2
C●20 C●0
dom,E2⊗2′
E1⊙1′ ,E2⊗2′
Fˆ F
E●0⊙0′
⇓η1⊙1′ ,id
⇓θ⊗2φ = C
2
1 , C
2
2 C
2
1 , C
2
2 C1, C2
C●20 C●20 C●0
L1,R2
Fˆ
dom,E′2
E′1,E′2
Fˆ F
L●0 E′0●
⇓θˆ2
⇓η′1,id
⇓φ
= C21 , C22 C21 , C22 C1, C2
C●20 C●20 C●0
L1,R2
Fˆ
dom,E′2
dom,cod
Fˆ F
L●0
cod●
E′0●
⇓θˆ2
⇓id,e′2
⇓γ1
⇓e′0●
= C21 , C22 C21 , C22 C1, C2
C●20 C●20 C●0
L1,R2
Fˆ
dom,E′2
dom,cod
F
id
L●0
cod●
E′0●
⇓γ1
⇓id,e′2
⇓e⃗●0 ⇓e′0●
= C21 , C22 C1, C2
C●20 C●0
dom,E2⊗2′
dom,cod
Fˆ F
cod●
E●0⊙0′
⇓id,e′2⊗2′
⇓γ1
⇓e●0⊙0′
(11.4)
Finally, given a n-ary vertical 1-cell F, the unit 2-cells IiF are simply given by
γi, which are easily verified to define 2-cells in FF(M).
110
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