aa r X i v : . [ m a t h . C T ] F e b DISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS ANDDECAGONS
CHARLES WALKER
Abstract.
We give alternative definitions of distributive laws and pseudodis-tributive laws involving the decagonal coherence conditions which naturallyarise when the involved monads and pseudomonads are presented in extensiveform. We then use these results to give a number of simplifications in thecoherence conditions for distributive laws and pseudodistributive laws.In particular, we show that five coherence axioms suffice in the usual def-inition of pseudodistributive laws, we give simple descriptions of distributivelaws and pseudodistributive laws in terms of (pseudo)algebra structure maps,and we give concise definitions of distributive laws and pseudodistributive lawsin no-iteration form.
Contents
1. Introduction 21.1. Structure of the paper 42. Distributive laws of monads 52.1. Monads in monoidal and extensive form 52.2. Monoidal definition of distributive laws 62.3. Decagon definition of distributive laws 62.4. Algebra definition of distributive laws 72.5. No-iteration definition of distributive laws 72.6. Mixed distributive law case 83. Pseudodistributive laws of pseudomonads 83.1. Pseudomonads in pseudomonoidal and pseudoextensive form 83.2. Monoidal definition of pseudodistributive laws 113.3. Decagon definition of pseudodistributive laws 153.4. Pseudoalgebra definition of pseudodistributive laws 173.5. No-iteration definition of pseudodistributive laws 184. Equivalence of definitions of pseudodistributive laws 204.1. Equivalence of monoidal and decagon definitions 214.2. Explanation of redundant coherence axioms 234.3. Equivalence of decagon and algebra definitions 234.4. Equivalence of algebra and no-iteration definitions 265. Future work 26References 26
Mathematics Subject Classification.
Key words and phrases. distributive laws, pseudodistributive laws, no-iteration.This work was supported by the Operational Programme Research, Development and Educa-tion Project “Postdoc@MUNI” (No. CZ.02.2.69/0.0/0.0/18_053/0016952). Introduction
Monads are one of the fundamental constructions in category theory, and inrecent years have also become more prevalent in computer science [2, 23, 8]. Typi-cally, a monad on a category C is defined as an endofunctor T : C → C along withnatural transformations u : 1 C → T and m : T → T satisfying three coherenceaxioms.Distributive laws of monads were introduced by Beck [1] and give a concisedescription of the data and coherence conditions needed to compose two monads( T, u, m ) and (
P, η, µ ). More precisely, Beck defines a distributive law of monadsas a natural transformation λ : T P → P T such that the below two triangles andtwo pentagons commute
T P λ / / P T T P λ / / P TP
P u rrrrrr uP e e ❑❑❑❑❑ T ηT ssssss T η e e ❑❑❑❑❑❑ T P mP (cid:15) (cid:15) T λ / / T P T λT / / P T T
P m (cid:15) (cid:15)
T P T µ (cid:15) (cid:15) λP / / P T P
P λ / / P T µT (cid:15) (cid:15) T P λ / / P T T P λ / / P T
It is not hard to arrive at this set of four axioms. Indeed, given a λ : T P → P T ifone works out what is required to extend the monad (
T, u, m ) to a monad (cid:16) e
T , e u, e m (cid:17) : Kl ( P, η, µ ) → Kl ( P, η, µ )on the Kleisli category of P , they arrive at two of these axioms from the nullaryand binary functoriality conditions of e T , and the other two from naturality of e u and e m .It turns out that one may take a different approach to distributive laws, based onthe “extensive” (also called “Kleisli triple”) presentation of monads as studied byManes [13], which in fact dates back to early work of Walters [25]. In this extensiveform, a monad on a category C is defined as an assignation on objects T : C ob → C ob with a family of arrows u X : X → T X and functions C ( X, T Y ) → C ( T X, T Y )typically called “extension operators”. This data is then required to satisfy threedifferent coherence axioms. It is an interesting fact that the functoriality and nat-urality conditions automatically follow these three conditions. This simplificationwhich happens in
Cat when monads are presented extensively is explained in detailby Marmolejo [19], as a consequence of any functor having a right adjoint in thebicategory of profunctors
Prof . If one works out what is needed to extend the monad (
T, u, m ) the Kleisli cat-egory of P , with this extension now defined in extensive form, they will naturallyarrive at three coherence conditions for distributive laws corresponding to the threeaxioms for a monad in extensive form. These three axioms are the two triangles ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 3 from earlier, but with the two pentagons replaced by a single decagon condition
T P T P T
T P λT / / λT P T (cid:15) (cid:15) T P T T P m / / T P T T µT / / T P T λT (cid:15) (cid:15) P T P T
P mP T (cid:15) (cid:15)
P T P m (cid:15) (cid:15)
P T P T
P λT / / P T P m / / P T µT / / P T
In one dimension, the difference between these two definitions of distributive law israther trivial. However, in two dimensions the difference becomes significant, as thismeans pseudodistributive laws can be naturally defined taking three modificationsas the basic data rather than the usual four [15]. Moreover, the reduction in thedata makes the coherence conditions for pseudodistributive laws much easier tounderstand conceptually. Interestingly, one recovers a variant of the triangle andpentagon axioms for monoidal categories.It is this understanding, along with Kelly’s results concerning coherence formonoidal categories [9], that allow us to deduce that three of Marmolejo and Wood’seight coherence axioms for pseudodistributive laws [18] are redundant in the sensethat they follow from the other five.Moreover, the reader will notice the composites λT · P m : T P T → P T appearingin the decagon, so that denoting this composite by α , the decagon may be seen asthe hexagon axiom T P T P T
T P α / / αP T (cid:15) (cid:15) T P T T µT / / T P T α (cid:15) (cid:15) P T P T
P α / / P T µT / / P T
These morphisms α : T P T → P T (or morphisms
P T P → P T in the dual situ-ation ) should be familiar to the reader, appearing in the characterization of dis-tributive laws in terms of Kleisli and Eilenberg-Moore objects [16]. Indeed, thesecharacterizations are often useful for considering distributive laws when Kleisli andEilenberg-Moore objects do not exist [16].Interestingly, this hexagon axiom leads to a significant simplification of thesecharacterizations. It turns out that distributive laws λ are in bijection with mor-phisms α : T P T → P T rendering commutative the diagrams(1.1)
T P T α / / P T T T ηT / / m (cid:15) (cid:15) T P T α (cid:15) (cid:15) T P T P T
T P α / / αP T (cid:15) (cid:15) T P T T µT / / T P T α (cid:15) (cid:15) P T uP T O O id < < ①①①①①①①①① T ηT / / P T P T P T
P α / / P T µT / / P T giving a definition of distributive law with just 12 non-identity sides and 3 coherenceaxioms. This definition is closely connected to the definition of distributive laws in One might denote such morphisms by res Tη : P T P → P T as they exhibit
T η as a P -embedding, using the “admissibility” point of view [24]. In [24] the dual problem of extending topseudo-algebras extensively was considered, though in the simpler lax-idempotent case [26, 12],and the decagon conditions were not recognized. ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 4 terms of an extension operators ( − ) λ due to Marmolejo [19, Theorem 6.2]. We willalso give a version of the above for pseudodistributive laws.In the case of pseudodistributive laws, there a number of problems which havenot practical to solve until now. Indeed, the definition of a pseudodistributive lawin terms of extension operators ( − ) λ would normally be impractical due to thecoherence conditions involved. However, using the pseudo version of (1.1), whichis already closely related to presentations in terms of extension operators ( − ) λ , weare able to find such a definition. This is especially important in the setting ofrelative pseudomonads [5], where one is forced to use extension operators.Another example is the generalization of pseudodistributive laws to the n -arycase, done for distributive laws by Cheng [3]. Here one should arrive at an analogueof the coherence axioms for “ n -ary pseudomonoids” using the approach of this paper(where one finds an analogue of the axioms for a pseudomonoid as the axioms fora 2-ary pseudodistributive law).1.1. Structure of the paper.
In Section 2 we recall the monoidal and extensivedefinitions of monads and give three definitions of distributive laws of monads;namely:(1) the “monoidal” definition. This is the usual definition due to Beck [1],involving two triangles and two pentagons;(2) the “Kleisli-decagon” definition. This involves the decagonal conditions onefinds for distributive laws when the involved monads are presented in ex-tensive form. The “Kleisli” prefix refers to the fact the decagon starts with
T P T P T , as happens when one extends to the Kleisli category extensively.Dual versions starting from
P T P T P will arise from extending to algebras;(3) the “algebra” definition in terms of maps α : T P T → P T . This is a reducedversion of the above in which a change of variables leads to a simplificationin the axioms. Moreover, this definition may be regarded as a “base case”for definitions of distributive laws in terms of extension operators ( − ) λ .(4) the “no-iteration” definition in terms of extension operators( − ) λ : ( ⋆, P T ∗ ) → ( T ⋆, P T ∗ ) . These no-iteration definitions are intended to avoid any iteration of thefunctor P , which is important in the “relative” case [5]. This is an analogueof a result of Marmolejo and Wood [19], though slightly modified so thatit will better generalize to the two dimensional case.In this simpler one dimensional setting, the results are not hard to verify directlyand so we will omit much of the proof. In this simpler setting we also briefly mentiona version for mixed distributive laws, though the mixed case appears somewhat morecomplicated and so most the of the analogous results will be left for future work.In Section 3 we give the two dimensional version of the above. Here the “monoidal”definition involves four modifications subject to five coherence axioms; the next twodefinitions involve three modifications subject to two coherence axioms (which thereader will recognize as a version of the triangle and pentagon equations of monoidalcategories). The no-iteration version involves three families of 2-cells subject to twocoherence axioms.In Section 4 we justify our four definitions of pseudodistributive law by provingan equivalence with extensions to the Kleisli bicategory of a pseudomonad. In the ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 5 case of the monoidal definition, we will also explain how one recovers the threeredundant pseudodistributive law axioms.2.
Distributive laws of monads
We start by recalling two equivalent definitions of monad. We then give threeequivalent definitions of distributive laws of monads. It is not worth giving theproof of these definitions here, as they are easily verified directly, and moreoverthey will be shown in the more general two dimensional setting in the followingsections. We will only briefly mention the mixed distributive law case.2.1.
Monads in monoidal and extensive form.
The usual definition of monadis based on the fact that a monad on a category C is a monoid in the monoidalcategory End ( C ) with tensor given by composition, and so we refer to this as the“monoidal form” of a monad. Definition 2.1.1. A monad (in monoidal form) on a category C consists of afunctor equipped with natural transformations as below T : C → C , u : 1 C → T, m : T → T subject to the three coherence axioms T uT / / id ❇❇❇❇❇❇❇❇❇ T m (cid:15) (cid:15) T T u o o id ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦⑦ T T m / / mT (cid:15) (cid:15) T m (cid:15) (cid:15) T T m / / T The other definition of monad is the so called “extensive” form due Walters [25],also known as the “Kleisli triple” presentation. This is the form most often used incomputer science.
Definition 2.1.2 (Walters, Manes [25, 13]) . A monad (in extensive form) on acategory C consists of • an assignation on objects C ob → C ob : X T X ; • for each X ∈ C , a u X : X → T X ; • for each f : X → T Y , a map f T : T X → T Y ;such that:(1) for each f : X → T Y , we have f T · u X = f ;(2) for each X ∈ C , we have u TX = id T X ;(3) for each f : X → T Y and g : Y → T Z , we have (cid:0) g T · f (cid:1) T = g T · f T .It is not hard to check these two definitions of monad are in bijection. Indeedgiven a monad ( T, u, m ) in monoidal form one defines for any f : X → T Y the“extension” f T : T X → T Y as the composite
T X
T f / / T Y mY / / T Y
In the other direction, one can recover the multiplication data mX : T X → T X as the extension (id
T X ) T of the identity on T X . One can then verify the relevantcoherence conditions hold.
Definition 2.1.3.
The
Kleisli category of a monad (
T, u, m ) on a category C ,denoted Kl ( T ) is the category with: ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 6 • the same objects as C ; • a morphism f : X Y in Kl ( T ) is a morphism f : X → T Y in C ; • the identity id X : X X on a object X is the unit uX : X → T X ; • for each f : X Y and g : Y Z the composite g · f : X Z is given asthe composite X f / / T Y
T g / / T Z mZ / / T Z .
Remark . The reader will of course notice the composite above may be writtenas g T · f when the monad is presented in extensive form.2.2. Monoidal definition of distributive laws.
With monads presented in monoidalform, one naturally arrives at Beck’s definition of distributive laws. Indeed, thisdefinition may be recovered as the notion of a monad in the 2-category of monads[21]. The disadvantage of this approach (where monads are presented in monoidalform) will later become apparent in two dimensions, where one arrives at a numberof redundant coherence axioms.
Definition 2.2.1 (Beck [1]) . A distributive law (in monoidal form) between mon-ads ( T, u, m ) and (
P, η, µ ) is a natural transformation λ : T P → P T renderingcommutative the four diagrams below
T P λ / / P T T P λ / / P TP
P u rrrrrr uP e e ❑❑❑❑❑ T ηT ssssss T η e e ❑❑❑❑❑❑ T P mP (cid:15) (cid:15) T λ / / T P T λT / / P T T
P m (cid:15) (cid:15)
T P T µ (cid:15) (cid:15) λP / / P T P
P λ / / P T µT (cid:15) (cid:15) T P λ / / P T T P λ / / P T
Decagon definition of distributive laws.
Taking the basic data of a dis-tributive law to be a natural transformation λ : T P → P T as before; one naturallyarrives at the following conditions for extending a monad (
T, u, m ) to the Kleisli cat-egory of (
P, η, µ ), with this extension presented in extensive form. These “decagon”formulations serve as an intermediary between Beck’s monoidal definition, and thelater “algebra” definition.
Definition 2.3.1. A distributive law (in Kleisli-decagon form) between monads( T, u, m ) and (
P, η, µ ) is a natural transformation λ : T P → P T rendering commu-tative the three diagrams below
T P λ / / P T T P λ / / P TP
P u ttttt uP f f ▲▲▲▲▲▲ T ηT ; ; ✇✇✇✇✇ T η d d ■■■■■ T P T P T
T P λT / / λT P T (cid:15) (cid:15) T P T T P m / / T P T T µT / / T P T λT (cid:15) (cid:15) P T P T
P mP T (cid:15) (cid:15)
P T P m (cid:15) (cid:15)
P T P T
P λT / / P T P m / / P T µT / / P T
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 7
Algebra definition of distributive laws.
The following definition is in-tended to define distributive laws using the least total number of sides and coher-ence conditions possible. The closest result to this would be [19, Theorem 6.2],though the assumption of the multiplication algebra axiom is redundant with therewritten condition involving ηT . Indeed, one may view the below definition as a“base case” of [19, Theorem 6.2], with extension operators applied to identities. Definition 2.4.1. A distributive law (in algebra form) between monads ( T, u, m )and (
P, η, µ ) is a natural transformation α : T P T → P T rendering commutativethe three diagrams below
T P T α / / P T T T ηT / / m (cid:15) (cid:15) T P T α (cid:15) (cid:15) T P T P T
T P α / / αP T (cid:15) (cid:15) T P T T µT / / T P T α (cid:15) (cid:15) P T uP T O O id < < ①①①①①①①①① T ηT / / P T P T P T
P α / / P T µT / / P T
This definition may also be regarded as a simplification of the following resultof Marmolejo, Rosebrugh and Wood, which assumes five coherence axioms on themorphism α : T P T → P T . Proposition 2.4.2. [16, Prop. 3.5]
Distributive laws λ : T P → P T of monads ( T, u, m ) and ( P, η, µ ) are in bijection with T -algebras α : T P T → P T renderingcommutative the three diagrams
T P T αT (cid:15) (cid:15) T P m / / T P T α (cid:15) (cid:15) T T ηT / / m (cid:15) (cid:15) T P T α (cid:15) (cid:15) T P T T P uP T / / T µT (cid:15) (cid:15)
T P T P T αP T / / P T P T
P α / / P T µT (cid:15) (cid:15) P T P m / / P T T ηT / / P T T P T α / / P T
No-iteration definition of distributive laws.
The following defines a dis-tributive law in terms of extension operators ( − ) λX,Y , which are families of functions( − ) λX,Y : C ( X, P T Y ) → C ( T X, P T Y ) , X, Y ∈ C induced by pasting with a diagram of the form C T ❅❅❅❅❅❅❅❅ α (cid:11) (cid:19) C P T > > ⑦⑦⑦⑦⑦⑦⑦⑦ P T / / C This allows for the following definition of distributive law involving no iteration ofthe monad P . This is essentially [19, Theorem 6.2], though the multiplicative alge-bra axiom has been omitted as it becomes redundant with the rewritten conditioninvolving ηT . Definition 2.5.1. A distributive law (in no-iteration form) between monads ( T, u, m )and (cid:16)
P, η, ( − ) P (cid:17) on a category C is a pasting operator( − ) λX,Y : C ( X, P T Y ) → C ( T X, P T Y ) , X, Y ∈ C ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 8 such that all f : X → P T Y and g : Y → P T Z render commutative X uX (cid:15) (cid:15) f (cid:22) (cid:22) T X mX (cid:15) (cid:15) ( ηT X ) λ (cid:23) (cid:23) T X f λ (cid:15) (cid:15) (cid:0) ( g λ ) P f (cid:1) λ (cid:23) (cid:23) T X f λ / / P T Y T X ηT X / / P T X P T Y ( g λ ) P / / P T Z
Mixed distributive law case.
In the case of distributive laws, the decagoncondition may be regarded as a restriction of the associativity condition of thecomposite monad
P T along units. From this perspective it is perhaps slightlysurprising we have an analogue in the mixed case, where no composite monads arepresent.
Definition 2.6.1. A mixed distributive law (in decagon form) between a comonad( L, ε, δ ) and monad (
R, η, µ ) is a natural transformation λ : LR → RL renderingcommutative the two triangles LR λ / / εR " " ❉❉❉❉ RL Rε } } ③③③③ LR λ / / RLR L ηL = = ③③③③ Lη a a ❉❉❉❉ and the decagon RLR
RδR / / RL R RLλ / / RLRL
RλL / / R L µL % % ❏❏❏❏ LR λR tttt δR % % ❏❏❏❏ RL L R LλR / / LRLR
LRλ / / LR L LµL / / LRL λL ttttt Remark . Whilst these decagon conditions do show up in the mixed case, theydo so less nicely. This additional complexity is the reason the mixed case is left forfuture work. This is likely related to the fact that the definition of a distributivelaw in terms of extension operators is more complex in the mixed case [17].3.
Pseudodistributive laws of pseudomonads
We start this section by recalling two equivalent definitions of pseudomonad,including the three axioms which are known to be redundant by results of Kelly[9]. We then list three equivalent characterizations of pseudodistributive laws. Thejustification of these definitions is left to the next section, where we will use these re-dundant pseudomonad axioms to explain why three of the usual pseudodistributivelaw axioms are redundant.3.1.
Pseudomonads in pseudomonoidal and pseudoextensive form.
In or-der to define pseudomonads, we first need the notions of pseudonatural transforma-tions and modifications. The notion of pseudonatural transformation is the (weak)2-categorical version of natural transformation. Modifications, defined below, takethe place of morphisms between pseudonatural transformations.
Definition 3.1.1. A pseudonatural transformation between pseudofunctors t : F → G : A → B where A and B are bicategories provides for each 1-cell f : A → B in ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 9 A , 1-cells t A and t B and an invertible 2-cell t f in B as below F A F f / / t A (cid:15) (cid:15) F B t B (cid:15) (cid:15) tf = ⇒ G A Gf / / G B satisfying coherence conditions outlined in [11, Definition 2.2]. Given two pseudo-natural transformations t, s : F → G : A → B as above, a modification α : s → t consists of, for every object A ∈ A , a 2-cell α A : t A → s A such that for each 1-cell f : A → B in A we have the equality α B · F f · t f = s f · Gf · α A .By considering pseudomonads as pseudomonoids in a Gray-monoid of endo-pseudofunctors one naturally arrives at the following definition. Definition 3.1.2. A pseudomonad (in pseudomonoidal form) on a bicategory C consists of a pseudofunctor equipped with pseudonatural transformations as below T : C → C , u : 1 C → T, m : T → T along with three invertible modifications T uT / / id ❇❇❇❇❇❇❇❇❇ T m (cid:15) (cid:15) T T u o o id ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦⑦ T T m / / mT (cid:15) (cid:15) T m (cid:15) (cid:15) α ⇐ = β ⇐ = T T m / / T γ ⇐ = subject to the two coherence axioms T m & & ▲▲▲▲▲▲▲ T T m & & ▼▼▼▼▼▼ T T uT / / T T m qqqqqq mT & & ▼▼▼▼▼▼ ⇓ γ T = T T uT qqqqqq T uT & & ▼▼▼▼▼▼ id / / ⇓ T α ⇓ βT T m / / TT m rrrrrrr T mT qqqqqq T T m / / mT (cid:15) (cid:15) T mT ❇❇❇❇ ❇❇❇❇ T T m ❇❇❇❇❇❇❇❇ T T m / / mT (cid:15) (cid:15) T T m ❇❇❇❇❇❇❇❇ mT (cid:15) (cid:15) Tγ ⇐ = m − m ⇐ = T mT ❇❇❇❇❇❇❇❇ γT ⇐ = T T m / / mT (cid:15) (cid:15) T m (cid:15) (cid:15) = T mT ❇❇❇❇❇❇❇❇ T m / / T m ❇❇❇❇❇❇❇❇❇ γ ⇐ = T m (cid:15) (cid:15) γ ⇐ = γ ⇐ = T m / / T T m / / T Remark . One should note here that there are three useful consequences ofthese pseudomonad axioms [14, Proposition 8.1] originally due to Kelly [9]. Theseare T m % % ▲▲▲▲▲▲▲ T uT % % ▲▲▲▲▲▲▲ C u / / T uT rrrrrrr T u % % ▲▲▲▲▲▲▲ id / / ⇓ α ⇓ β T = 1 C u rrrrrrr u % % ▲▲▲▲▲▲▲ ⇓ u − u T m / / TT m rrrrrrr T T u rrrrrrr ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 10 T uT / / id ❆❆❆❆❆❆❆❆❆ T T m / / mT (cid:15) (cid:15) αT ⇐ = T m (cid:15) (cid:15) T T m / / ⇓ u m T m / / α ⇐ = T γ ⇐ = = T m / / T T uT O O m / / T uT O O id ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ T T u / / id ❆❆❆❆❆❆❆❆❆ T mT / / T m (cid:15) (cid:15) Tβ − ⇐ = T m (cid:15) (cid:15) T mT / / ⇓ m − u T m / / β − ⇐ = T γ − ⇐ = = T m / / T T T u O O m / / T T u O O id ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ We only mention these redundant axioms as they will be important in this paper.Indeed, these three redundant axioms appear in the coherence conditions for apseudodistributive law. The first appears directly, the appearance of the other twoonly becomes apparent when one uses decagon axioms instead of the usual pentagonaxioms.The definition of a pseudomonad in extensive form is due to Marmolejo andWood [20]. However, it will be more convenient to use the presentation givenby Fiore, Gambino, Hyland and Winskel [5] for relative pseudomonads (with the“relative” part taken to be an identity).
Definition 3.1.4. [20, 5] A pseudomonad (in pseudoextensive form) on a bicategory C consists of • an assignation on objects C ob → C ob : X T X ; • for each X ∈ C , a 1-cell u X : X → T X ; • for each X, Y ∈ C a functor ( − ) TX,Y : C ( X, T Y ) → C ( T X, T Y ); • for each f : X → T Y , an isomorphism φ f : f ⇒ f T · u X natural in f ; • for each X ∈ C , an isomorphism θ X : u TX ⇒ id T X ; • for each f : X → T Y and g : Y → T Z , an isomorphism δ g,f : (cid:0) g T · f (cid:1) T ⇒ g T · f T natural in f and g ;satisfying the two coherence conditions:(1) each f : X → T Y renders commutative f T φ Tf / / unitor . . (cid:0) f T u X (cid:1) T δ f,uX / / f T u TXf T θ X (cid:15) (cid:15) f T · id ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 11 (2) each f : X → T Y , g : Y → T Z , and h : Z → T V renders commutative (cid:16)(cid:0) h T g (cid:1) T f (cid:17) T δ hT g,f ' ' ❖❖❖❖❖ ( δ h,g f ) T v v ♥♥♥♥♥ (cid:0)(cid:0) h T g T (cid:1) f (cid:1) T assoc. (cid:15) (cid:15) (cid:0) h T g (cid:1) T f Tδ h,g f T (cid:15) (cid:15) (cid:0) h T (cid:0) g T f (cid:1)(cid:1) Tδ hT ,gT f (cid:15) (cid:15) (cid:0) h T g T (cid:1) f T assoc. (cid:15) (cid:15) h T (cid:0) g T f (cid:1) T h T δ g,f / / h T (cid:0) g T f T (cid:1) Remark . The three useful consequences of the pseudomonad axioms listedearlier in Remark 3.1.3 now become the assertion [5, Lemma 3.2] which states thatany morphisms f : X → T Y and g : Y → T Z render commutative u X id + + φ uX / / u TX u Xθ X u X (cid:15) (cid:15) (cid:0) u TY f (cid:1) T δ uY ,f / / ( θ Y f ) T , , u TY f Tθ Y f T (cid:15) (cid:15) g T f g T φ f $ $ ■■■■■■■■■■ φ gT f / / (cid:0) g T f (cid:1) T u Xδ g,f u X (cid:15) (cid:15) u X f T g T f T u X Remark . Note that the Kleisli bicategory of a pseudomonad is defined simi-larly to the one dimensional case in Definition 2.1.3. The technicality here is thatwe only have a bicategory, and so we must also give the unitality and associativitydata. For the full details of this construction, see [4, Definition 4.1].3.2.
Monoidal definition of pseudodistributive laws.
Even when dealing withstrict 2-monads, it is often the case that one has no strict distributive law betweenthem, but only a pseudo distributive law where the usual diagrams only commuteup to invertible modifications [4]. Work on these “pseudo” versions of distributivelaws started with Kelly [10], who considered the case where the usual axioms heldstrictly with the exception of one of the pentagons.Later, pseudodistributive laws were considered in the general case (where allfour axioms only hold up to isomorphism) by Marmolejo [15], who imposed ninecoherence conditions on the four invertible modifications. It was then later shownby Marmolejo and Wood [18], that one of the original nine axioms, in addition to atenth axiom introduced by Tanaka [22], are redundant, thus reducing the numberof coherence axioms to eight.We now give another reduction in the coherence axioms, using just five to definea pseudodistributive law.
Remark . Note that the usual nine (or ten ) coherence axioms for a pseudodis-tributive law come from understanding the structure of a pseudomonad in the Graycategory of pseudomonads [6]. From there, it is a matter of working out which areredundant. Definition 3.2.2. A pseudodistributive law (in pseudomonoidal form) betweenpseudomonads (with their modification data suppressed) ( T, u, m ) and (
P, η, µ ) is If one includes the redundant coherence axiom for a pseudomonad morphism they shouldarrive at ten axioms.
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 12 a pseudonatural transformation λ : T P → P T and four invertible modifications asbelow T P λ / / P T T P λ / / P TP
P u ttttttt uP d d ❏❏❏❏❏❏❏ ω [ c ❄❄❄❄ T ηT : : ttttttt T η d d ❏❏❏❏❏❏❏ ω (cid:27) ❄❄❄❄ T P mP (cid:15) (cid:15) T λ / / T P T λT / / ω { (cid:3) ⑧⑧⑧⑧ P T T
P m (cid:15) (cid:15)
T P T µ (cid:15) (cid:15) λP / / P T P
P λ / / ω ; C ⑧⑧⑧⑧ P T µT (cid:15) (cid:15) T P λ / / P T T P λ / / P T satisfying the following five coherence axioms . The first two axioms are the unitaryaxioms of a pseudomonad morphism and pseudomonad opmorphism(W1) T P
T uP / / T ω ; C ⑧⑧⑧⑧ id (cid:15) (cid:15) id % % T P mP / / T λ (cid:15) (cid:15)
T P λ (cid:15) (cid:15) T P
T P u / / λ (cid:15) (cid:15) T P T ω ; C ⑧⑧⑧⑧ λT (cid:15) (cid:15) = id λ P T
P T u / / id : : P T P m / / P T (W2)
T P
T ηP / / id (cid:15) (cid:15) id % % ω P { (cid:3) ⑧⑧⑧⑧ T P T µ / / λP (cid:15) (cid:15) T P λ (cid:15) (cid:15) T P ηT P / / λ (cid:15) (cid:15) P T P
P λ (cid:15) (cid:15) ω { (cid:3) ⑧⑧⑧⑧ = id λ P T ηP T / / id : : P T µT / / P T The directions of the modifications below are chosen such that they will naturally composeinto decagons later on, and such that the directions of the induced pseudomonad modificationswill match with that of a pseudomonad in extensive form as in Definition 3.1.4 (which is definedas in [5]). Though these choices of directions do not matter in the sense that these modificationsare invertible, it will make the later proofs easier to follow. The presentation of the axioms is chosen such that later proofs will be easier to understand.
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 13
The next two axioms are the associativity axioms of a pseudomonad morphism andpseudomonad opmorphism(W3) T P T mP (cid:15) (cid:15) T λ / / mT P (cid:2) (cid:2) T P T
T λT / / T ω { (cid:3) ⑧⑧⑧⑧ T P T T P m (cid:15) (cid:15) λT / / P T P T m (cid:15) (cid:15) T P mP ) ) T P mP (cid:15) (cid:15) T λ / / T P T λT / / ω { (cid:3) ⑧⑧⑧⑧ P T P m (cid:15) (cid:15)
T P λ / / P T = T P T λ / / mT P (cid:15) (cid:15) T P T
T λT / / mP T (cid:15) (cid:15) T P T λT / / ω T { (cid:3) ⑧⑧⑧⑧ P T P mT (cid:15) (cid:15)
P T m (cid:28) (cid:28) T P mP (cid:15) (cid:15) T λ / / T P T λT / / ω { (cid:3) ⑧⑧⑧⑧ P T P m (cid:15) (cid:15)
P T P m u u T P λ / / P T (W4)
T P T µP (cid:15) (cid:15) λP / / T P µ (cid:2) (cid:2)
P T P P λP / / ω P ; C ⑧⑧⑧⑧ P T P µT P (cid:15) (cid:15) P λ / / P T µP T (cid:15) (cid:15) T P T µ ) ) T P T µ (cid:15) (cid:15) λP / / P T P
P λ / / ω ; C ⑧⑧⑧⑧ P T µT (cid:15) (cid:15) T P λ / / P T = T P λP / / T P µ (cid:15) (cid:15)
P T P P λP / / P T µ (cid:15) (cid:15) P T P P λ / / P ω ; C ⑧⑧⑧⑧ P T P µT (cid:15) (cid:15) µP T (cid:28) (cid:28)
T P T µ (cid:15) (cid:15) λP / / P T P
P λ / / ω ; C ⑧⑧⑧⑧ P T µT (cid:15) (cid:15) P T µT u u T P λ / / P T
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 14
The last axiom ensures that the pentagons ω and ω are compatible, and asks (W5) T P T λ / / T ω (cid:11) (cid:19) T P T λT $ $ ■■■■■■ T P T λP T & & ▲▲▲▲▲▲▲ T µT ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ P T P m ●●●●●● T P T λP / / T µ rrrrrrrrrrrrrrrrrr mP % % ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ T P T P
T P λ rrrrrrr λT P & & ▲▲▲▲▲▲ P T P T
P λT / / ω T (cid:11) (cid:19) P ω (cid:11) (cid:19) P T µT : : ✉✉✉✉✉✉ P m $ $ ■■■■■■ P TP T P P T λ rrrrrr P mP * * P T µT ; ; ✇✇✇✇✇ T P λP / / ω P (cid:11) (cid:19) P T P
P λ : : ✉✉✉✉✉✉ is equal to T P mP $ $ ■■■■■■ T λ / / T P T λT / / P T P m ●●●●●● ω (cid:11) (cid:19) T P T µ : : ✉✉✉✉✉✉ mP $ $ ■■■■■■ T P λ / / P TT P T µ : : ✉✉✉✉✉✉ λP / / P T P
P λ / / P T µT ; ; ✇✇✇✇✇ ω (cid:11) (cid:19) For convenience and easy reference, we also list the five redundant coherenceconditions of a pseudodistributive law. Note the redundancy of the first two is dueto Marmolejo and Wood [18].
Theorem 3.2.3.
Given a pseudodistributive law ( λ, ω , ω , ω , ω ) : T P → P T inpseudomonoidal form, the following five conditions are derivable. (W6)
T P uT P / / λ (cid:15) (cid:15) id % % T P mP / / T λ (cid:15) (cid:15)
T P λ (cid:15) (cid:15) P T uP T / / id (cid:15) (cid:15) ω T ; C ⑧⑧⑧⑧ T P T ω ; C ⑧⑧⑧⑧ λT (cid:15) (cid:15) = id λ P T
P uT / / id : : P T P m / / P T One might take the octagon is this condition as the data for a pseudodistributive law, but itis not clear if that would give easily understood coherence axioms.
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 15 (W7)
T P
T P η / / λ (cid:15) (cid:15) id % % T P T µ / / λP (cid:15) (cid:15) T P λ (cid:15) (cid:15) P T
P T η / / id (cid:15) (cid:15) P ω { (cid:3) ⑧⑧⑧⑧ P T P
P λ (cid:15) (cid:15) ω { (cid:3) ⑧⑧⑧⑧ = id λ P T
P ηT / / id : : P T µT / / P T (W8) T P T λ / / mP $ $ ■■■■■■ T P T λT / / P T P m ●●●●●● T P T λ / / T P T λT / / P T P m ●●●●●● T T η < < ②②②②②② m ❋❋❋❋❋❋ T P λ / / ω (cid:11) (cid:19) ω (cid:11) (cid:19) P T = T T η < < ②②②②②② m ❋❋❋❋❋❋ T ηT < < ηT = = T ω (cid:11) (cid:19) ω T (cid:11) (cid:19) P TT ηT T η : : ✉✉✉✉✉✉✉ T ηT (W9) P P u & & uP $ $ ■■■■■■■ P P u & & P µ ; ; ①①①①①① uP " " ❋❋❋❋❋ T P λ / / ω (cid:11) (cid:19) ω (cid:11) (cid:19) P T = P µ ; ; ①①①①①① uP " " ❋❋❋❋❋ P uP P u " " ω P (cid:11) (cid:19) P ω (cid:11) (cid:19) P TT P λP / / T µ : : ✉✉✉✉✉✉ P T P
P λ / / P T µT ; ; ✇✇✇✇✇✇ T P λP / / P T P
P λ / / P T µT ; ; ✇✇✇✇✇✇ (W10) P uP ! ! ❈❈❈❈❈❈❈❈ P u ! ! P P u ! ! ❈❈❈❈❈❈❈❈ u (cid:31) (cid:31) ❄❄❄❄❄❄❄ η ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ T P λ / / ω (cid:11) (cid:19) ω (cid:11) (cid:19) P T = u (cid:31) (cid:31) ❄❄❄❄❄❄❄ η ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ P TT
T η = = ④④④④④④④④ ηT = = T ηT = = ④④④④④④④④ Remark . We will leave the explanation of these redundant axioms until Sub-section 4.2, as this explanation relies on the later mentioned decagon conditions.Note that this is the best that one might hope for, in that only one compatiblyaxiom is needed between the pseudomonad morphism and opmorphism data. Statedmore precisely, this becomes the following result.
Theorem 3.2.5.
A pseudodistributive law ( λ, ω , ω , ω , ω ) : T P → P T is equiv-alently a pseudomonad morphism ( λ, ω , ω ) : T → T along P , and a pseudomonadopmorphism ( λ, ω , ω ) : P → P along T , such that ω and ω satisfy axiom (W5). Decagon definition of pseudodistributive laws.
The following is the def-inition of pseudodistributive law one finds working out the conditions on a pseudo-natural transformation λ : T P → P T needed for extending a pseudomonad (
T, u, m )to the Kleisli bicategory of a pseudomonad (
P, η, µ ) in pseudoextensive form. In
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 16 practice one would likely not use this definition, but it will be needed for the laterproofs and explanation of redundant coherence axioms.
Definition 3.3.1. A pseudodistributive law (in Kleisli-decagon form) between pseu-domonads (with their modification data suppressed) ( T, u, m ) and (
P, η, µ ) is apseudonatural transformation λ : T P → P T and three invertible modifications com-prising the two triangles
T P λ / / P T T P λ / / P TP
P u = = ③③③③③③ uP a a ❉❉❉❉❉❉ ω [ c ❄❄❄❄ T ηT = = ④④④④④④ T η a a ❈❈❈❈❈❈ ω (cid:27) ❄❄❄❄ and the decagon T P T P T
T P λT / / λT P T (cid:15) (cid:15) T P T (cid:11) (cid:19) T P m / / T P T T µT / / T P T λT (cid:15) (cid:15) P T P T
P mP T (cid:15) (cid:15)
P T P m (cid:15) (cid:15)
P T P T
P λT / / P T P m / / P T µT / / P T satisfying the following two coherence axioms(D1)
T P T
TηPT / / TuPT (cid:15) (cid:15) id & & id (cid:27) (cid:27) T P T TPuPT (cid:15) (cid:15) id / / TPω T (cid:11) (cid:19) T P T TP uT (cid:15) (cid:15) id / / T P T TµT / / T P T λT (cid:15) (cid:15) T P T
TηTPT / / id (cid:15) (cid:15) ω TPT (cid:11) (cid:19)
T P T P T
TPλT / / λTPT (cid:15) (cid:15) T P T (cid:11) (cid:19) TP m A A T P T ηT PT / / mPT (cid:15) (cid:15) P T P T
PmPT (cid:15) (cid:15)
P T Pm (cid:15) (cid:15) = id Pm · λT T P T ηTPT / / λT ) ) P T P T
PλT / / P T P m / / P T µT / / P TP T ηPT rrrrrrrrrr Pm / / P T ηPT tttttttttt id ; ; ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 17 and(D2)
T P T P T P T
TPTPλT / / λTPTPT (cid:15) (cid:15) T P T P T λTP T (cid:15) (cid:15) TPTP m / / T P T P T λTP T (cid:15) (cid:15) TPTµT / / T P T P T λTPT (cid:15) (cid:15)
TPλT ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ P T P T P T
PmPTPT (cid:15) (cid:15) P TPλT / / P T P T PmP T (cid:15) (cid:15) PT P m / / P T P T PmP T (cid:15) (cid:15) PT µT / / P T P T
PmPT (cid:15) (cid:15)
T P T TP m (cid:15) (cid:15) P T P T P T
PTPλT / / PλTPT (cid:15) (cid:15) P Ω (cid:11) (cid:19) P T P T PTP m / / P T P T PTµT / / P T P T
PλT (cid:15) (cid:15)
T P T TµT (cid:15) (cid:15) P T P T P mPT (cid:15) (cid:15) P T P m (cid:15) (cid:15) Ω (cid:11) (cid:19) T P T λT (cid:15) (cid:15) P T P T P λT / / µTPT ' ' PPPPPPPPPPPP P T P m / / µPT ' ' ❖❖❖❖❖❖❖❖❖❖❖ P T PµT / / µPT & & ▼▼▼▼▼▼▼▼▼▼▼ P T µT ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ P T Pm (cid:15) (cid:15) P T P T
PλT / / P T P m / / P T µT / / P T is equal to(D2)
T P T P T P T
TPλTPT ' ' PPPPPPPPPPPP
TPTPλT / / λTPTPT (cid:15) (cid:15) T P T P T TPTP m / / T P T P T TPTµT / / T P T P T
TPλT & & ▲▲▲▲▲▲▲▲▲▲ P T P T P T
PmPTPT (cid:15) (cid:15)
T P T P T TP mPT ' ' ❖❖❖❖❖❖❖❖❖❖❖ TP Ω (cid:11) (cid:19) Ω PT (cid:11) (cid:19) T P T TP m % % ❏❏❏❏❏❏❏❏❏ P T P T P T
PλTPT (cid:15) (cid:15)
T P T P T TP λT / / TµTPT (cid:15) (cid:15)
T P T TP m / / TµPT (cid:15) (cid:15) T P T TPµT / / TµPT (cid:15) (cid:15)
T P T TµT (cid:15) (cid:15) P T P T P mPT ' ' PPPPPPPPPPPP
T P T P T
TPλT / / λTPT (cid:15) (cid:15) Ω (cid:11) (cid:19) T P T TP m / / T P T TµT / / T P T λT (cid:15) (cid:15) P T P T µTPT ' ' ❖❖❖❖❖❖❖❖❖❖❖ P T P T
PmPT (cid:15) (cid:15)
P T Pm (cid:15) (cid:15) P T P T
PλT / / P T P m / / P T µT / / P T
Pseudoalgebra definition of pseudodistributive laws.
The following def-inition is intended to provide a definition of pseudodistributive laws which is likelyminimal in its number of 2-cells and 3-cells. This result is closely related to thelater mentioned no-iteration definition of a pseudodistributive law. Without thefollowing it would likely be impractical to find the later mentioned definition of apseudodistributive law in no-iteration form.
Definition 3.4.1. A pseudodistributive law (in pseudoalgebra form) between pseu-domonads (with their modification data suppressed) ( T, u, m ) and (
P, η, µ ) is apseudonatural transformation α : T P T → P T and three invertible modifications
T P T α / / ψ [ c ❄❄❄❄ P T T T ηT / / m (cid:15) (cid:15) ψ { (cid:3) ⑧⑧⑧⑧ T P T α (cid:15) (cid:15) T P T P T
T P α / / αP T (cid:15) (cid:15) Ψ (cid:11) (cid:19) T P T T µT / / T P T α (cid:15) (cid:15) P T uP T O O id A A ✄✄✄✄✄✄✄✄ T ηT / / P T P T P T
P α / / P T µT / / P T
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 18 satisfying the two coherence axioms(M1)
T P T
TηPT / / TuPT (cid:15) (cid:15) id ) ) id (cid:27) (cid:27) T P T TPuPT (cid:15) (cid:15) id / / TPψ (cid:11) (cid:19) T P T TµT / / Ψ (cid:11) (cid:19) T P T α (cid:15) (cid:15) T P T
TηTPT / / mPT (cid:15) (cid:15) ψ PT (cid:11) (cid:19) T P T P T αPT (cid:15) (cid:15)
TPα = id α T P T ηTPT / / α - - P T P T Pα / / P T µT / / P TP T ηPT ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ id and(M2) T P T P T P T αPTPT (cid:15) (cid:15)
TPT Pα / / T P T P T TPTµT / / αP T (cid:15) (cid:15) T P T P T
TPα ) ) ❘❘❘❘❘❘❘❘❘❘ αPT (cid:15) (cid:15) T P T T µT ( ( ◗◗◗◗◗◗◗◗◗ Ψ (cid:11) (cid:19) P T P T P T
PαPT * * ❯❯❯❯❯❯❯❯❯❯❯ PT Pα / / P T P T PTµT / / P Ψ (cid:11) (cid:19) P T P T Pα ) ) ❘❘❘❘❘❘❘❘❘❘❘ T P T α (cid:15) (cid:15) P T P T µTPT ) ) ❚❚❚❚❚❚❚❚❚❚❚ P α / / P T PµT / / µPT ) ) ❘❘❘❘❘❘❘❘❘❘❘ P T µT ( ( ◗◗◗◗◗◗◗◗◗◗ P T P T Pα / / P T µT / / P T is equal to
T P T P T P T αPTPT (cid:15) (cid:15)
TPTPα / / TPαPT * * ❯❯❯❯❯❯❯❯❯❯❯ T P T P T TPTµT / / T P T P T
TPα ) ) ❘❘❘❘❘❘❘❘❘❘ TP Ψ (cid:11) (cid:19) T P T P T
TµTPT ) ) ❚❚❚❚❚❚❚❚❚❚ Ψ PT (cid:11) (cid:19) TP α / / T P T TPµT / / TµPT ) ) ❘❘❘❘❘❘❘❘❘❘ T P T TµT ( ( ◗◗◗◗◗◗◗◗◗ P T P T P T
PαPT * * ❯❯❯❯❯❯❯❯❯❯❯ T P T P T αPT (cid:15) (cid:15)
TPα / / T P T TµT / / Ψ (cid:11) (cid:19) T P T α (cid:15) (cid:15) P T P T µTPT ) ) ❚❚❚❚❚❚❚❚❚❚❚ P T P T Pα / / P T µT / / P T
No-iteration definition of pseudodistributive laws.
The following de-fines a pseudodistributive law in terms of extension operators ( − ) λX,Y , thus allowingfor a definition of pseudodistributive laws of the type λ : T P → P T involving noiteration of the pseudomonad P . This definition of pseudodistributive law appearslikely to generalize to case where P is a relative pseudomonad (the setting of [5]),though it is unclear how much of the formal theory holds in the relative case withthis definition. Note the following only assumes the existence of the composite P T ,not
T P . ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 19
Definition 3.5.1. A pseudodistributive law (in no-iteration form) between pseu-domonads (with their modification data suppressed) ( T, u, m ) and (cid:16)
P, η, ( − ) P (cid:17) ona 2-category C is a 2-pasting operator( − ) λX,Y : C ( X, P T Y ) → C ( T X, P T Y ) , X, Y ∈ C along with for all f : X → P T Y and g : Y → P T Z a family of invertible 2-cells X uX (cid:15) (cid:15) f (cid:22) (cid:22) T X mX (cid:15) (cid:15) ( ηT X ) λ (cid:22) (cid:22) T X f λ (cid:15) (cid:15) (cid:0) ( g λ ) P f (cid:1) λ (cid:22) (cid:22) ψ f { (cid:3) ⑧⑧⑧⑧ ψ X { (cid:3) ⑧⑧⑧⑧ Ψ f,g { (cid:3) ⑧⑧⑧⑧ T X f λ / / P T Y T X ηT X / / P T X P T Y ( g λ ) P / / P T Z natural in f and g , such that for all g : X → P T Y we have(I1)
T X g λ / / id (cid:24) (cid:24) P T Y ( ψ g ) λ (cid:11) (cid:19) T X
T uX (cid:15) (cid:15) ( g λ · uX ) λ / / P T YT X ( g λ ) λ / / P T Y = id g λ T X mX (cid:15) (cid:15) T X (cid:0) ( g λ ) P · ηT X (cid:1) λ / / ( ηT X ) λ (cid:15) (cid:15) P T Y ψ X { (cid:3) ⑧⑧⑧⑧ Ψ ηTX,g { (cid:3) ⑧⑧⑧⑧ T X ηT X / / g λ P T X ( g λ ) P / / P T Y and for all f : X → P T Y , g : Y → P T Z and h : Z → P T W (I2)
T X (cid:16)(cid:16)(cid:0) ( h λ ) P · g (cid:1) λ (cid:17) P · f (cid:17) λ / / P T W Ψ Pf, ( hλ ) P · g (cid:11) (cid:19) T X f λ / / P T Y (cid:16)(cid:0) ( h λ ) P · g (cid:1) λ (cid:17) P / / P T W Ψ Pg,h (cid:11) (cid:19)
T X f λ / / P T Y (cid:0) ( h λ ) P · g λ (cid:1) P / / P T WT X f λ / / P T Y ( g λ ) P / / P T Z ( h λ ) P / / P T W
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 20 is equal to
T X (cid:16)(cid:16)(cid:0) ( h λ ) P · g (cid:1) λ (cid:17) P · f (cid:17) λ / / P T W ( Ψ Pg,h · f ) λ (cid:11) (cid:19) T X (cid:16)(cid:0) ( h λ ) P · g λ (cid:1) P · f (cid:17) λ / / P T WT X (cid:0) ( h λ ) P · ( g λ ) P · f (cid:1) λ / / P T W Ψ( gλ ) P · f,h (cid:11) (cid:19) T X (cid:0) ( g λ ) P · f (cid:1) / / P T Z ( h λ ) P / / P T W Ψ f,g (cid:11) (cid:19) T X f λ / / P T Y ( g λ ) P / / P T Z ( h λ ) P / / P T W
Remark . The reader will notice that this definition involves iteration of T ,and it is unclear if this problem can nicely be avoided. The technicality is thatthe data ψ is quite strong (the weaker version coming from restricting this along T u X ). Whilst this weaker version should avoid iteration of T , one would thenneed to give additional data (corresponding to the H of Prop. 4.3.1) to recoverthe equivalence of the definitions. This additional data would then need to satisfycoherence conditions.4. Equivalence of definitions of pseudodistributive laws
As all four of our definitions of pseudodistributive laws are new, we must justifythem by showing they are equivalent to a pseudodistributive law in the sense ofMarmolejo [15]. This is the purpose of proving the following theorem, which makesuse of the equivalence between Marmolejo’s definition of pseudodistributive lawand extensions of a pseudomonad to the Kleisli bicategory shown in [4], in order tojustify our four definitions of pseudodistributive law.
Theorem 4.0.1.
Given two pseudomonads (with their modification data suppressed) ( T, u, m ) and ( P, η, µ ) on a 2-category C , the following are in equivalence: (1) a pseudodistributive law λ : T P → P T in pseudomonoidal form; (2) a pseudodistributive law λ : T P → P T in Kleisli-decagon form; (3) a pseudodistributive law α : T P T → P T in pseudoalgebra form; (4) a pseudodistributive law ( − ) λ : C ( ⋆, P T ∗ ) → C ( T ⋆, P T ∗ ) in no-iterationform; (5) an extension of ( T, u, m ) to a pseudomonad on the Kleisli bicategory of ( P, η, µ ) .Remark . We will not burden this paper with the definitions of morphisms ofpseudodistributive laws, as these are simply modifications λ ⇒ λ ′ or α ⇒ α ′ suchthat the obvious pasting diagrams agree. ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 21
Equivalence of monoidal and decagon definitions.
In order the showthe equivalence of the monoidal and decagon definitions, we first establish thecorrespondence between the data of those definitions.
Lemma 4.1.1.
For a given λ , the data ( ω , ω , ω , ω ) with axioms (W1) and (W2) is in bijection with the data ( ω , ω , Ω) with axiom (D1) .Proof. From the modifications comprising the pentagons ω and ω , the decagonΩ is constructed as the pasting diagram T P T P T
T P λT / / λT P T (cid:15) (cid:15) T P T T P m / / λP T (cid:15) (cid:15) T P T T µT / / λP T (cid:15) (cid:15) T P T λT (cid:15) (cid:15) P T P T
P mP T (cid:15) (cid:15)
P T λT / / P T P T P λT (cid:15) (cid:15) P T P m / / P T P T
P λT (cid:15) (cid:15) ω T (cid:11) (cid:19) P ω T (cid:11) (cid:19) P T P mT (cid:15) (cid:15) P T m / / P T P m (cid:15) (cid:15) µT / / P T P m (cid:15) (cid:15)
P T P T
P λT / / P T P m / / P T µT / / P T
Conversely, given the decagon Ω one recovers the pentagon ω as T P
T P u (cid:15) (cid:15) λ (cid:27) (cid:27) T P T P u / / λP (cid:15) (cid:15) T µ + + T P T T P uP T / / T P uT ' ' λP T (cid:15) (cid:15) id T P T P T
T P λT / / λT P T (cid:15) (cid:15) Ω (cid:11) (cid:19) T P ω T (cid:11) (cid:19) T P T T P m / / T P T T µT / / T P T λT (cid:15) (cid:15) P T
P T u | | id w w P T P
P T P u / / P λ $ $ P T P T
P T uP T / / id ) ) P T P T
P mP T (cid:15) (cid:15)
P T P m (cid:15) (cid:15) P T P T u id P T P T
P λT / / P T P m / / P T µT / / P T
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 22 and the pentagon ω as T P T
T P T u / / id ( ( T P T T ηP T & & ▲▲▲▲▲▲▲▲▲▲ T P m / / T P T
T ηP T % % ❏❏❏❏❏❏❏❏❏ id T P T P u / / mP (cid:15) (cid:15) T λ : : ✈✈✈✈✈✈✈✈✈ T P T
T ηT P T / / ηT P T % % mP T (cid:15) (cid:15) T λT ssssssssss T P T P T
T P λT / / λT P T (cid:15) (cid:15) Ω (cid:11) (cid:19) T P T T P m / / T P T T µT / / T P T λT (cid:15) (cid:15) T P
T P u / / λ (cid:15) (cid:15) ω T P T (cid:11) (cid:19)
T P T ηT P T % % ❑❑❑❑❑❑❑❑❑❑ λT (cid:15) (cid:15) P T P T
P mP T (cid:15) (cid:15)
P T P m (cid:15) (cid:15)
P T
P T u / / id - - P T ηP T P m % % ▲▲▲▲▲▲▲▲▲▲ P T P T
P λT / / P T P m / / P T µT / / P TP T ηP T ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ id It is then routine to verify that that these are inverse processes. (cid:3)
Lemma 4.1.2.
In the statement of Theorem 4.0.1, we have fully faithful assigna-tions (1) (2) , (2) (5) and an equivalence (5) (1) .Proof. For (1) (2) we need only note Lemma 4.1.1, and the fact that axiom (D2)involves only the decagons (which are constructed from only the pentagons), andso it routine to check that (D2) follows from the axioms (W3), (W4), (W5) whichconcern the coherence conditions of only pentagons.For (2) (5). Suppose we are given a pseudodistributive law λ : T P → P T inKleisli-decagon form. We will define a pseudomonad e T in pseudoextensive form (asin Definition 3.1.4) on the Kleisli bicategory of ( P, η, µ ). We define e T to have thesame action on objects as T . For each X ∈ Kl ( P ), we take our unit e u X : X T X to be the composite X u X / / T X η TX / / P T X
Each functor Kl ( P ) ( X, T Y ) → Kl ( P ) ( T X, T Y ), that is C ( X, P T Y ) → C ( T X, P T Y ),is defined by sending an f : X → P T Y to P mY · λT · T f : T X → P T Y . For each f : X → P T Y we take the 2-cell φ f : f ⇒ f e T · e u X , as the pasting X uX " " ❊❊❊❊❊❊❊❊ f / / P T Y uP T Y $ $ ❏❏❏❏❏❏❏❏❏ P uT Y ' ' id / / P T Y ηP T Y % % ❑❑❑❑❑❑❑❑❑ id / / P T YT X
T f / / ηT X $ $ ■■■■■■■■■■ ω T Y (cid:11) (cid:19)
T P T Y λT Y / / ηT P T Y & & ▲▲▲▲▲▲▲▲▲▲ P T Y P mY qqqqqqqqqq ηP T Y & & ▲▲▲▲▲▲▲▲▲▲ P T Y µT Y : : ✉✉✉✉✉✉✉✉✉ P T X
P T f / / P T P T Y
P λT Y / / P T Y P mY sssssssss ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 23 and for each X we take the 2-cell θ X : ( e u X ) e T ⇒ id e T X to be
T X
T uX / / id . . T X T ηT X / / ηT X mX & & T P T X λT X / / ω T X (cid:11) (cid:19)
P T X P mX / / P T XT X ηT X ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ and for each f : X → P T Y and g : Y → P T Z we take δ g,f : (cid:0) g T · f (cid:1) T ⇒ g T · f T as T X
T f / / T P T Y
T P T g / / λT Y (cid:15) (cid:15) T P T P T Z λT P T Z (cid:15) (cid:15)
T P λT Z / / T P T Z T P mZ / / T P T Z
T µT Z / / T P T Z λT Z (cid:15) (cid:15)
P T Y P mY (cid:15) (cid:15)
P T g / / P T P T Z
P mP T Z (cid:15) (cid:15) Ω Z (cid:11) (cid:19) P T Z P mZ (cid:15) (cid:15)
P T Y P T g / / P T P T Z
P λT Z / / P T Z P mZ / / P T Z µT Z / / P T Z
Naturality is clear in the above definitions. Moreover, the two axioms (D1) and(D2) ensure the two coherence conditions of a pseudomonad in extensive form aresatisfied. That these assignations are fully faithful is trivial. The fact that (1) (5)is well defined means it is an equivalence, since extensions to the Kleisli bicategoryand pseudodistributive laws in the sense of Marmolejo are in equivalence [4]. (cid:3) As it is clear the composite assignation (1) (2) (5) ≃ (1) is isomorphic tothe identity, we have the following. Corollary 4.1.3.
In the statement of Theorem 4.0.1, the data (1) , (2) and (5) arein equivalence. Explanation of redundant coherence axioms.
Let us now consider theredundant axioms of a pseudomonad in extensive form as in Remark 3.1.5. It is triv-ial to see that the leftmost axiom of this remark, which asks that a φ (constructedfrom ω ) followed by a θ (constructed from ω ) is the identity, is equivalent to thecondition (W10).Moreover, it is not hard to see the remaining two axioms of Remark 3.1.5 arerespectively equivalent to the two conditions (W8) and (W9), if one replaces thepentagons ω and ω by their definitions in terms of the decagon Ω as in Lemma4.1.1.Finally, the redundancy of (W6) and (W7) is shown directly by Marmolejo andWood [18]. However, this result can be seen more easily by noting that pseu-domonad morphisms can be seen as instances of pseudoalgebras (as is well knownin one dimension [16, 19]), and that one of the unitality axioms for a pseudoalgebrais redundant [14, Lemma 9.1]. Curiously the methods of this paper give anotherproof of the redundancy, though this proof would be less strong as it uses additionalpseudodistributive law axioms.4.3. Equivalence of decagon and algebra definitions.
We now give the nextpart of the proof of Theorem 4.0.1, by showing the equivalence of the Kleisli-decagonand pseudoalgebra formulations of a pseudodistributive law.
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 24
Proposition 4.3.1.
In the statement of Theorem 4.0.1, the data (2) and (3) arein equivalence.Proof.
We show that the ( λ, ω , ω , Ω) ( α, ψ , ψ , Ψ) defines an equivalence,where λ : T P → P T is sent to
P m · λT : T P T → P T , the units ω and ω aresent to the slightly modified units ψ and ψ , and the decagon Ω collapses into thehexagon Ψ.We first define the data which will be needed to exhibit α as an “op-homomorphism”.Note that with the order of composition reversed below, this would be the data forexhibiting α as a T -homomorphism. T P T
T P ηT % % id % % T P ψ − (cid:11) (cid:19) T P T T P T ηT / / αT (cid:15) (cid:15) T P m T P T P T
T P α / / αP T (cid:15) (cid:15) T P T T µT / / Ψ (cid:11) (cid:19) T P T α (cid:15) (cid:15) H := P T P T ηT / / P m - - P T P T
P α / / P ψ (cid:11) (cid:19) P T µT / / P TP T
P ηT : : id : : We then deduce the three consequences of the triangle and pentagon equations P T uP T $ $ ■■■■■■ id $ $ ψ (cid:11) (cid:19) P T id ! ! T ηT = = ④④④④④④ uT ! ! ❈❈❈❈❈ T P T α / / ψ (cid:11) (cid:19) P T = T ηT = = ④④④④④④ uT ! ! ❈❈❈❈❈ id P TT T ηT ; ; ✈✈✈✈✈✈ m / / T ηT = = ④④④④④④ T m / / T ηT = = ④④④④④④ P T P T Pα / / uPTPT (cid:15) (cid:15) P T µT / / uP T (cid:15) (cid:15) P T uPT (cid:15) (cid:15) id | | P T P T Pα / / uPTPT (cid:15) (cid:15) id y y P T µT / / P T id | | T P T P T
TPα / / αPT (cid:15) (cid:15) Ψ (cid:11) (cid:19) T P T TµT / / T P T α (cid:15) (cid:15) ψ { (cid:3) ⑧⑧⑧⑧ = T P T P T αPT (cid:15) (cid:15) ψ PT k s P T P T Pα / / P T µT / / P T P T P T Pα / / P T µT / / P T
T PT TPTηT / / αT (cid:15) (cid:15) T PT PT
TPα / / αPT (cid:15) (cid:15) Ψ (cid:11) (cid:19) T P T TµT / / T PT α (cid:15) (cid:15) T PT TPTηT / / αT (cid:15) (cid:15) TPm $ $ ■■■■■■■■■■■■■■ T PT PT
TPα / / TPψ (cid:11) (cid:19) T P T TµT / / T PT α (cid:15) (cid:15) = PT PTηT / / Pm $ $ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏ PT PT Pα / / Pψ (cid:11) (cid:19) P T µT / / PT PT Pm $ $ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏ H (cid:11) (cid:19) T PT
TPηT : : ✉✉✉✉✉✉✉✉✉✉✉✉✉✉ id : : α (cid:15) (cid:15) PTPT
PηT : : tttttttttttttt id : : PT id : : Curiously, one of the axioms involves H and is trivially redundant. Perhaps this happens asthis result has no lax version, i.e. the invertibility of ψ is required to construct H . ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 25
From which we show that ( α, H ) satisfies the two coherence axioms of an “opho-momorphism” below T P T T P mT / / αT (cid:15) (cid:15) T P T αT (cid:15) (cid:15) T P m $ $ ❍❍❍❍❍❍❍❍❍❍ H T { (cid:3) ⑧⑧⑧⑧ T P T T P mT / / αT (cid:15) (cid:15) T P T m $ $ ■■■■■■■■■■ T P T T P m $ $ ❍❍❍❍❍❍❍❍❍❍ P T P mT / / P T m $ $ ■■■■■■■■■■ P T P m $ $ ❍❍❍❍❍❍❍❍❍❍ T P T H { (cid:3) ⑧⑧⑧⑧ α (cid:15) (cid:15) = P T P T m $ $ ■■■■■■■■■■ T P T T P m / / αT (cid:15) (cid:15) T P T α (cid:15) (cid:15) H { (cid:3) ⑧⑧⑧⑧ P T P m / / P T P T P m / / P T
T P T
T P T u $ $ ■■■■■■■■■ α (cid:15) (cid:15) id T P T α (cid:15) (cid:15) id " " P T
P T u $ $ ❍❍❍❍❍❍❍❍❍❍ T P T T P m / / αT (cid:15) (cid:15) T P T α (cid:15) (cid:15) H { (cid:3) ⑧⑧⑧⑧ = P T
P T u ●●●●●●●●● id " " T P T α (cid:15) (cid:15) P T P m / / P T P T P m / / P T which are needed to show essential surjectivity. Note that here for a given α werecover λ as α · T P u : T P → P T . We then verify the modifications ψ , ψ , and Ψare themselves “ophomorphisms” meaning they satisfy the axioms P T uPT ❍❍❍❍❍❍❍❍❍ id / / P T Pm / / P T P T Pm / / uPT ❍❍❍❍❍❍❍❍❍ P T id / / uPT $ $ ❍❍❍❍❍❍❍❍❍❍ P T = T P T αT : : ✉✉✉✉✉✉✉✉✉✉ ψ T (cid:11) (cid:19) TPm / / T P T α < < ①①①①①①①①① H − (cid:11) (cid:19) T P T TPm / / T P T α < < ①①①①①①①①① ψ (cid:11) (cid:19) T T ηT / / mT (cid:15) (cid:15) ψ T { (cid:3) ⑧⑧⑧⑧ T P T αT (cid:15) (cid:15) T P m $ $ ❍❍❍❍❍❍❍❍❍❍ T T m " " ❋❋❋❋❋❋❋❋❋ mT (cid:15) (cid:15) T ηT / / T P T T P m $ $ ❍❍❍❍❍❍❍❍❍❍ T ηT / / m ❋❋❋❋❋❋❋❋❋❋ P T P m $ $ ■■■■■■■■■■ T P T α (cid:15) (cid:15) H { (cid:3) ⑧⑧⑧⑧ = T m ❋❋❋❋❋❋❋❋❋❋ T T ηT / / m (cid:15) (cid:15) ψ { (cid:3) ⑧⑧⑧⑧ T P T α (cid:15) (cid:15) T ηT / / P T T ηT / / P T
T P T P T TPαT / / αPT (cid:15) (cid:15) Ψ T { (cid:3) ⑧⑧⑧⑧ T P T TµT / / T P T αT (cid:15) (cid:15) TPm $ $ ❍❍❍❍❍❍❍❍❍❍ T P T P T TPTPm & & ▼▼▼▼▼▼▼▼▼▼▼▼ αPT (cid:15) (cid:15) TPαT / / T P T TµT / / TP m % % ❑❑❑❑❑❑❑❑❑❑ TP H − { (cid:3) ⑧⑧⑧⑧ T P T TPm $ $ ❍❍❍❍❍❍❍❍❍❍ P T P T PαT / / PTPm & & ▲▲▲▲▲▲▲▲▲▲▲ P T µT / / P m $ $ ❏❏❏❏❏❏❏❏❏❏ P H − { (cid:3) ⑧⑧⑧⑧ P T Pm $ $ ❍❍❍❍❍❍❍❍❍❍ T P T H { (cid:3) ⑧⑧⑧⑧ α (cid:15) (cid:15) = P T P T PTPm & & ▼▼▼▼▼▼▼▼▼▼▼▼ T P T P T
TPα / / αPT (cid:15) (cid:15) Ψ { (cid:3) ⑧⑧⑧⑧ T P T TµT / / T P T α (cid:15) (cid:15) P T P T Pα / / P T µT / / P T P T P T Pα / / P T µT / / P T which are precisely the conditions needed to show fully faithfulness. Note that itis not worth giving the full details of the proof here, as ultimately this is just an
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 26 application of well known coherence results for monoidal categories [9], though ina modified form. (cid:3)
Equivalence of algebra and no-iteration definitions.
We now finish theproof of Theorem 4.0.1, by showing the equivalence of the pseudoalgebra and no-iteration formulations of a pseudodistributive law.
Proposition 4.4.1.
In the statement of Theorem 4.0.1, the data (3) and (4) arein equivalence.Proof.
Firstly, note that given (4) one may take f and g to be identities. From thisthe data and coherence conditions of (3) are recovered. Note also that ψ , ψ andΨ become modifications by a similar argument to [20, Prop. 3.4 and 3.5].Conversely, given (3) one recovers (4) by defining the operator ( − ) λ : ( ⋆, P T ∗ ) → ( T ⋆, P T ∗ ) to send an f : X → P T Y to the composite
T X
T f / / T P T Y αY / / P T Y
It is then easy to construct ψ f from ψ , and Ψ f,g from Ψ. Note ψ has the samedata in both cases. The coherence conditions are straightforward but tedious toverify. (cid:3) Future work
Some possible directions for future work include:(1) To understand these results in terms of the formal theory of monads [21].This may involve the profunctorial explanation of the Kleisli presentationof monads given in [19].(2) Reduce the coherence axioms for mixed pseudodistributive laws, as listedin [7]. It is expected these will also reduce to five axioms, though the proofwill likely be slightly more complex.(3) Understand how this relates with the definition of a pseudodistributive lawof a pseudomonad and a KZ pseudomonad [24]. Curiously, two of thecoherence axioms used in [24] are redundant given the five axioms in thisapproach. However, the result of [24] is still likely minimal.
References [1]
J. Beck , Distributive laws , in Sem. on Triples and Categorical Homology Theory (ETH,Zürich, 1966/67), Springer, Berlin, 1969, pp. 119–140.[2]
G. L. Cattani and G. Winskel , Profunctors, open maps and bisimulation , Math. StructuresComput. Sci., 15 (2005), pp. 553–614.[3]
E. Cheng , Iterated distributive laws , Math. Proc. Cambridge Philos. Soc., 150 (2011),pp. 459–487.[4]
E. Cheng, M. Hyland, and J. Power , Pseudo-distributive laws , Electronic Notes in Theo-retical Computer Science, 83 (2003).[5]
M. Fiore, N. Gambino, M. Hyland, and G. Winskel , Relative pseudomonads, Kleislibicategories, and substitution monoidal structures , Selecta Math. (N.S.), 24 (2018), pp. 2791–2830.[6]
N. Gambino and G. Lobbia , On the formal theory of pseudomonads and pseudodistributivelaws , Theory Appl. Categ., 37 (2021), pp. No. 2, 14–56.[7]
R. Garner , Polycategories via pseudo-distributive laws , Adv. Math., 218 (2008), pp. 781–827.[8]
M. Hyland, G. Plotkin, and J. Power , Combining Computational Effects: Commutativityand Sum , Springer US, Boston, MA, 2002, pp. 474–484.
ISTRIBUTIVE LAWS, PSEUDODISTRIBUTIVE LAWS AND DECAGONS 27 [9]
G. M. Kelly , On MacLane’s conditions for coherence of natural associativities, commuta-tivities, etc , J. Algebra, 1 (1964), pp. 397–402.[10] ,
Coherence theorems for lax algebras and for distributive laws , in Category Seminar(Proc. Sem., Sydney, 1972/1973), 1974, pp. 281–375. Lecture Notes in Math., Vol. 420.[11] ,
On clubs and doctrines , in Category Seminar (Proc. Sem., Sydney, 1972/1973),Springer, Berlin, 1974, pp. 181–256. Lecture Notes in Math., Vol. 420.[12]
A. Kock , Monads for which structures are adjoint to units , J. Pure Appl. Algebra, 104(1995), pp. 41–59.[13]
E. G. Manes , Algebraic theories , Springer-Verlag, New York-Heidelberg, 1976. GraduateTexts in Mathematics, No. 26.[14]
F. Marmolejo , Doctrines whose structure forms a fully faithful adjoint string , Theory Appl.Categ., 3 (1997), pp. No. 2, 24–44.[15] ,
Distributive laws for pseudomonads , Theory Appl. Categ., 5 (1999), pp. No. 5, 91–147.[16]
F. Marmolejo, R. D. Rosebrugh, and R. J. Wood , A basic distributive law , J. Pure Appl.Algebra, 168 (2002), pp. 209–226. Category theory 1999 (Coimbra).[17]
F. Marmolejo and A. Vázquez-Márquez , No-iteration mixed distributive laws , Math.Structures Comput. Sci., 27 (2017), pp. 1–16.[18]
F. Marmolejo and R. J. Wood , Coherence for pseudodistributive laws revisited , TheoryAppl. Categ., 20 (2008), pp. No. 5, 74–84.[19] ,
Monads as extension systems—no iteration is necessary , Theory Appl. Categ., 24(2010), pp. No. 4, 84–113.[20] ,
No-iteration pseudomonads , Theory Appl. Categ., 28 (2013), pp. No. 14, 371–402.[21]
R. Street , The formal theory of monads , J. Pure Appl. Algebra, 2 (1972), pp. 149–168.[22]
M. Tanaka , Pseudo-Distributive Laws and a Unified Framework for Variable Binding , PhDthesis, University of Edinburgh, 2004.[23]
M. Tanaka and J. Power , Pseudo-distributive laws and axiomatics for variable binding ,Higher-Order and Symbolic Computation, 19 (2006).[24]
C. Walker , Distributive laws via admissibility , Appl. Categ. Structures, 27 (2019), pp. 567–617.[25]
R. F. C. Walters , A categorical approach to universal algebra , PhD thesis, Australian Na-tional University, 1970.[26]
V. Zöberlein , Doctrines on -categories , Math. Z., 148 (1976), pp. 267–279. Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, Brno61137, Czech Republic
Email address ::