Theory and Applications of Categories, Vol. 26, No. 2, 2012, pp. 30–59.

ON THE ITERATION OF WEAK WREATH PRODUCTS

GABRIELLA BÖHM

Abstract. Based on a study of the 2-category of weak distributive laws, we describe a method of iterating Street’s weak wreath product construction. That is, for any 2-category K and for any non-negative integer n, we introduce 2-categories Wdl^{(n)}(K), of (n + 1)-tuples of monads in K pairwise related by weak distributive laws obeying the Yang-Baxter equation. The first instance Wdl^{(0)}(K) coincides with Mnd(K), the usual 2-category of monads in K, and for other values of n, Wdl^{(n)}(K) contains Mnd^{n+1}(K) as a full 2-subcategory. For the local idempotent closure \overline{K} of K, extending the multiplication of the 2-monad Mnd, we equip these 2-categories with n possible ‘weak wreath product’ 2-functors Wdl^{(n)}(\overline{K}) → Wdl^{(n−1)}(\overline{K}), such that all of their possible n-fold composites Wdl^{(n)}(\overline{K}) → Wdl^{(0)}(\overline{K}) are equal; that is, such that the weak wreath product is ‘associative’. Whenever idempotent 2cells in K split, this leads to pseudofunctors Wdl^{(n)}(K) → Wdl^{(n−1)}(K) obeying the associativity property up-to isomorphism. We present a practically important occurrence of an iterated weak wreath product: the algebra of observable quantities in an Ising type quantum spin chain where the spins take their values in a dual pair of finite weak Hopf algebras. We also construct a fully faithful embedding of Wdl^{(n)}(\overline{K}) into the 2-category of commutative n + 1 dimensional cubes in Mnd(\overline{K}) (hence into the 2-category of commutative n + 1 dimensional cubes in K whenever K has Eilenberg-Moore objects and its idempotent 2-cells split). Finally we give a sufficient and necessary condition on a monad in \overline{K} to be isomorphic to an n-ary weak wreath product.

Introduction

At the heart of the iteration of wreath products in the work [Cheng (2011)] of Eugenia Cheng, lies the 2-monad Mnd on the 2-category 2-Cat of 2-categories, 2-functors and 2-natural transformations, first discussed in [Street (1972)]. For any 2-category K, the iteration of its associative multiplication Mnd^{n}(K) → Mnd^{n−1}(K) → … → Mnd(K) takes an (n + 1)-tuple of monads, pairwise related by distributive laws obeying the Yang-Baxter equality, to a unique monad in K. The resulting monad (that appeared already in [Jara Martínez et al. (2008)] in the case when K is the monoidal category of vector spaces) can be interpreted as an iterated wreath product.

The aim of this paper is to find a similar iteration process for weak wreath products introduced by Ross Street in [Street (2009)] and by Stefaan Caenepeel and Erwin De Groot in [Caenepeel, De Groot (2000)].

These weak wreath products are defined in 2-categories in which idempotent 2-cells split, see [Street (2009)]. They are induced by weak distributive laws; that is, certain

Received by the editors 2011-10-09 and, in revised form, 2012-01-25. Transmitted by R.J. Wood. Published on 2012-01-30. 2000 Mathematics Subject Classification: 18C15, 18D05, 16W30. Key words and phrases: monad, weak distributive law, n-ary weak wreath product, Yang-Baxter

equation, quantum spin chain.

\begin{matrix}c \\ ◯ \end{matrix} Gabriella Böhm, 2012. Permission to copy for private use granted.

2-cells relating two monads. They obey the usual compatibility conditions of distributive laws with the multiplications of the monads, but the compatibility conditions with the units are weakened [Caenepeel, De Groot (2000)], [Street (2009)]. Making weak distributive laws conceptually different from their non-weak counterparts, they are not known to be monads in any 2-category. (However, a weak distributive law can be characterized as a pair of monads in 2-categories extending Mnd(K) and its variant Mnd^{∗}(K), respectively, see [Böhm (2010)]).

In Section 2, for any 2-category K, we construct a 2-category Wdl^{(n)}(K) for every non-negative integer n. Its objects are (n + 1)-tuples of monads in K pairwise related by weak distributive laws obeying the Yang-Baxter equation. The first one, Wdl^{(0)}(K) is isomorphic to Mnd(K), the 2-category of monads in K as defined in [Street (1972)]. The next one, Wdl^{(1)}(K) is the 2-category of weak distributive laws, obtained by dualizing the definition in [Böhm, Lack, Street (2011)]. For every n, Wdl^{(n)}(K) contains Mnd^{n+1}(K) as a full 2-subcategory. But, in contrast to the classical (that is, non-weak) case, Wdl^{(n)}(K) is not known to arise by the (n + 1)-fold application of some 2-monad. Although in this way we can not interpret them as multiplications of some 2-monad, for each value of n we describe n different 2-functors Wdl^{(n)}(\overline{K}) → Wdl^{(n−1)}(\overline{K}) (where \overline{K} denotes the local idempotent closure of K). They extend the n possible multiplications Mnd^{n+1}(\overline{K}) → Mnd^{n}(\overline{K}). They give rise to a unique composite Wdl^{(n)}(\overline{K}) → Wdl^{(0)}(\overline{K}) whose value on an object of Wdl^{(n)}(\overline{K}) is regarded as the (associatively iterated) weak wreath product of the n + 1 occurring monads in \overline{K}. Whenever idempotent 2-cells in K split; that is, \overline{K} and K are biequivalent, our construction yields pseudofunctors Wdl^{(n)}(K) → Wdl^{(n−1)}(K) giving rise to a composite Wdl^{(n)}(K) → Wdl^{(0)}(K) which is unique up-to a pseudonatural equivalence in the choice of the biequivalence \overline{K} → K.

Our motivation to study iterated weak wreath products comes from mathematical physics. The Ising model is a quantum spin chain in which the spins take their values in the sign group ℤ(2). In its various generalizations, the spins may take their values in arbitrary finite groups [Szlachányi, Vecsernyés (1993)], in finite dimensional Hopf algebras [Nill, Szlachányi (1997)] or in finite dimensional weak Hopf algebras [Nill, Szlachányi, Wiesbrock (preprint 1998)], [Böhm (1997)]. In all of these, except the last quoted family of models, the algebra of the observable quantities in any finite interval is given by an iterated wreath product. In quantum spin chains based on weak Hopf algebras, however, the algebras of observables are iterated weak wreath products. In Section 3 we present this example in some detail.

The definition of Wdl^{(n)}(\overline{K}) is further motivated in Section 4 by a fully faithful embedding of it into the 2-category of n + 1 dimensional cubes in the 2-category of monads in \overline{K}. Whenever idempotent 2-cells in K split, this yields a fully faithful embedding of Wdl^{(n)}(K) into the 2-category of n + 1 dimensional cubes in the 2category of monads in K; and also into the 2-category of n + 1 dimensional cubes in K whenever in addition K admits Eilenberg-Moore objects for monads.

In our final Section 5 we analyze the n-ary factorization problem associated to weak distributive laws. That is, we give a complete characterization of those monads in the local idempotent closure of a 2-category which arise as iterated weak wreath products of n monads.

Throughout, for a technical simplification, we work with 2-categories. There is no difficulty, however, to extend our considerations to bicategories.

Acknowledgement I would like to express my gratitude to Ross Street and Kornél Szlachányi for their helpful comments on this work. Partial support by the Hungarian Scientific Research Fund OTKA K68195 is gratefully acknowledged.

1. Preliminaries on weak distributive laws

In this section we revisit some recent ‘weak’ generalizations of the formal theory of monads that will be used in the sequel.

1.1. Local idempotent closure To any 2-category K we associate another 2category \overline{K} by freely splitting idempotent 2-cells. In more detail, the 0-cells of \overline{K} are the same as those in K. The 1-cells in \overline{K} are pairs consisting of a 1-cell v and a 2-cell \overline{v} : v → v in K such that \overline{v}.\overline{v} = \overline{v}; that is, \overline{v} is idempotent. The 2-cells (v, \overline{v}) → (v′, \overline{v}′) in \overline{K} are 2-cells ω : v → v′ in K such that \overline{v}′.ω = ω = ω.\overline{v}. Horizontal and vertical compositions in \overline{K} are induced by those in K. The identity 2-cell is \overline{v} : (v, \overline{v}) → (v, \overline{v}).

Throughout, we shall use the notation seen above: If it is not otherwise stated, in a 1-cell in \overline{K}, for the idempotent 2-cell part we use the overlined version of the same symbol that denotes the 1-cell part.

For any 2-category K, there is an evident inclusion 2-functor K → \overline{K}, acting on the 0-cells as the identity map, taking a 1-cell v to (v, v) – that is, the 1-cell with identity 2-cell part – and acting on the 2-cells again as the identity map.

We say that idempotent 2-cells in a 2-category K split if, for any idempotent 2-cell θ : v → v there exist a 1-cell w and 2-cells ι : w → v and π : v → w such that π.ι = w and ι.π = θ. If the splitting exists then it is unique up-to an isomorphism of w. Clearly, in \overline{K} idempotent 2-cells split for any 2-category K.

Whenever in K idempotent 2-cells split, the inclusion K → \overline{K} becomes a biequivalence. (Since it acts on the 0-cells as the identity map, this simply means that it induces an equivalence of the hom categories.) Hence there is a pseudofunctor \overline{K} → K which is its inverse (in the sense of inverse biequivalences). On the 0-cells also this pseudofunctor acts as the identity map. On a 1-cell (v, \overline{v}) its action is constructed via a chosen splitting of the idempotent 2-cell \overline{v}. If (ι : w → v, π : v → w) is this chosen splitting, then the image of (v, \overline{v}) is w. A 2-cell ω : (v, \overline{v}) → (v′, \overline{v′}) is taken to [figure: w; ι; v; ω; v′; π′; w′] . Let us stress that the biequivalence \overline{K} → K is not a 2-functor in general and it is unique only up-to a pseudonatural equivalence arising from the choice of the splitting of each idempotent 2-cell.

1.2. Demimonads In simplest terms, a demimonad in a 2-category is a monad (A, (t, \overline{t})) in the local idempotent closure, cf. [Böhm, Lack, Street (2012)]. Explicitly, it is given by a 1-cell t : A → A and 2-cells μ : t^{2} → t and η : 1_{A} → t such that the following diagrams commute.

\xymatrix{t^{3} \ar[r]^{μt} \ar[d]_{tμ} & t^{2} \ar[d]^{μ} & t \ar[r]^{ηt} \ar[d]_{tη} & t^{2} \ar[d]^{μ} & 1_{A} \ar[r]^{ηη} \ar[dr]_{η} & t^{2} \ar[d]^{μ} & t^{2} \ar[r]^{ηt^{2}} \ar[dr]_{μ} & t^{3} \ar[d]^{μ^{2}} \\ t^{2} \ar[r]_{μ} & t & t^{2} \ar[r]_{μ} & t &  & t &  & t}

For a monad (A, (t, \overline{t})) in the local idempotent closure, the 2-cell η being a two-sided unit for μ means that the idempotent 2-cell \overline{t} must be equal to μ.tη = μ.ηt (hence it is a redundant information that will be often omitted in the sequel). This structure occurred in [Böhm (2010)] under the name ‘pre-monad’.

A demimonad (A, (t, \overline{t})) is the image of a monad under the inclusion K → \overline{K} if and only if \overline{t} is the identity 2-cell t.

1.3. Weak distributive laws Extending the notion of distributive law due to Jon Beck (see [Beck (1969)]), weak distributive laws in a 2-category were introduced by Ross Street in [Street (2009)] as follows. They consist of two monads (A, t) and (A, s) on the same object, and a 2-cell λ : ts → st such that the following diagrams commute.

\xymatrix{t^{2}s \ar[r]^{tλ} \ar[d]^{μs} & tst \ar[r]^{λt} & st^{2} \ar[d]_{sμ} & s \ar[rrr]^{ηs} \ar[d]^{sη} &  &  & ts \ar[d]_{λ} & ts^{2} \ar[r]^{λs} \ar[d]^{tμ} & sts \ar[r]^{sλ} & s^{2}t \ar[d]_{μt} & t \ar[rrr]^{tη} \ar[d]^{ηt} &  &  & ts \ar[d]_{λ} \\ ts \ar[rr]_{λ} &  & st & st \ar[r]_{stη} & sts \ar[r]_{sλ} & s^{2}t \ar[r]_{μt} & st & ts \ar[rr]_{λ} &  & st & st \ar[r]_{ηst} & tst \ar[r]_{λt} & st^{2} \ar[r]_{sμ} & st} (1)

The same set of axioms occurred also in [Caenepeel, De Groot (2000)]. By Proposition 2.2 in [Street (2009)], the second and fourth diagrams can be replaced by a single diagram

\xymatrix{st \ar[r]^{ηst} \ar[d]_{stη} & tst \ar[r]^{λt} & st^{2} \ar[d]^{sμ} \\ sts \ar[r]_{sλ} & s^{2}t \ar[r]_{μt} & st .}  (2)

The equal paths around (2) give rise to an idempotent 2-cell \overline{λ} : st → st (which occurs also in the bottom rows of the second and fourth diagrams in (1)). It is an identity if and only if λ is a distributive law in the strict sense. Since (2) generalizes the ‘unitality conditions’ λ.tη = ηt and λ.ηs = sη on a distributive law λ, we refer to it as the ‘weak unitality condition’.

Note that a weak distributive law in K is the same as a weak distributive law in K^{op}, the dual of K with respect to 1-cells.

A weak distributive law in \overline{K} is then given by demimonads (A, t) and (A, s) and a 2-cell λ : ts → st rendering commutative the diagrams in (1) and obeying in addition the normalization conditions

\xymatrix{ts \ar[r]^{tμ.tηs} \ar[d]_{λ} \ar[dr]^{λ} & ts \ar[d]^{λ} & ts \ar[r]^{μs.tηs} \ar[d]_{λ} \ar[dr]^{λ} & ts \ar[d]^{λ} \\ st \ar[r]_{μt.sηt} & st & st \ar[r]_{sμ.sηt} & st .}

In the sequel we shall need some identities on weak distributive laws (in \overline{K}). The axioms imply commutativity of the following diagrams, see [Street (2009)].

\xymatrix{ts \ar[r]^{λ} \ar[ddr]_{λ} & st \ar[dd]^{\overline{λ}} & s^{2}t^{2} \ar[r]^{s\overline{λ}t} \ar[dd]_{μμ} & s^{2}t^{2} \ar[dd]^{μμ} & sts \ar[r]^{sλ} \ar[d]_{\overline{λ}s} & s^{2}t \ar[dd]^{μt} & tst \ar[r]^{λt} \ar[d]_{t\overline{λ}} & st^{2} \ar[dd]^{sμ} \\  &  &  &  & sts \ar[d]_{sλ} &  & tst \ar[d]_{λt} \\  & st & st \ar[r]_{\overline{λ}} & st & s^{2}t \ar[r]_{μt} & st & st^{2} \ar[r]_{sμ} & st}  (3)

Moreover, by the associativity of μ, the left-bottom path in the last diagram in (1) commutes with the multiplication by t on the right; in the sense that sμ.\overline{λ}t.ηt^{2} = \overline{λ}.ηt.μ. Hence also the top-right path in the last diagram in (1) commutes with the right multiplication by t, meaning the commutativity of the first diagram in

\xymatrix{t^{2} \ar[d]_{μ} \ar[r]^{tηt} & tst \ar[r]^{λt} & st^{2} \ar[d]^{sμ} & s^{2} \ar[d]_{μ} \ar[r]^{sηs} & sts \ar[r]^{sλ} & s^{2}t \ar[d]^{μt} \\ t \ar[r]_{tη} & ts \ar[r]_{λ} & st & s \ar[r]_{ηs} & ts \ar[r]_{λ} & st.}  (4)

Commutativity of the second diagram follows symmetrically.

1.4. The 2-category of weak distributive laws Dualizing in the appropriate sense the definition of the 2-category of mixed weak distributive laws in the paper [Böhm, Lack, Street (2011)], the following 2-category Wdl(K) of weak distributive laws in K is obtained (see [Böhm, Gómez-Torrecillas (preprint 2011), Paragraph 1.9]). The 0-cells are the weak distributive laws λ : ts → st. The 1-cells λ → λ′ are triples consisting of a 1-cell v : A → A′ and 2-cells ξ : t′v → vt and ζ : s′v → vs in K, such that (v, ξ) : (A, t) → (A′, t′) and (v, ζ) : (A, s) → (A′, s′) are 1-cells in Mnd(K) (also called monad morphisms in [Street (1972)]) and the following diagram commutes.

\xymatrix{t′s′v \ar[rr]^{t′ζ} \ar[d]_{λ′v} &  & t′vs \ar[rr]^{ξs} &  & vts \ar[d]^{vλ} \\ s′t′v \ar[r]_{s′ξ} & s′vt \ar[rr]_{ζt} &  & vst \ar[r]_{v\overline{λ}} & vst}  (5)

The 2-cells (v, ξ, ζ) → (v′, ξ′, ζ′) are 2-cells ω : v → v′ in K which are 2-cells in Mnd(K) (called monad transformations in [Street (1972)]); both (v, ξ) → (v′, ξ′) and (v, ζ) → (v′, ζ′). Horizontal and vertical compositions are induced by those in K. As in [Böhm, Lack, Street (2011)], this definition can be interpreted in terms of (weak) liftings.

There is a fully faithful embedding Mnd^{2}(K) → Wdl(K) as follows. It takes a 0-cell ((A, t), (s, λ)) to the distributive law λ : ts → st, regarded as a weak distributive law. It takes a 1-cell ((v, ξ), ζ) to (v, ξ, ζ) and it acts on the 2-cells as the identity map.

1.5. Weak wreath product The weak wreath product induced by a weak distributive law in a 2-category in which idempotent 2-cells split, was discussed by Ross Street in [Street (2009), Theorem 2.4]. In the particular case of the monoidal category (that is, one object bicategory) of modules over a commutative ring, it appeared in [Caenepeel, De Groot (2000), Theorem 3.2].

For an arbitrary 2-category K, there is a weak wreath product 2-functor Wdl(\overline{K}) → Mnd(\overline{K}), which sends a weak distributive law λ : ts → st to the monad (st, \overline{λ}) in \overline{K}, with multiplication and unit

\xymatrix{(st)^{2} \ar[r]^{sλt} & s^{2}t^{2} \ar[r]^{μμ} & st & and & 1 \ar[r]^{ηη} & ts \ar[r]^{λ} & st .}

It sends a 1-cell ((v, \overline{v}), ξ, ζ) : λ → λ′ to the monad morphism with the same 1-cell part (v, \overline{v}) and the 2-cell part

\xymatrix{s′t′v \ar[r]^{s′ξ} & s′vt \ar[r]^{ζt} & vst \ar[r]^{v\overline{λ}} & vst .}

On the 2-cells it acts as the identity map.

Whenever idempotent 2-cells in K split, the biequivalence \overline{K} ≃ K induces a pseudofunctor Wdl(K) →^{≃} Wdl(\overline{K}) → Mnd(\overline{K}) →^{≃} Mnd(K). (It can be chosen, in fact, to be a 2-functor by choosing the biequivalence \overline{K} → K adopting the convention that we split identity 2-cells trivially; that is, via identity 2-cells.) Its object map yields Street’s weak wreath product in K.

1.6. Binary factorization Let K be any 2-category. As proved in the work [Böhm, Gómez-Torrecillas (preprint 2011)], a demimonad (A, r) is isomorphic to a weak wreath product induced by some weak distributive law ts → st in \overline{K} if and only if the following hold.

(a) There are 1-cells in Mnd(\overline{K}) with trivial 1-cell parts

\xymatrix{(A, (t, \overline{t})) & (A, (r, \overline{r})) \ar[l]_{((A,A),α)} \ar[r]^{((A,A),β)} & (A, (s, \overline{s})) ;}

(b) The 2-cell

π := ( \xymatrix{(st, \overline{s}\overline{t}) \ar[r]^{βα} & (rr, \overline{r} \overline{r}) \ar[r]^{μ} & (r, \overline{r})} )

in \overline{K} possesses section ι (meaning π.ι = \overline{r} ≡ μ.rη) which is an s-t bimodule morphism with respect to the t- and s-actions induced on r by α and β, respectively.

Indeed, for the weak wreath product induced by a weak distributive law λ : ts → st, we have 1-cells

\xymatrix{(A, (t, \overline{t})) & (A, (st, \overline{λ})) \ar[l]_{((A,A),λ.tη)} \ar[r]^{((A,A),λ.ηs)} & (A, (s, \overline{s}))}

in Mnd(\overline{K}). Moreover, the 2-cell π in part (b) comes out as

( \xymatrix{(st, \overline{s}\overline{t}) \ar[r]^{λλ.ηstη} & (stst, \overline{λ} \overline{λ}) \ar[r]^{μμ.sλt} & (st, \overline{λ})} ) = ( \xymatrix{(st, \overline{s}\overline{t}) \ar[r]^{\overline{λ}} & (st, \overline{λ})} ) ,

which is split by the bimodule morphism \overline{λ} : (st, \overline{λ}) → (st, \overline{s}\overline{t}).

Conversely, if properties (a) and (b) hold, then

\xymatrix{ts \ar[r]^{αβ} & rr \ar[r]^{μ} & r \ar[r]^{ι} & st}

is a weak distributive law with corresponding idempotent equal to ι.π : st → st. The isomorphism between the induced weak wreath product and (A, r) is provided by \xymatrix{(st, ι.π) \ar[r]^{π} & (r, \overline{r}) \ar[l]^{ι}} in \overline{K}. For the details of the proof we refer to the article [Böhm, Gómez-Torrecillas (preprint 2011)].

2. 2-categories of weak distributive laws and the iterated weak wreath product

Throughout this section, K is an arbitrary 2-category and \overline{K} stands for its local idempotent closure. For any non-negative integer n, we define a 2-category Wdl^{(n)}(K). Its objects are (n + 1)-tuples of monads pairwise related by weak distributive laws obeying the Yang-Baxter condition (see (6) below). For each value of n, we construct n different 2-functors Wdl^{(n)}(\overline{K}) → Wdl^{(n−1)}(\overline{K}) corresponding to taking the weak wreath product of two consecutive monads of the n + 1 occurring ones. We show that these 2-functors give rise to a unique composite Wdl^{(n)}(\overline{K}) → Wdl^{(0)}(\overline{K}) = Mnd(\overline{K}). We regard its object map as the n-ary weak wreath product of the involved monads.

2.1. The 2-category Wdl^{(n)}(K) For any non-negative integer n, a 0-cell is given by n + 1 monads s_{0}, s_{1}, …, s_{n} together with weak distributive laws λ_{i,j} : s_{j}s_{i} → s_{i}s_{j} for all 0 ≤ i < j ≤ n, obeying for all 0 ≤ i < j < k ≤ n the Yang-Baxter relation

\xymatrix{s_{k}s_{j}s_{i} \ar[r]^{λ_{j,k}s_{i}} \ar[d]_{s_{k}λ_{i,j}} & s_{j}s_{k}s_{i} \ar[r]^{s_{j}λ_{i,k}} & s_{j}s_{i}s_{k} \ar[d]^{λ_{i,j}s_{k}} \\ s_{k}s_{i}s_{j} \ar[r]_{λ_{i,k}s_{j}} & s_{i}s_{k}s_{j} \ar[r]_{s_{i}λ_{j,k}} & s_{i}s_{j}s_{k}.}  (6)

The 1-cells consist of a 1-cell v and 2-cells ξ_{i} : s′_{i}v → vs_{i} in K for all 0 ≤ i ≤ n, such that (v, ξ_{i}, ξ_{j}) is a 1-cell λ_{i,j} → λ′_{i,j} in Wdl(K) (see Paragraph 1.4), for all 0 ≤ i < j ≤ n. The 2-cells are those 2-cells ω : v → v′ in K which are 2-cells (v, ξ_{i}, ξ_{j}) → (v′, ξ′_{i}, ξ′_{j}) in Wdl(K) (in the sense of Paragraph 1.4), for all 0 ≤ i < j ≤ n. Since Wdl(K) is closed under the horizontal and vertical compositions in K, so is Wdl^{(n)}(K). Hence it is a 2-category with the horizontal and vertical compositions induced by those in K.

Recall from [Cheng (2011)] that a 0-cell in Mnd^{n}(K) is given by n monads, pairwise related by distributive laws obeying the Yang-Baxter condition. The 1-cells consist of n monad morphisms for the n involved monads with a common underlying 1-cell, obeying (5) (though in that case the occurring idempotents are identities). The 2cells are those 2-cells in K which are monad transformations for all of the n monad morphisms. With this description in mind, extending that in Paragraph 1.4, there is an evident fully faithful embedding Mnd^{n+1}(K) → Wdl^{(n)}(K).

Taking any m + 1-element subset of {0, 1, …, n} induces an evident 2-functor Wdl^{(n)}(K) → Wdl^{(m)}(K).

2.2. Lemma. Take any object \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{0≤i<j≤2} of Wdl^{(2)}(\overline{K}). In addition to \overline{s}_{i} : s_{i} → s_{i} for 0 ≤ i ≤ 2, and \overline{λ}_{i,j} : s_{i}s_{j} → s_{i}s_{j} for 0 ≤ i < j ≤ 2, let us introduce the the following idempotent 2-cells in K.

\xymatrix{\overrightarrow{λ}_{0,p,q} := (s_{0}s_{p}s_{q} \ar[r]^{s_{0}s_{p}s_{q}η_{0}} & s_{0}s_{p}s_{q}s_{0} \ar[r]^{s_{0}s_{p}λ_{0,q}} & s_{0}s_{p}s_{0}s_{q} \ar[r]^{s_{0}λ_{0,p}s_{q}} & s_{0}^{2}s_{p}s_{q} \ar[r]^{μ_{0}s_{p}s_{q}} & s_{0}s_{p}s_{q}), \\ \overleftarrow{λ}_{k,l,2} := (s_{k}s_{l}s_{2} \ar[r]^{η_{2}s_{k}s_{l}s_{2}} & s_{2}s_{k}s_{l}s_{2} \ar[r]^{λ_{k,2}s_{l}s_{2}} & s_{k}s_{2}s_{l}s_{2} \ar[r]^{s_{k}λ_{l,2}s_{2}} & s_{k}s_{l}s_{2}^{2} \ar[r]^{s_{k}s_{l}μ_{2}} & s_{k}s_{l}s_{2}),}

for p = 1, q = 2 and p = 2, q = 1; and for k = 0, l = 1 and k = 1, l = 0. They obey the following equalities.

\overrightarrow{λ}_{0,p,q}.\overline{λ}_{0,p}s_{q} = \overrightarrow{λ}_{0,p,q} = \overline{λ}_{0,p}s_{q}.\overrightarrow{λ}_{0,p,q}  (7)
\overrightarrow{λ}_{0,1,2}.s_{0}λ_{1,2} = s_{0}λ_{1,2}.\overrightarrow{λ}_{0,2,1}  (8)
\overrightarrow{λ}_{0,p,q}.λ_{0,p}s_{q} = λ_{0,p}s_{q}.s_{p}\overline{λ}_{0,q}  (9)
\overleftarrow{λ}_{k,l,2}.s_{k}\overline{λ}_{l,2} = \overleftarrow{λ}_{k,l,2} = s_{k}\overline{λ}_{l,2}.\overleftarrow{λ}_{k,l,2}  (10)
\overleftarrow{λ}_{0,1,2}.λ_{0,1}s_{2} = λ_{0,1}s_{2}.\overleftarrow{λ}_{1,0,2}  (11)
\overleftarrow{λ}_{k,l,2}.s_{k}λ_{l,2} = s_{k}λ_{l,2}.\overline{λ}_{k,2}s_{l}  (12)
\overleftarrow{λ}_{0,1,2}.\overrightarrow{λ}_{0,1,2} = \overleftarrow{λ}_{0,1,2}.\overline{λ}_{0,1}s_{2} = \overrightarrow{λ}_{0,1,2}.s_{0}\overline{λ}_{1,2} = \overrightarrow{λ}_{0,1,2}.\overleftarrow{λ}_{0,1,2} .  (13)

In what follows, we shall denote by \overline{λ}_{012} the equal 2-cells in (13) (we shall see later the irrelevance of inserting any comma between the labels).

Proof. We only present a proof of (13), verification of the other equalities is left to the reader.

The first and the last expressions in (13) are equal by commutativity of the following diagram.

[figure: s_{0}s_{1}s_{2}; s_{0}s_{1}s_{2}η_{0}; η_{2}s_{0}s_{1}s_{2}; s_{0}s_{1}s_{2}s_{0}; s_{0}s_{1}λ_{0,2}; s_{0}s_{1}s_{0}s_{2}; s_{0}λ_{0,1}s_{2}; s_{0}^{2}s_{1}s_{2}; μ_{0}s_{1}s_{2}; s_{0}s_{1}s_{2}; η_{2}s_{0}s_{1}s_{2}; s_{2}s_{0}s_{1}s_{2}; s_{2}s_{0}s_{1}s_{2}η_{0}; λ_{0,2}s_{1}s_{2}; s_{2}s_{0}s_{1}s_{2}s_{0}; s_{2}s_{0}s_{1}λ_{0,2}; s_{2}s_{0}s_{1}s_{0}s_{2}; s_{2}s_{0}λ_{0,1}s_{2}; s_{2}s_{0}^{2}s_{1}s_{2}; s_{2}μ_{0}s_{1}s_{2}; λ_{0,2}s_{0}s_{1}s_{2}; (1); s_{2}s_{0}s_{1}s_{2}; λ_{0,2}s_{1}s_{2}; s_{0}s_{2}s_{1}s_{2}; s_{0}s_{2}s_{1}s_{2}η_{0}; s_{0}λ_{1,2}s_{2}; s_{0}s_{2}s_{1}s_{2}s_{0}; s_{0}s_{2}s_{1}λ_{0,2}; s_{0}s_{2}s_{1}s_{0}s_{2}; s_{0}s_{2}λ_{0,1}s_{2}; s_{0}λ_{1,2}s_{0}s_{2}; (YB); s_{0}s_{2}s_{0}s_{1}s_{2}; s_{0}λ_{0,2}s_{1}s_{2}; s_{0}s_{1}s_{2}^{2}; s_{0}s_{1}s_{2}^{2}η_{0}; s_{0}s_{1}μ_{2}; s_{0}s_{1}s_{2}^{2}s_{0}; s_{0}s_{1}s_{2}λ_{0,2}; s_{0}s_{1}μ_{2}s_{0}; (1); s_{0}s_{1}s_{2}s_{0}s_{2}; s_{0}s_{1}λ_{0,2}s_{2}; s_{0}^{2}s_{2}s_{1}s_{2}; μ_{0}s_{2}s_{1}s_{2}; s_{0}^{2}λ_{1,2}s_{2}; s_{0}s_{2}s_{1}s_{2}; s_{0}λ_{1,2}s_{2}; s_{0}s_{1}s_{0}s_{2}^{2}; s_{0}λ_{0,1}s_{2}^{2}; s_{0}s_{1}s_{0}μ_{2}; s_{0}^{2}s_{1}s_{2}^{2}; μ_{0}s_{1}s_{2}^{2}; s_{0}^{2}s_{1}μ_{2}; s_{0}s_{1}s_{2}^{2}; s_{0}s_{1}μ_{2}; s_{0}s_{1}s_{2}; s_{0}s_{1}s_{2}η_{0}; s_{0}s_{1}s_{2}s_{0}; s_{0}s_{1}λ_{0,2}; s_{0}s_{1}s_{0}s_{2}; s_{0}λ_{0,1}s_{2}; s_{0}^{2}s_{1}s_{2}; μ_{0}s_{1}s_{2}; s_{0}s_{1}s_{2}]

The first and the third expressions in (13) are equal by commutativity of the following diagram.

[figure: s_{0}s_{1}s_{2}; s_{0}s_{1}s_{2}η_{0}; s_{0}η_{2}s_{1}s_{2}; s_{0}s_{1}s_{2}s_{0}; s_{0}s_{1}λ_{0,2}; s_{0}s_{1}s_{0}s_{2}; s_{0}λ_{0,1}s_{2}; s_{0}^{2}s_{1}s_{2}; μ_{0}s_{1}s_{2}; s_{0}η_{2}s_{0}s_{1}s_{2}; (4); s_{0}s_{1}s_{2}; η_{2}s_{0}s_{1}s_{2}; s_{0}s_{2}s_{1}s_{2}; s_{0}s_{2}s_{1}s_{2}η_{0}; s_{0}λ_{1,2}s_{2}; s_{0}s_{2}s_{1}s_{2}s_{0}; s_{0}s_{2}s_{1}λ_{0,2}; s_{0}s_{2}s_{1}s_{0}s_{2}; s_{0}s_{2}λ_{0,1}s_{2}; s_{0}λ_{1,2}s_{0}s_{2}; (YB); s_{0}s_{2}s_{0}s_{1}s_{2}; s_{0}λ_{0,2}s_{1}s_{2}; s_{2}s_{0}s_{1}s_{2}; λ_{0,2}s_{1}s_{2}; s_{0}s_{1}s_{2}^{2}; s_{0}s_{1}s_{2}^{2}η_{0}; s_{0}s_{1}μ_{2}; s_{0}s_{1}s_{2}^{2}s_{0}; s_{0}s_{1}s_{2}λ_{0,2}; s_{0}s_{1}μ_{2}s_{0}; (1); s_{0}s_{1}s_{2}s_{0}s_{2}; s_{0}s_{1}λ_{0,2}s_{2}; s_{0}^{2}s_{2}s_{1}s_{2}; μ_{0}s_{2}s_{1}s_{2}; s_{0}^{2}λ_{1,2}s_{2}; s_{0}s_{2}s_{1}s_{2}; s_{0}λ_{1,2}s_{2}; s_{0}s_{1}s_{0}s_{2}^{2}; s_{0}λ_{0,1}s_{2}^{2}; s_{0}s_{1}s_{0}μ_{2}; s_{0}^{2}s_{1}s_{2}^{2}; μ_{0}s_{1}s_{2}^{2}; s_{0}s_{1}s_{2}^{2}; s_{0}s_{1}μ_{2}; s_{0}s_{1}s_{2}; s_{0}s_{1}s_{2}η_{0}; s_{0}s_{1}s_{2}s_{0}; s_{0}s_{1}λ_{0,2}; s_{0}s_{1}s_{0}s_{2}; s_{0}λ_{0,1}s_{2}; s_{0}^{2}s_{1}s_{2}; μ_{0}s_{1}s_{2}; s_{0}s_{1}s_{2}]

Equality of the second and last expressions in (13) follows symmetrically.

2.3. Lemma. For any object \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{0≤i<j≤2} of Wdl^{(2)}(\overline{K}), consider the monads (s_{0}s_{1}, \overline{λ}_{0,1}) and (s_{1}s_{2}, \overline{λ}_{1,2}) in \overline{K}, induced by the weak distributive laws λ_{0,1} and λ_{1,2}, respectively. There are weak distributive laws

[figure: λ_{01,2} := (s_{2}(s_{0}s_{1}); λ_{0,2}s_{1}; s_{0}s_{2}s_{1}; s_{0}λ_{1,2}; s_{0}s_{1}s_{2}; ^{\overrightarrow{λ}^{0,1,2}}; (s_{0}s_{1})s_{2}); λ_{01,2} = (s_{2}(s_{0}s_{1}); s_{2}\overline{λ}_{01}; s_{2}s_{0}s_{1}; λ_{0,2}s_{1}; s_{0}s_{2}s_{1}; s_{0}λ_{1,2}; (s_{0}s_{1})s_{2}); and; λ_{0,12} := ((s_{1}s_{2})s_{0}; s_{1}λ_{0,2}; s_{1}s_{0}s_{2}; λ_{0,1}s_{2}; s_{0}s_{1}s_{2}; _{\overleftarrow{λ}_{0,1,2}}; s_{0}(s_{1}s_{2})); λ_{0,12} = ((s_{1}s_{2})s_{0}; \overline{λ}_{12}s_{0}; s_{1}s_{2}s_{0}; s_{1}λ_{0,2}; s_{1}s_{0}s_{2}; λ_{0,1}s_{2}; s_{0}(s_{1}s_{2}))]

in \overline{K}. Moreover, their induced monads (s_{0}s_{1}s_{2}, \overline{λ}_{01,2}) and (s_{0}s_{1}s_{2}, \overline{λ}_{0,12}) are equal.

Proof. Both given forms of λ_{01,2} are equal by (8) and (9). It is a 2-cell in \overline{K} by (7). Compatibility with the multiplication of s_{2} holds since both λ_{0,2} and λ_{1,2} are compatible with it. With the normalization conditions \overline{λ}_{01}.μ_{01} = μ_{01} = μ_{01}.\overline{λ}_{01}s_{0}s_{1} = μ_{01}.s_{0}s_{1}\overline{λ}_{01} at hand, compatibility of λ_{01,2} with the multiplication of s_{0}s_{1} follows by the compatibilities of λ_{0,2} with μ_{0} and of λ_{1,2} with μ_{1} and the Yang-Baxter condition. The weak unitality condition (2) follows by the equality of the second and third expressions in (13) (so that \overline{λ}_{01,2} = \overline{λ}_{012}). This proves that λ_{01,2} is a weak distributive law and λ_{0,12} can be handled symmetrically (in particular, \overline{λ}_{0,12} = \overline{λ}_{012}).

Equality of the units in the induced monads follows immediately by the Yang-Baxter condition. Concerning the multiplications, composing the equal paths around

[figure: s_{1}s_{2}s_{1}s_{0}s_{1}; s_{1}s_{2}λ_{0,1}s_{1}; s_{1}λ_{1,2}s_{0}s_{1}; (YB); s_{1}s_{2}s_{0}s_{1}^{2}; s_{1}s_{2}s_{0}μ_{1}; s_{1}λ_{0,2}s_{1}^{2}; s_{1}s_{2}s_{0}s_{1}; s_{1}λ_{0,2}s_{1}; s_{1}s_{0}s_{2}s_{1}; λ_{0,1}s_{2}s_{1}; s_{1}s_{0}s_{2}s_{1}^{2}; λ_{0,1}s_{2}s_{1}^{2}; s_{1}s_{0}λ_{1,2}s_{1}; s_{0}s_{1}s_{2}s_{1}^{2}; s_{0}s_{1}s_{2}μ_{1}; s_{0}s_{1}λ_{1,2}s_{1}; s_{0}s_{1}s_{2}s_{1}; s_{0}s_{1}λ_{1,2}; s_{1}^{2}s_{2}s_{0}s_{1}; s_{1}^{2}λ_{0,2}s_{1}; μ_{1}s_{2}s_{0}s_{1}; s_{1}^{2}s_{0}s_{2}s_{1}; s_{1}λ_{0,1}s_{2}s_{1}; μ_{1}s_{0}s_{2}s_{1}; s_{1}s_{0}s_{1}s_{2}s_{1}; λ_{0,1}s_{1}s_{2}s_{1}; s_{0}s_{1}^{2}s_{2}s_{1}; s_{0}s_{1}^{2}λ_{1,2}; s_{0}μ_{1}s_{2}s_{1}; s_{0}s_{1}^{3}s_{2}; s_{0}s_{1}μ_{1}s_{2}; s_{0}μ_{1}s_{1}s_{2}; s_{0}s_{1}^{2}s_{2}; s_{0}μ_{1}s_{2}; s_{1}s_{2}s_{0}s_{1}; s_{1}λ_{0,2}s_{1}; s_{1}s_{0}s_{2}s_{1}; λ_{0,1}s_{2}s_{1}; s_{0}s_{1}s_{2}s_{1}; s_{0}s_{1}λ_{1,2}; s_{0}s_{1}^{2}s_{2}; s_{0}μ_{1}s_{2}; s_{0}s_{1}s_{2}]

by s_{1}s_{2}η_{1}s_{0}s_{1} on the right, we obtain

s_{0}μ_{1}s_{2}.λ_{0,1}λ_{1,2}.s_{1}λ_{0,2}s_{1}.\overline{λ}_{12}s_{0}s_{1} = s_{0}μ_{1}s_{2}.λ_{0,1}λ_{1,2}.s_{1}λ_{0,2}s_{1}.s_{1}s_{2}\overline{λ}_{01}.  (14)

Inserting these equal 2-cells s_{1}s_{2}s_{0}s_{1} → s_{0}s_{1}s_{2} into μ_{0}s_{1}μ_{2}.s_{0}(−)s_{2}, we conclude the equality of the multiplications induced on s_{0}s_{1}s_{2} by λ_{01,2} and λ_{0,12}, respectively.

2.4. On the Yang-Baxter condition Actually, also a sort of converse of Lemma 2.3 holds. Consider weak distributive laws \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{0≤i<j≤2} in \overline{K}. Assume that

\xymatrix{λ_{01,2} := (s_{2}(s_{0}s_{1}) \ar[r]^{s_{2}\overline{λ}_{01}} & s_{2}s_{0}s_{1} \ar[r]^{λ_{0,2}s_{1}} & s_{0}s_{2}s_{1} \ar[r]^{s_{0}λ_{1,2}} & (s_{0}s_{1})s_{2}) & and \\ λ_{0,12} := ((s_{1}s_{2})s_{0} \ar[r]^{\overline{λ}_{12}s_{0}} & s_{1}s_{2}s_{0} \ar[r]^{s_{1}λ_{0,2}} & s_{1}s_{0}s_{2} \ar[r]^{λ_{0,1}s_{2}} & s_{0}(s_{1}s_{2}))}

are weak distributive laws inducing equal monads (s_{0}s_{1}s_{2}, \overline{λ}_{01,2}) and (s_{0}s_{1}s_{2}, \overline{λ}_{0,12}). Then the Yang-Baxter condition holds.

Indeed, equality of the multiplications μ_{0,12} and μ_{01,2} is equivalent to (14). With this identity at hand, from the compatibility of λ_{0,12} with μ_{12} we obtain

λ_{0,12}.μ_{12}s_{0} = s_{0}μ_{1}μ_{2}.λ_{0,1}s_{1}s_{2}^{2}.s_{1}s_{0}λ_{1,2}s_{2}.s_{1}λ_{0,2}s_{1}s_{2}.s_{1}s_{2}λ_{0,12} .

Precomposing this equality with η_{1}s_{2}s_{1}η_{2}s_{0}, we conclude that

λ_{0,1}s_{2}.s_{1}λ_{0,2}.λ_{1,2}s_{0} = s_{0}s_{1}μ_{2}.s_{0}λ_{1,2}s_{2}.λ_{0,2}s_{1}s_{2}.s_{2}λ_{0,1}s_{2}.s_{2}s_{1}λ_{0,2}.s_{2}λ_{1,2}s_{0}.s_{2}η_{2}s_{1}s_{0} . (15)

Symmetrically,

s_{0}λ_{1,2}.λ_{0,2}s_{1}.s_{2}λ_{0,1} = μ_{0}s_{1}s_{2}.s_{0}λ_{0,1}s_{2}.s_{0}s_{1}λ_{0,2}.s_{0}λ_{1,2}s_{0}.λ_{0,2}s_{1}s_{0}.s_{2}λ_{0,1}s_{0}.s_{2}s_{1}η_{0}s_{0} . (16)

It follows by the associativity of μ_{1} that s_{0}\overline{λ}_{1,2}.\overline{λ}_{0,1}s_{2} = \overline{λ}_{0,1}s_{2}.s_{0}\overline{λ}_{1,2}. Precomposing with η_{0}s_{1}η_{2}, we obtain from this s_{0}λ_{1,2}.s_{0}η_{2}s_{1}.λ_{0,1}.s_{1}η_{0} = λ_{0,1}s_{2}.s_{1}η_{0}s_{2}.λ_{1,2}.η_{2}s_{1}. Inserting these latter equal expressions into

s_{0}s_{1}μ_{2}.s_{0}λ_{1,2}s_{2}.λ_{0,2}s_{1}s_{2}.s_{2}μ_{0}s_{1}s_{2}.s_{2}s_{0}λ_{0,1}s_{2}.s_{2}s_{0}s_{1}λ_{0,2}.s_{2}(−)s_{0}

and using (15) and (16) to simplify both sides of the resulting equality, we obtain the Yang-Baxter condition.

2.5. Lemma. For any 1-cell \{ξ_{i} : s′_{i}v → vs_{i}\}_{0≤i≤2} in Wdl^{(2)}(\overline{K}), the following yield 1-cells in Wdl(\overline{K}) between the 0-cells described in Lemma 2.3.

\{ξ_{01} := ( \xymatrix{s′_{0}s′_{1}v \ar[r]^{s′_{0}ξ_{1}} & s′_{0}vs_{1} \ar[r]^{ξ_{0}s_{1}} & vs_{0}s_{1} \ar[r]^{v\overline{λ}_{01}} & vs_{0}s_{1}} ), \xymatrix{s′_{2}v \ar[r]^{ξ_{2}} & vs_{2}} \} and

\{ \xymatrix{s′_{0}v \ar[r]^{ξ_{0}} & vs_{0}} , ξ_{12} := ( \xymatrix{s′_{1}s′_{2}v \ar[r]^{s′_{1}ξ_{2}} & s′_{1}vs_{2} \ar[r]^{ξ_{1}s_{2}} & vs_{1}s_{2} \ar[r]^{v\overline{λ}_{12}} & vs_{1}s_{2}} )\}

Proof. By Paragraph 1.5, ξ_{01} and ξ_{12} are 1-cells in Mnd(\overline{K}). Moreover, ξ_{01} and ξ_{2} obey (5) by commutativity of the following diagram.

[figure: s′_{2}s′_{0}s′_{1}v; s′_{2}ξ_{01}; s′_{2}\overline{λ′}_{01}v; s′_{2}s′_{0}s′_{1}v; λ′_{0,2}s′_{1}v; s′_{2}s′_{0}ξ_{1}; s′_{0}s′_{2}s′_{1}v; s′_{0}λ′_{1,2}v; s′_{0}s′_{2}ξ_{1}; (5); s′_{0}s′_{1}s′_{2}v; s′_{0}s′_{1}ξ_{2}; s′_{2}s′_{0}vs_{1}; λ′_{0,2}vs_{1}; s′_{2}ξ_{0}s_{1}; (5); s′_{0}s′_{2}vs_{1}; s′_{0}ξ_{2}s_{1}; s′_{0}s′_{1}vs_{2}; s′_{0}ξ_{1}s_{2}; s′_{2}vs_{0}s_{1}; ξ_{2}s_{0}s_{1}; s′_{2}vs_{0}s_{1}; s′_{2}v\overline{λ}_{01}; ξ_{2}s_{0}s_{1}; s′_{0}vs_{2}s_{1}; s′_{0}vλ_{1,2}; ξ_{0}s_{2}s_{1}; s′_{0}vs_{1}s_{2}; ξ_{0}s_{1}s_{2}; s′_{0}vs_{1}s_{2}; s′_{0}v\overline{λ}_{12}; ξ_{0}s_{1}s_{2}; vs_{2}s_{0}s_{1}; vλ_{0,2}s_{1}; vs_{2}\overline{λ}_{01}; (9); vs_{0}s_{2}s_{1}; v\overrightarrow{λ}_{0,2,1}; (7); vs_{0}s_{2}s_{1}; vs_{0}λ_{1,2}; v\overrightarrow{λ}_{0,2,1}; v\overline{λ}_{02}s_{1}; (8); vs_{0}s_{1}s_{2}; v\overrightarrow{λ}_{0,1,2}; (13); vs_{0}s_{1}s_{2}; vs_{0}\overline{λ}_{12}; v\overline{λ}_{012}; vs_{2}s_{0}s_{1}; vs_{2}\overline{λ}_{01}; vs_{2}s_{0}s_{1}; vλ_{0,2}s_{1}; vs_{0}s_{2}s_{1}; vs_{0}s_{2}s_{1}; vs_{0}λ_{1,2}; vs_{0}s_{1}s_{2}; vs_{0}s_{1}s_{2}]

The top-left region commutes since ξ_{01} is a 2-cell in Mnd(\overline{K}) of domain (s′_{0}s′_{1}v, \overline{λ′}_{0,1}\overline{v}). Also ξ_{0} and ξ_{12} obey (5), hence constitute a 1-cell in Wdl(\overline{K}), by commutativity of the similar diagram below. The bottom-left region commutes by the normalization of ξ_{12}.

[figure: s′_{1}s′_{2}s′_{0}v; \overline{λ′}_{12}s′_{0}v; s′_{1}s′_{2}ξ_{0}; s′_{1}s′_{2}s′_{0}v; s′_{1}λ′_{0,2}v; s′_{1}s′_{2}ξ_{0}; (5); s′_{1}s′_{0}s′_{2}v; λ′_{0,1}s′_{2}v; s′_{1}s′_{0}ξ_{2}; s′_{0}s′_{1}s′_{2}v; s′_{0}s′_{1}ξ_{2}; s′_{1}s′_{2}vs_{0}; ξ_{12}s_{0}; \overline{λ′}_{12}vs_{0}; s′_{1}s′_{2}vs_{0}; s′_{1}ξ_{2}s_{0}; s′_{1}s′_{0}vs_{2}; λ′_{0,1}vs_{2}; s′_{1}ξ_{0}s_{2}; (5); s′_{0}s′_{1}vs_{2}; s′_{0}ξ_{1}s_{2}; s′_{1}vs_{2}s_{0}; s′_{1}vλ_{0,2}; ξ_{1}s_{2}s_{0}; s′_{1}vs_{0}s_{2}; ξ_{1}s_{0}s_{2}; s′_{1}vs_{0}s_{2}; s′_{1}v\overline{λ}_{02}; ξ_{1}s_{0}s_{2}; s′_{0}vs_{1}s_{2}; ξ_{0}s_{1}s_{2}; vs_{1}s_{2}s_{0}; vs_{1}λ_{0,2}; v\overline{λ}_{12}s_{0}; (12); vs_{1}s_{0}s_{2}; v\overleftarrow{λ}_{1,0,2}; (10); vs_{1}s_{0}s_{2}; vλ_{0,1}s_{2}; v\overleftarrow{λ}_{1,0,2}; vs_{1}\overline{λ}_{02}; (11); vs_{0}s_{1}s_{2}; v\overleftarrow{λ}_{0,1,2}; (13); vs_{0}s_{1}s_{2}; v\overline{λ}_{01}s_{2}; v\overline{λ}_{012}; vs_{1}s_{2}s_{0}; v\overline{λ}_{12}s_{0}; vs_{1}s_{2}s_{0}; vs_{1}λ_{0,2}; vs_{1}s_{0}s_{2}; vs_{1}s_{0}s_{2}; vλ_{0,1}s_{2}; vs_{0}s_{1}s_{2}; vs_{0}s_{1}s_{2}]

2.6. Theorem. For any 2-category K, and any positive integer n, there are n different 2-functors C_{k} : Wdl^{(n)}(\overline{K}) → Wdl^{(n−1)}(\overline{K}), for 1 ≤ k ≤ n, as follows. They take a 0-cell {λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}}_{0≤i<j≤n} to

\{ [figure: s_{j}s_{i}; λ_{i,j}; s_{i}s_{j}; if i, j ∉ \{k − 1, k\}; s_{j}(s_{k−1}s_{k}); s_{j} \overline{λ}_{k−1,k}; s_{j}s_{k−1}s_{k}; λ_{k−1,j}s_{k}; s_{k−1}s_{j}s_{k}; s_{k−1}λ_{k,j}; (s_{k−1}s_{k})s_{j}; if k < j; (s_{k−1}s_{k})s_{i}; \overline{λ}_{k−1,k}s_{i}; s_{k−1}s_{k}s_{i}; s_{k−1}λ_{i,k}; s_{k−1}s_{i}s_{k}; λ_{i,k−1}s_{k}; s_{i}(s_{k−1}s_{k}); if i < k − 1] \}

(17)

where (s_{k−1}s_{k}, \overline{λ}_{k−1,k}) is the monad in \overline{K} induced by the weak distributive law λ_{k−1,k}. They send a 1-cell {ξ_{i} : s′_{i}v → vs_{i}}_{0≤i≤n} to

\{ [figure: s′_{i}v; ξ_{i}; vs_{i}; if 0 ≤ i < k − 1; (s′_{k−1}s′_{k})v; s′_{k−1}ξ_{k}; s′_{k−1}vs_{k}; ξ_{k−1}s_{k}; vs_{k−1}s_{k}; v\overline{λ}_{k−1,k}; v(s_{k−1}s_{k}); s′_{i+1}v; ξ_{i+1}; vs_{i+1}; if k − 1 < i < n] \} (18)

On the 2-cells they act as the identity map.

Proof. By Lemma 2.3, each line in (17) is a weak distributive law in \overline{K}. We only need to check the Yang-Baxter conditions. For 0 ≤ i < j < k−1 < n the Yang-Baxter condition follows by commutativity of

[figure: s_{k−1}s_{k}s_{j}s_{i}; \overline{λ}_{k−1,k}s_{j}s_{i}; \overline{λ}_{k−1,k}s_{j}s_{i}; s_{k−1}s_{k}λ_{i,j}; s_{k−1}s_{k}s_{j}s_{i}; s_{k−1}λ_{j,k}s_{i}; s_{k−1}s_{j}s_{k}s_{i}; λ_{j,k−1}s_{k}s_{i}; s_{j}s_{k−1}s_{k}s_{i}; s_{j} \overline{λ}_{k−1,k}s_{i}; s_{k−1}s_{k}s_{i}s_{j}; \overline{λ}_{k−1,k}s_{i}s_{j}; \overline{λ}_{k−1,k}s_{i}s_{j}; s_{k−1}s_{k}s_{j}s_{i}; s_{k−1}λ_{j,k}s_{i}; s_{k−1}s_{k}λ_{i,j}; (YB); s_{k−1}s_{j}s_{k}s_{i}; λ_{j,k−1}s_{k}s_{i}; s_{k−1}s_{j}λ_{i,k}; s_{j}s_{k−1}s_{k}s_{i}; s_{j}s_{k−1}λ_{i,k}; s_{k−1}s_{k}s_{i}s_{j}; s_{k−1}λ_{i,k}s_{j}; s_{k−1}s_{k}s_{i}s_{j}; s_{k−1}λ_{i,k}s_{j}; s_{k−1}s_{j}s_{i}s_{k}; λ_{j,k−1}s_{i}s_{k}; s_{k−1}λ_{i,j}s_{k}; (YB); s_{j}s_{k−1}s_{i}s_{k}; s_{j}λ_{i,k−1}s_{k}; s_{k−1}s_{i}s_{k}s_{j}; λ_{i,k−1}s_{k}s_{j}; s_{k−1}s_{i}s_{k}s_{j}; s_{k−1}s_{i}λ_{j,k}; λ_{i,k−1}s_{k}s_{j}; s_{k−1}s_{i}s_{j}s_{k}; λ_{i,k−1}s_{j}s_{k}; s_{j}s_{i}s_{k−1}s_{k}; λ_{i,j}s_{k−1}s_{k}; s_{i}s_{k−1}s_{k}s_{j}; s_{i}\overline{λ}_{k−1,k}s_{j}; s_{i}s_{k−1}s_{k}s_{j}; s_{i}s_{k−1}λ_{j,k}; s_{i}s_{k−1}s_{j}s_{k}; s_{i}λ_{j,k−1}s_{k}; s_{i}s_{j}s_{k−1}s_{k} .]

The top-right region and the bottom-left region commute by Lemma 2.3. The 0 < k < i < j ≤ n case is treated symmetrically. The Yang-Baxter condition in the last case, when 0 ≤ i < k − 1 and k < j ≤ n, follows by commutativity of the similar diagram

[figure: s_{j}s_{k−1}s_{k}s_{i}; s_{j} \overline{λ}_{k−1,k}s_{i}; s_{j} \overline{λ}_{k−1,k}s_{i}; s_{j} \overline{λ}_{k−1,k}s_{i}; s_{j}s_{k−1}s_{k}s_{i}; λ_{k−1,j}s_{k}s_{i}; s_{k−1}s_{j}s_{k}s_{i}; s_{k−1}λ_{k,j}s_{i}; s_{k−1}s_{k}s_{j}s_{i}; s_{k−1}s_{k}λ_{i,j}; \overline{λ}_{k−1,k}s_{j}s_{i}; s_{k−1}s_{k}s_{i}s_{j}; \overline{λ}_{k−1,k}s_{i}s_{j}; s_{j}s_{k−1}s_{k}s_{i}; s_{j}s_{k−1}λ_{i,k}; s_{j}s_{k−1}s_{k}s_{i}; λ_{k−1,j}s_{k}s_{i}; s_{j}s_{k−1}λ_{i,k}; s_{k−1}s_{j}s_{k}s_{i}; s_{k−1}λ_{k,j}s_{i}; s_{k−1}s_{j}λ_{i,k}; (YB); s_{k−1}s_{k}s_{j}s_{i}; s_{k−1}s_{k}λ_{i,j}; s_{k−1}s_{k}s_{i}s_{j}; s_{k−1}λ_{i,k}s_{j}; s_{j}s_{k−1}s_{i}s_{k}; s_{j}λ_{i,k−1}s_{k}; s_{j}s_{k−1}s_{i}s_{k}; λ_{k−1,j}s_{i}s_{k}; s_{j}λ_{i,k−1}s_{k}; (YB); s_{k−1}s_{j}s_{i}s_{k}; s_{k−1}λ_{i,j}s_{k}; s_{k−1}s_{i}s_{j}s_{k}; s_{k−1}s_{i}λ_{k,j}; λ_{i,k−1}s_{j}s_{k}; s_{k−1}s_{i}s_{k}s_{j}; λ_{i,k−1}s_{k}s_{j}; s_{j}s_{i}s_{k−1}s_{k}; s_{j}s_{i}\overline{λ}_{k−1,k}; λ_{i,j}s_{k−1}s_{k}; s_{j}s_{i}s_{k−1}s_{k}; λ_{i,j}s_{k−1}s_{k}; s_{i}s_{j}s_{k−1}s_{k}; s_{i}s_{j} \overline{λ}_{k−1,k}; s_{i}s_{j}s_{k−1}s_{k}; s_{i}λ_{k−1,j}s_{k}; s_{i}s_{k−1}s_{j}s_{k}; s_{i}s_{k−1}λ_{k,j}; s_{i}s_{k−1}s_{k}s_{j} .]

Both regions at the top-left commute by Lemma 2.3. This proves that (17) describes a 0-cell in Wdl^{(n−1)}(\overline{K}). By Lemma 2.5, (18) is a 1-cell in Wdl^{(n−1)}(\overline{K}). Evidently, 2-cells in Wdl^{(n)}(\overline{K}) are 2-cells in Wdl^{(n−1)}(\overline{K}) as well. Hence the stated maps define 2-functors C_{k} which clearly preserve the horizontal and vertical compositions.

Via the fully faithful embedding Mnd^{n+1}(\overline{K}) → Wdl^{(n)}(\overline{K}) in Paragraph 2.1, the 2-functors in Theorem 2.6 extend the multiplication C of the 2-monad Mnd. That is, the following diagram commutes, for all 1 ≤ k ≤ n.

\xymatrix{Mnd^{n+1}(\overline{K}) \ar[r]^{Mnd^{k−1}CMnd^{n−k}(\overline{K})} \ar[d] & Mnd^{n}(\overline{K}) \ar[d] \\ Wdl^{(n)}(\overline{K}) \ar[r]_{C_{k}} & _{Wdl_{(n−1)}}}

By associativity of the 2-monad Mnd, the n-fold iteration of the 2-functor in the top row; that is,

[figure: Mnd^{n+1}(\overline{K}); Mnd^{kn−1}CMnd^{n−kn}(\overline{K}); …; Mnd^{k_{2}−1}CMnd^{2−k_{2}}(\overline{K}); Mnd^{2}(\overline{K}); C; Mnd(\overline{K})]

does not depend on the values of k_{i} ∈ {1, …, i}, for 1 ≤ i ≤ n. That is, its object map describes an ‘associative’ wreath product of monads. Although the 2-functor in the bottom row is not known to correspond to the multiplication in any 2-monad, in the rest of this section we show that it describes an associative weak wreath product in an appropriate sense.

2.7. Lemma. For any integer n > 1, and for any 0-cell {λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}}_{0≤i<j≤n} of Wdl^{(n)}(\overline{K}), consider the idempotent 2-cell

\xymatrix{\overleftarrow{λ}_{0,1,...,n} := (s_{0}s_{1} … s_{n} \ar[r]^{η_{n}s_{0}s_{1}...s_{n}} & s_{n}s_{0}s_{1} … s_{n} \ar[r] & s_{0}s_{1} … s_{n−1}s_{n}^{2} \ar[r]^{s_{0}s_{1}...s_{n−1}μ_{n}} & s_{0}s_{1} … s_{n})}

in K (where the unlabelled arrow denotes the unique composite of λ_{i,j}s of the given domain and codomain) and

\overline{λ}_{01...n} := \overleftarrow{λ}_{0,1,...,n}.\overleftarrow{λ}_{0,1,...,n−1}s_{n}. · · · .\overleftarrow{λ}_{0,1,2}s_{3}s_{4} … s_{n}.\overline{λ}_{01}s_{2}s_{3} … s_{n} .  (19)

This construction in (19) associates the same idempotent 2-cell to any 0-cell {λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{0≤i<j≤n} of Wdl^{(n)}(\overline{K}) and to its image under any of the 2-functors C_{k} in Theorem 2.6. That is, for all 1 ≤ k ≤ n, \overline{λ}_{01...n} = \overline{λ}_{01...k−2(k−1,k)k+1...n}.

Proof. Using commutativity of

[figure: s_{k+1}s_{0} … s_{k}; s_{k+1}η_{k}s_{0}...s_{k}; s_{k+1}\overleftarrow{λ}_{0,...,k}; s_{k+1}\overleftarrow{λ}_{0,...,k}; s_{k+1}s_{k}s_{0} … s_{k}; s_{k+1}s_{0} … s_{k−1}s_{k}^{2}; s_{k+1}s_{0}...s_{k−1}μ_{k}; (1); s_{k+1}s_{0} … s_{k}; s_{k}s_{0} … s_{k+1}; (YB); s_{0} … s_{k−1}s_{k}^{2}s_{k+1}; s_{0}...s_{k−1}μ_{k}s_{k+1}; \overleftarrow{λ}_{0,...,k}s_{k}s_{k+1}; s_{0} … s_{k+1}; \overleftarrow{λ}_{0,...,k}s_{k+1}; s_{k+1}s_{0} … s_{k−1}s_{k}^{2}; s_{k+1}s_{0}...s_{k−1}μ_{k}; (YB); (1); s_{0} … s_{k−1}s_{k}^{2}s_{k+1}; s_{0}...s_{k−1}μ_{k}s_{k+1}; s_{k+1}s_{0} … s_{k}; s_{0} … s_{k+1}; s_{0} … s_{k+1}]

(20)

for any 1 ≤ k < n, it follows easily that \overline{λ}_{0...n} is idempotent. By (17) on one hand, and by (20) on the other,

\overleftarrow{λ}_{0,...,k−2,(k−1,k),k+1,...m} = \overleftarrow{λ}_{0,...,m}.s_{0} … s_{k−2}\overline{λ}_{k−1,k}s_{k+1} … s_{m}

= s_{0} … s_{k−2}\overline{λ}_{k−1,k}s_{k+1} … s_{m}.\overleftarrow{λ}_{0,...,m}.s_{0} … s_{k−2}\overline{λ}_{k−1,k}s_{k+1} … s_{m}

for all k < m ≤ n. Moreover, by commutativity of

[figure: s_{0} … s_{k}; η_{k}η_{k−1}s_{0}...s_{k}; η_{k−1}s_{0}...s_{k}; s_{k}s_{k−1}s_{0} … s_{k}; λ_{k−1,k}s_{0}...s_{k}; λ_{k−1,k}s_{0}...s_{k}; (3); s_{k−1}s_{k}s_{0} … s_{k}; \overline{λ}_{k−1,k}s_{0}...s_{k}; s_{k−1}s_{0} … s_{k}; (YB); s_{k−1}s_{k}s_{0} … s_{k}; s_{0} … s_{k−2}(s_{k−1}s_{k})^{2}; s_{0} … s_{k−2}s_{k−1}^{2}s_{k}; η_{k}s_{0}...s_{k−2}s_{k−1}^{2}s_{k}; s_{0}...s_{k−2}μ_{k−1}s_{k}; s_{k}s_{0} … s_{k−2}s_{k−1}^{2}s_{k}; s_{k}s_{0}...s_{k−2}μ_{k−1}s_{k}; (1); s_{0} … s_{k−2}s_{k−1}^{2}s_{k}^{2}; s_{0}...s_{k−2}μ_{k−1}s_{k}^{2}; s_{0} … s_{k}; η_{k}s_{0}...s_{k}; s_{k}s_{0} … s_{k}; s_{0} … s_{k−1}s_{k}^{2}]

we obtain \overleftarrow{λ}_{0,...,k−2,(k−1,k)} = \overleftarrow{λ}_{0,...,k}.\overleftarrow{λ}_{0,...,k−1}s_{k} = s_{0} … s_{k−2}\overline{λ}_{k−,k}.\overleftarrow{λ}_{0,...,k}.\overleftarrow{λ}_{0,...,k−1}s_{k}. Combining these identities we conclude the claim.

2.8. Lemma. In terms of the 2-functors in Theorem 2.6, for any integer n > 1, the composite

\xymatrix{Wdl^{(n)}(\overline{K}) \ar[r]^{C_{1}} & Wdl^{(n−1)}(\overline{K}) \ar[r]^{C_{1}} & … \ar[r]^{C_{1}} & Wdl(\overline{K})}

takes a 0-cell {λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}}_{0≤i<j≤n} to the weak distributive law

λ_{0...n−1,n} := ( \xymatrix{s_{n}(s_{0} … s_{n−1}) \ar[r]^{s_{n}\overline{λ}_{0...n−1}} & s_{n}s_{0} … s_{n−1} \ar[r] & (s_{0} … s_{n−1})s_{n}} ), (21)

where the unlabelled arrow denotes the unique combination of λ_{i,j}s with the given domain and codomain.

Proof. We proceed by induction in n. For n = 2 the claim follows by Theorem 2.6. Assume that it holds for some n ≥ 2. Then the following diagram commutes.

[figure: s_{0} … s_{n}; η_{n}s_{0}...s_{n}; \overline{λ}_{0...n−1}s_{n}; s_{n}s_{0} … s_{n}; s_{n}\overline{λ}_{0...n−1}s_{n}; λ_{0...n−1,n}s_{n}; (∗); s_{0} … s_{n}; η_{n}s_{0}...s_{n}; _{\overleftarrow{λ}_{0,...,n}}; s_{n}s_{0} … s_{n}; s_{0} … s_{n−1}s_{n}^{2}; s_{0}...s_{n−1}μ_{n}; s_{0} … s_{n}]

The region marked by (∗) commutes by the induction hypothesis. The left bottom path is equal to \overline{λ}_{0...n}, see (19). Hence we conclude that \overline{λ}_{0...n} is equal to \overline{λ}_{0...n−1,n}; that is, the idempotent associated to the weak distributive law λ_{0...n−1,n}. Using this observation and (17) (for the monads s_{0} … s_{n−1}, s_{n} and s_{n+1}), in the top-right path of the following diagram we recognize λ_{0...n,n+1}.

[figure: s_{n+1}s_{0} … s_{n}; s_{n+1}\overline{λ}_{0...n}; s_{n+1}\overline{λ}_{0...n}; s_{n+1}s_{0} … s_{n}; s_{n+1}\overline{λ}_{0...n−1}s_{n}; λ_{0...n−1,n+1}s_{n}; (∗); s_{n+1}s_{0} … s_{n}; s_{0} … s_{n−1}s_{n+1}s_{n}; s_{0}...s_{n−1}λ_{n,n+1}; s_{0} … s_{n+1}]

The region marked by (∗) commutes by the induction hypothesis and the triangle on the left commutes by (20).

Applying (8) and (9) to the monads s_{0} … s_{n−2}, s_{n−1} and s_{n}, from (21) we obtain the equal expression

λ_{0...n−1,n} = ( \xymatrix{s_{n}(s_{0} … s_{n−1}) \ar[r] & s_{0} … s_{n} \ar[r]^{^{\overline{λ}^{0...n}}} & (s_{0} … s_{n−1})s_{n}} ).  (22)

2.9. Lemma. In terms of the 2-functors in Theorem 2.6, for any positive integer n, consider the composite

\xymatrix{Wdl^{(n)}(\overline{K}) \ar[r]^{C_{1}} & Wdl^{(n−1)}(\overline{K}) \ar[r]^{C_{1}} & … \ar[r]^{C_{1}} & Wdl(\overline{K}) \ar[r]^{C_{1}} & Mnd(\overline{K}) .}  (23)

(1) It takes a 0-cell \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{0≤i<j≤n} to the monad (s_{0}s_{1} … s_{n}, \overline{λ}_{01...n}) in \overline{K}, with multiplication μ_{0...n} equal to

\xymatrix{s_{0}s_{1} … s_{n}s_{0}s_{1} … s_{n} \ar[r] & s_{0}^{2}s_{1}^{2} … s_{n}^{2} \ar[r]^{μ_{0}μ_{1}...μ_{n}} & s_{0}s_{1} … s_{n} \ar[r]^{^{\overline{λ}^{01...n}}} & s_{0}s_{1} … s_{n}}

(24)

and unit η_{0...n} equal to

\xymatrix{1 \ar[r]^{η_{n}η_{n−1}...η_{0}} & s_{n}s_{n−1} … s_{0} \ar[r] & s_{0}s_{1} … s_{n} \ar[r]^{^{\overline{λ}^{01...n}}} & s_{0}s_{1} … s_{n},}  (25)

where the unlabelled arrows denote the unique (by the Yang-Baxter condition) combinations of λ_{i,j}s with the given domain and codomain.

(2) It takes a 1-cell {ξ_{i} : s′_{i}v → vs_{i}}_{0≤i≤n} to

\xymatrix{s′_{0}s′_{1} … s′_{n}v \ar[r]^{s′_{0}s′_{1}...s′_{n−1}ξ_{n}} & s′_{0}s′_{1} … s′_{n−1}vs_{n} \ar[r]^{s′_{0}s′_{1}...s′_{n−2}ξ_{n−1}s_{n}} & … \ar[r]^{ξ_{0}s_{1}s_{2}...s_{n}} & vs_{0}s_{1} … s_{n} \ar[r]^{v\overline{λ}_{01...n}} & vs_{0}s_{1} … s_{n},}

to be denoted by ξ_{01...n}.

(3) On the 2-cells it acts as the identity map.

Proof. (1) We proceed by induction in n. For n = 1 the claim holds by Paragraph 1.5. Assume that it holds for some n ≥ 1. Then μ_{0...n+1} occurs in the top-right path of the following diagram.

[figure: (s_{0} … s_{n+1})^{2}; s_{0}...s_{n}λ_{0...n,n+1}s_{n+1}; (22); (s_{0} … s_{n})^{2}s_{n+1}^{2}; (s_{0} … s_{n+1})^{2}; (s_{0} … s_{n})^{2}s_{n+1}^{2}; (∗); (s_{0} … s_{n})^{2}s_{n+1}^{2}; s_{0}...s_{n}\overline{λ}_{0...n+1}s_{n+1}; μ_{0...n}μ_{n+1}; (s_{0} … s_{n})^{2}s_{n+1}^{2}; μ_{0...n}μ_{n+1}; s_{0}^{2} … s_{n+1}^{2}; μ_{0}...μ_{n+1}; s_{0} … s_{n+1}; \overline{λ}_{0...n}s_{n+1}; s_{0} … s_{n+1}; _{\overline{λ}_{0...n+1}}; s_{0} … s_{n+1}]

The region marked by (∗) commutes by the induction hypothesis and the bottom-right region commutes by the bilinearity of \overline{λ}_{0...n+1} = \overline{λ}_{0...n,n+1}, cf. (3). The composite of the last two arrows in the bottom row is equal to \overline{λ}_{0...n+1} by the explicit form in (19).

Similarly, η_{0...n+1} occurs in the top-right path of the following diagram.

[figure: A; η_{n+1}η_{0...n}; (∗); s_{n+1}s_{0} … s_{n}; A; η_{n+1}...η_{0}; s_{n+1} … s_{0}; s_{n+1}s_{0} … s_{n}; s_{n+1}\overline{λ}_{0...n}; λ_{0...n,n+1}; s_{n+1}s_{0} … s_{n}; λ_{0...n,n+1}; s_{0} … s_{n+1}; _{\overline{λ}_{0...n+1}}; (22); s_{0} … s_{n+1}]

The region marked by (∗) commutes by the induction hypothesis.

Part (2) is easily proved by induction in n and part (3) is trivial.

2.10. Theorem. For any 2-category K, and any positive integer n, the 2-functors in Theorem 2.6 give rise to a unique composite

\xymatrix{Wdl^{(n)}(\overline{K}) \ar[r]^{C_{kn}} & Wdl^{(n−1)}(\overline{K}) \ar[r]^{C_{k_{n−1}}} & … \ar[r]^{C_{k_{2}}} & Wdl(\overline{K}) \ar[r]^{C_{k_{1}}} & Mnd(\overline{K})}

which is independent of the choice of the index set k_{i} ∈ {1, 2, …, i}, for 1 ≤ i ≤ n.

Proof. We proceed by induction in n. For n = 1 the claim is trivial: there is only one 2-functor C_{1} : Wdl^{(1)}(\overline{K}) ≡ Wdl(\overline{K}) → Wdl^{(0)}(\overline{K}) ≡ Mnd(\overline{K}), that recalled in Paragraph 1.5. Assume now that the claim holds for some positive integer n; that is, C_{1}C_{k_{2}} … C_{k_{n}} does not depend on k_{i} ∈ {1, 2, …, i}, for 1 ≤ i ≤ n. With the explicit form of the 2-functor (23) in Lemma 2.9, and the explicit form of C_{k} : Wdl^{(n+1)}(\overline{K}) → Wdl^{(n)}(\overline{K}) in (17) and (18) at hand, the equality C_{k}C_{1}C_{1} … C_{1} = C_{1}C_{1}C_{1} … C_{1} of the n + 1-fold composites follows by Lemma 2.7, for all 1 ≤ k ≤ n + 1.

2.11. If idempotent 2-cells split Let us take a 2-category K which is locally idempotent complete; that is, K ≃ \overline{K}. Then we may consider the pseudofunctors

\xymatrix{Wdl^{(n)}(K) \ar[r]^{≃} & Wdl^{(n)}(\overline{K}) \ar[r]^{C_{k}} & Wdl^{(n−1)}(\overline{K}) \ar[r]^{≃} & Wdl^{(n−1)}(K),}

for all values of 1 ≤ k ≤ n. (Choosing the biequivalence \overline{K} → K by adopting the convention that we split identity 2-cells trivially, that is, via identity 2-cells; they become in fact 2-functors.) Their n-fold iteration is pseudonaturally equivalent to

\xymatrix{Wdl^{(n)}(K) \ar[r]^{≃} & Wdl^{(n)}(\overline{K}) \ar[r] & Mnd(\overline{K}) \ar[r]^{≃} & Mnd(K)}

where the unlabelled arrow stands for the 2-functor in Theorem 2.10. This pseudofunctor (or in fact 2-functor with an appropriate choice) takes an object {λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{0≤i<j≤n} of Wdl^{(n)}(K), considered as an object of Wdl^{(n)}(\overline{K}), to the image of the idempotent \overline{λ}_{0...n} in (19). This is regarded as the weak wreath product of the monads s_{0}, s_{1}, …, s_{n} in K. It is unique – that is, the weak wreath product is associative – up-to an isomorphism arising from the chosen splittings of the occurring idempotents.

3. Examples from Ising type spin chains

In this section K := Vec will be the one-object 2-category (in fact bicategory); that is, monoidal category of vector spaces over a given field F. Thus there is only one 0-cell F; the 1-cells are the F -vector spaces and the 2-cells are the linear maps. The horizontal composition (that is, monoidal product) is given by the tensor product ⊗ and the vertical composition is given by the composition of linear maps. Monads are just the F -algebras. We shall make use of the fact that the monoidal category of vector spaces is symmetric; the symmetry natural isomorphism (that is, the flip map) will be denoted by σ. Clearly, Vec is idempotent complete.

Our aim is to present an object of Wdl^{(n)}(Vec) (for any positive integer n) in terms of a finite dimensional weak bialgebra. We start with recalling the notion of weak bialgebra from [Nill (preprint 1998)], [Böhm, Nill, Szlachányi (1999)].

3.1. Definition. A weak bialgebra is a vector space H equipped with an algebra (that is, monad) structure μ : H ⊗ H → H, η : F → H and a coalgebra (that is, comonad) structure Δ : H → H ⊗ H, ε : H → F such that the following diagrams commute.

\xymatrix{H^{⊗2} \ar[r]^{Δ⊗Δ} \ar[d]_{μ} & H^{⊗4} \ar[r]^{H⊗σ⊗H} & H^{⊗4} \ar[d]^{μ⊗μ} \\ H \ar[rr]_{Δ} &  & H^{⊗2}}

\xymatrix{F \ar[r]^{η⊗η} \ar[dr]^{η} \ar[d]_{η⊗η} & H^{⊗2} \ar[r]^{Δ⊗Δ} & H^{⊗4} \ar[dd]^{H⊗μ⊗H} & H^{⊗3} \ar[r]^{H⊗Δ⊗H} \ar[dr]^{μ^{2}} \ar[dd]_{H⊗Δ⊗H} & H^{⊗4} \ar[r]^{H⊗σ⊗H} & H^{⊗4} \ar[d]^{μ⊗μ} \\ H^{⊗2} \ar[d]_{Δ⊗Δ} & H \ar[dr]^{Δ^{2}} &  &  & H \ar[dr]^{ε} & H^{⊗2} \ar[d]^{ε⊗ε} \\ H^{⊗4} \ar[r]_{H⊗σ⊗H} & H^{⊗4} \ar[r]_{H⊗μ⊗H} & H^{⊗3} & H^{⊗4} \ar[r]_{μ⊗μ} & H^{⊗2} \ar[r]_{ε⊗ε} & F}

This definition can be generalized from Vec to any braided monoidal category, see [Alonso Álvarez et al. (2008)], [Pastro, Street (2009)]. Note that the axioms of a weak bialgebra are self-dual in the sense that they are closed under the reversing of the arrows in the diagrams depicting them.

3.2. Duals of weak bialgebras Whenever the 1-cell underlying a monad in a 2-category possesses a (left or right) adjoint, this adjoint comes equipped with the canonical structure of a comonad. Conversely, the adjoint of a comonad is a monad. In the monoidal category of vector spaces, a 1-cell H – that is, a vector space – possesses a (left and right) adjoint if and only if it is finite dimensional over F; in which case the adjoint is the linear dual Ĥ := Hom(H, F); that is, the vector space of linear maps from H to F. In particular, the dual of a finite dimensional weak bialgebra is both an algebra – with multiplication μ̂ := Hom(Δ, F) : Ĥ ⊗ Ĥ ≅ Hom(H ⊗ H, F) → Ĥ and unit η̂ := Hom(ε, F) : F → Ĥ – and a coalgebra – with comultiplication Δ̂ := Hom(μ, F) and counit ε̂:= Hom(η, F). That is to say, the (co)algebra structure of Ĥ is defined by the following commutative diagrams

\xymatrix{Ĥ ⊗ H^{⊗2} \ar[r]^{Δ̂⊗H^{⊗2}} \ar[d]^{Ĥ⊗σ} & Ĥ^{⊗2} ⊗ H^{⊗2} \ar[d]_{Ĥ⊗ev⊗H} & Ĥ \ar[r]^{Ĥ⊗η} \ar[ddr]_{ε̂} & Ĥ ⊗ H \ar[dd]_{ev} & Ĥ^{⊗2} ⊗ H \ar[r]^{σ⊗H} \ar[d]^{Ĥ^{⊗2}⊗Δ} & Ĥ^{⊗2} ⊗ H \ar[d]_{μ̂⊗H} & H \ar[r]^{η̂⊗H} \ar[ddr]_{ε} & Ĥ ⊗ H \ar[dd]_{ev} \\ Ĥ ⊗ H^{⊗2} \ar[d]^{Ĥ⊗μ} & Ĥ ⊗ H \ar[d]_{ev} &  &  & Ĥ^{⊗2} ⊗ H^{⊗2} \ar[d]^{Ĥ⊗ev⊗H} & Ĥ ⊗ H \ar[d]_{ev} \\ Ĥ ⊗ H \ar[r]_{ev} & F &  & F & Ĥ ⊗ H \ar[r]_{ev} & F &  & F}

(26)

where ev : Ĥ ⊗ H → F stands for the evaluation map (that is, the counit of the adjunction Ĥ ⊣ H). What is more, by self-duality of the weak bialgebra axioms, Ĥ is a weak bialgebra again with the above algebra and coalgebra structures.

3.3. The iterated weak wreath product of a finite weak bialgebra and its dual In terms of a finite dimensional weak bialgebra H, an object of Wdl^{(n)}(Vec) is given as follows. If 0 ≤ i ≤ n is even, then let s_{i} be the algebra underlying H and if i is odd then let s_{i} be the algebra underlying Ĥ. If j − i > 1 then let λ_{i,j} be given by the flip map σ. If i is odd, then let λ_{i,i+1} be equal to λ defined as

\xymatrix{H ⊗ Ĥ \ar[r]^{σ} & Ĥ ⊗ H \ar[r]^{Δ̂⊗Δ} & Ĥ ⊗ Ĥ ⊗ H ⊗ H \ar[r]^{Ĥ⊗ev⊗H} & Ĥ ⊗ H,}

and if i is even then let λ_{i,i+1} be equal to λ̂ given by

\xymatrix{Ĥ ⊗ H \ar[r]^{σ} & H ⊗ Ĥ \ar[r]^{Δ⊗Δ̂} & H ⊗ H ⊗ Ĥ ⊗ Ĥ \ar[r]^{H⊗σ⊗Ĥ} & H ⊗ Ĥ ⊗ H ⊗ Ĥ \ar[r]^{H⊗ev⊗Ĥ} & H ⊗ Ĥ.}

The symmetry σ : X ⊗ Y → Y ⊗ X, for X, Y ∈ {H, Ĥ}, is a distributive law hence a weak distributive law. We show that λ : H ⊗ Ĥ → Ĥ ⊗ H is a weak distributive law. The morphism

ξ := ( \xymatrix{Ĥ ⊗ H \ar[r]^{Δ̂⊗H} & Ĥ^{⊗2} ⊗ H \ar[r]^{Ĥ⊗ev} & Ĥ} )

is an associative (and evidently unital) action in the sense of commutativity of

\xymatrix{Ĥ ⊗ H^{⊗2} \ar[rr]^{Δ̂⊗H^{⊗2}} \ar[dd]_{Ĥ⊗σ} \ar@{-->}[dr]^{Δ̂⊗H^{⊗2}} &  & Ĥ^{⊗2} ⊗ H^{⊗2} \ar[r]^{Ĥ⊗ev⊗H} \ar[d]^{Δ̂⊗Ĥ⊗H^{⊗2}} & Ĥ ⊗ H \ar[d]^{Δ̂⊗H} \\  & Ĥ^{⊗2} ⊗ H^{⊗2} \ar[r]^{Ĥ⊗Δ̂⊗H^{⊗2}} \ar[d]^{Ĥ^{⊗2}⊗σ} & Ĥ^{⊗3} ⊗ H^{⊗2} \ar[r]^{Ĥ^{⊗2}⊗ev⊗H} & Ĥ^{⊗2} ⊗ H \ar[dd]^{Ĥ⊗ev} \\ Ĥ ⊗ H^{⊗2} \ar[r]^{Δ̂⊗H^{⊗2}} \ar[d]_{Ĥ⊗μ} & Ĥ^{⊗2} ⊗ H^{⊗2} \ar[d]^{Ĥ^{⊗2}⊗μ} \\ Ĥ ⊗ H \ar[r]_{Δ̂⊗H} & Ĥ^{⊗2} ⊗ H \ar[rr]_{Ĥ⊗ev} &  & Ĥ}  (27)

where the bottom-right region commutes by the first identity in (26). In terms of ξ,

λ = ( \xymatrix{H ⊗ Ĥ \ar[r]^{σ} & Ĥ ⊗ H \ar[r]^{Ĥ⊗Δ} & Ĥ ⊗ H^{⊗2} \ar[r]^{ξ⊗H} & Ĥ ⊗ H} ).

Using this form of λ, its compatibility with the multiplication of H follows by commutativity of the diagram below.

[figure: H^{⊗2} ⊗ Ĥ; H⊗σ; H⊗Δ⊗Ĥ; σ_{H⊗H,Ĥ}; μ⊗Ĥ; H ⊗ Ĥ ⊗ H; H⊗Ĥ⊗Δ; H ⊗ Ĥ ⊗ H^{⊗2}; H⊗ξ⊗H; σ_{H,Ĥ⊗H}⊗H; H ⊗ Ĥ ⊗ H; σ⊗H; H^{⊗3} ⊗ Ĥ; H⊗σ_{H⊗H,Ĥ}; Δ⊗H^{⊗2}⊗Ĥ; Ĥ ⊗ H^{⊗3}; ξ⊗H^{⊗2}; Ĥ⊗H⊗Δ⊗H; Ĥ ⊗ H^{⊗2}; Ĥ⊗Δ⊗H; H^{⊗4} ⊗ Ĥ; σ_{H^{⊗4},Ĥ}; Ĥ ⊗ H^{⊗4}; ξ⊗H^{⊗3}; Ĥ⊗σ⊗H^{⊗2}; (27); Ĥ ⊗ H^{⊗3}; ξ⊗H^{⊗2}; Ĥ ⊗ H^{⊗2}; Ĥ⊗Δ⊗Δ; Ĥ⊗μ; Ĥ ⊗ H^{⊗4}; Ĥ⊗H⊗σ⊗H; Ĥ⊗σ_{H⊗H,H}⊗H; Ĥ ⊗ H^{⊗4}; Ĥ⊗μ⊗H^{⊗2}; Ĥ ⊗ H^{⊗3}; ξ⊗H^{⊗2}; Ĥ⊗H⊗μ; Ĥ ⊗ H^{⊗2}; Ĥ⊗μ; H ⊗ Ĥ; σ; Ĥ ⊗ H; Ĥ⊗Δ; Ĥ ⊗ H^{⊗2}; ξ⊗H; Ĥ ⊗ H]

The region at the middle of the bottom row commutes by the first weak bialgebra axiom. Symmetrically, in terms of ζ := (ev ⊗ H).(Ĥ ⊗ Δ), we can write λ = (Ĥ ⊗ ζ).(Δ̂ ⊗ H).σ. With this form of λ at hand, its compatibility with the multiplication of Ĥ follows symmetrically. It remains to check the weak unitality condition (2). For that consider the (idempotent) morphism

ε̄_{s} := ( \xymatrix{H \ar[r]^{η⊗H} & H^{⊗2} \ar[r]^{Δ⊗H} & H^{⊗3} \ar[r]^{H⊗μ} & H^{⊗2} \ar[r]^{H⊗ε} & H} ).

Recall from [Böhm, Caenepeel, Janssen (2011)] (equations (4) and (8), respectively) that the following diagrams involving ε̄_{s} commute.

\xymatrix{H^{⊗2} \ar[r]^{Δ⊗H} \ar[d]_{H⊗ε̄_{s}} & H^{⊗3} \ar[r]^{H⊗μ} & H^{⊗2} \ar[d]^{H⊗ε} & H \ar[r]^{η⊗H} \ar[d]_{Δ} & H^{⊗2} \ar[r]^{Δ⊗H} & H^{⊗3} \ar[d]^{H⊗μ} \\ H^{⊗2} \ar[rr]_{μ} &  & H & H^{⊗2} \ar[rr]_{ε̄_{s}⊗H} &  & H^{⊗2}}  (28)

Using the definitions in (26), the first identity in (28) is equivalent to ξ.(Ĥ ⊗ ε̄_{s}) = μ̂.(Ĥ ⊗ ξ).(Ĥ ⊗ η̂ ⊗ H), implying commutativity of the bottom-right region in

[figure: Ĥ ⊗ H; η⊗Ĥ⊗H; Ĥ⊗η⊗H; H ⊗ Ĥ ⊗ H; σ⊗H; Ĥ ⊗ H^{⊗2}; Ĥ⊗Δ⊗H; (28); Ĥ ⊗ H^{⊗3}; ξ⊗H^{⊗2}; Ĥ⊗H⊗μ; Ĥ ⊗ H^{⊗2}; Ĥ⊗μ; Ĥ ⊗ H; Ĥ⊗Δ; Ĥ⊗H⊗η̂; Ĥ⊗η̂⊗H; Ĥ ⊗ H^{⊗2}; Ĥ⊗ε̄_{s}⊗H; Ĥ⊗η̂⊗H^{⊗2}; Ĥ ⊗ H^{⊗2}; ξ⊗H; Ĥ ⊗ H; Ĥ ⊗ H ⊗ Ĥ; Ĥ⊗σ; Ĥ^{⊗2} ⊗ H; Ĥ^{⊗2}⊗Δ; Ĥ^{⊗2} ⊗ H^{⊗2}; Ĥ⊗ξ⊗H; Ĥ^{⊗2} ⊗ H; μ̂⊗H; Ĥ ⊗ H .]

Any path in this diagram yields an alternative expression of the idempotent \overline{λ} : Ĥ ⊗ H → Ĥ ⊗H, proving that λ is a weak distributive law. By symmetrical considerations so is λ̂.

With some routine computations using the weak bialgebra axioms, one checks that \overline{λ} is equal to the identity map – that is, λ is a distributive law in the strict sense – if and only if Δ.η = η ⊗ η; that is, H is a bialgebra in the strict sense.

Our next task is to check the Yang-Baxter conditions. The symmetry operators among themselves obey the Yang-Baxter condition, hence for {i, j, k} such that j−i > 1 and k − j > 1 we are done. For {i − 1, i, j} and {i, j, j + 1}, such that j − i > 1, the Yang-Baxter conditions follow by naturality of the symmetry. So we are left with the case {i − 1, i, i + 1}. Assume first that i is odd. Then the Yang-Baxter condition follows by commutativity of

\xymatrix{H ⊗ Ĥ ⊗ H \ar[r]^{σ⊗H} \ar[dd]^{H⊗σ} & Ĥ ⊗ H^{⊗2} \ar[r]^{Δ̂⊗Δ⊗H} \ar[dd]^{σ_{Ĥ⊗H,H}} & Ĥ^{⊗2} ⊗ H^{⊗3} \ar[r]^{Ĥ⊗ev⊗H^{⊗2}} \ar[dd]^{σ_{Ĥ^{⊗2}⊗H^{⊗2},H}} & Ĥ ⊗ H^{⊗2} \ar[d]_{Ĥ⊗σ} \\  &  &  & Ĥ ⊗ H^{⊗2} \ar[d]_{σ⊗H} \\ H^{⊗2} ⊗ Ĥ \ar[r]^{σ_{H,H⊗Ĥ}} \ar[d]^{H⊗Δ⊗Δ̂} & H ⊗ Ĥ ⊗ H \ar[r]^{H⊗Δ̂⊗Δ} \ar[d]^{Δ⊗Δ̂⊗H} & H ⊗ Ĥ^{⊗2} ⊗ H^{⊗2} \ar[r]^{H⊗Ĥ⊗ev⊗H} \ar[d]^{Δ⊗Δ̂⊗Ĥ⊗H^{⊗2}} & H ⊗ Ĥ ⊗ H \ar[d]_{Δ⊗Δ̂⊗H} \\ H^{⊗3} ⊗ Ĥ^{⊗2} \ar[d]^{H^{⊗2}⊗ev.σ⊗Ĥ} & H^{⊗2} ⊗ Ĥ^{⊗2} ⊗ H \ar[r]^{H^{⊗2}⊗Ĥ⊗Δ̂⊗Δ} \ar[dd]^{H⊗ev.σ⊗Ĥ⊗H} & H^{⊗2} ⊗ Ĥ^{⊗3} ⊗ H^{⊗2} \ar[dd]^{H⊗⊗ev.σ⊗Ĥ^{⊗2}⊗H^{⊗2}} & H^{⊗2} ⊗ Ĥ^{⊗2} ⊗ H \ar[dd]_{H⊗ev.σ⊗Ĥ⊗H} \\ H^{⊗2} ⊗ Ĥ \ar@{-->}[dr]^{σ_{H,H⊗Ĥ}} \ar[d]^{σ⊗Ĥ} \\ H^{⊗2} ⊗ Ĥ \ar[r]_{H⊗σ} & H ⊗ Ĥ ⊗ H \ar[r]_{H⊗Δ̂⊗Δ} & H ⊗ Ĥ^{⊗2} ⊗ H^{⊗2} \ar[r]_{H⊗Ĥ⊗ev⊗H} & H ⊗ Ĥ ⊗ H .}

The case when i is even is treated symmetrically. This proves that the construction in this paragraph yields an object in Wdl^{(n)}(Vec) (which is an object of Mnd^{n+1}(Vec) if and only if H is a bialgebra in the strict sense). Hence by Theorem 2.10 there is a corresponding weak wreath product monad (that is, F -algebra) given as the image of the idempotent (19). Since σ is unital, one obtains the following explicit forms of this idempotent. If n is odd, then it is

\xymatrix{(H ⊗ Ĥ)^{⊗ \frac{n+1}{2}} \ar[r]^{^{\overline{λ̂} ^{⊗ \frac{n+1}{2}}}} & H ⊗ (Ĥ ⊗ H)^{⊗ \frac{n−1}{2}} ⊗ Ĥ \ar[r]^{H⊗\overline{λ} ^{⊗ \frac{n−1}{2}} ⊗Ĥ} & (H ⊗ Ĥ)^{⊗ \frac{n+1}{2}}}

and if n is even, then it is

\xymatrix{(H ⊗ Ĥ)^{⊗ \frac{n}{2}} ⊗ H \ar[r]^{\overline{λ̂} ^{⊗ \frac{n}{2}} ⊗H} & H ⊗ (Ĥ ⊗ H)^{⊗ \frac{n}{2}} \ar[r]^{H⊗\overline{λ} ^{⊗ \frac{n}{2}}} & (H ⊗ Ĥ)^{⊗ \frac{n}{2}} ⊗ H.}

If H is a bialgebra in the strict sense (for instance it is the linear span of a finite group), then these idempotents become identity maps and so the above weak wreath products reduce to wreath products in the strict sense (see [Nill, Szlachányi (1997)] and [Jara Martínez et al. (2008)]).

In the quantum spin chains in [Nill, Szlachányi, Wiesbrock (preprint 1998)] and [Böhm (1997)], where the spins take their values in a dual pair of finite dimensional weak Hopf algebras, this (n+1)-ary weak wreath product is regarded as the algebra of observable quantities localized in the interval [0, n] of the one dimensional lattice. In particular, in spin chains built on dual pairs of finite dimensional Hopf algebras (for instance on pairs of a finite group algebra and the algebra of linear functions on this group), the observable algebra is a proper (n+1)-ary wreath product. In the classical Ising model – where the spins only have ‘up’ and ‘down’ positions – these dual Hopf algebras are both isomorphic to the linear span of the sign group ℤ(2).

4. A fully faithful embedding

In this section we show that, for any 2-category K, and any non-negative integer n, Wdl^{(n)}(\overline{K}) admits a fully faithful embedding into the power 2-category Mnd(\overline{K})^{2^{n+1}}. Whenever idempotent 2-cells in K split, this gives rise to a fully faithful embedding Wdl^{(n)}(K) → Mnd(K)^{2^{n+1}}. If in addition K admits Eilenberg-Moore objects, this amounts to a fully faithful embedding Wdl^{(n)}(K) → K^{2^{n+1}}.

4.1. The 2-category K^{2^{n}} The 2-category 2 has two 0-cells 0 and 1; an only nonidentity 1-cell 1 → 0; and all of its 2-cells are identities. For any 2-category K, there is a 2-category K^{2} of 2-functors 2 → K, 2-natural transformations and modifications. Iteratively, for n > 1 we define K^{2^{n}} as (K^{2^{n−1}})^{2}. That is, K^{2^{n}} is isomorphic to the 2-category of 2-functors from the n-fold Cartesian product 2 × … × 2 to K, 2-natural transformations and modifications. An explicit description is given as follows. The 0-cells are the n dimensional oriented cubes whose 2-faces are commutative squares of 1-cells in K. A 1-cell from an n-cube of edges \{v_{\underline{p},\underline{q}} : A\underline{p} → A\underline{q}\} to \{v′_{\underline{p},\underline{q}} : A′ ; \underline{p} → A′ ; \underline{q}\} consists of 1-cells \{u\underline{p} : A\underline{p} → A′ ; \underline{p}\} in K such that the n + 1-cube \{v_{\underline{p},\underline{q}} : A\underline{p} → A\underline{q}, u\underline{p} : A\underline{p} → A′ ; \underline{p}, v′_{\underline{p},\underline{q}} : A′ ; \underline{p} → A′ ; \underline{q}\} is commutative. That is, for all values of \underline{p} and \underline{q}, v′_{\underline{p},\underline{q}}.u\underline{p} = u\underline{q}.v_{\underline{p},\underline{q}}. Finally, 2-cells consist of 2-cells ω\underline{p} : u\underline{p} → ũ\underline{p} in K such that v′_{\underline{p},\underline{q}}.ω\underline{p} = ω\underline{q}.v_{\underline{p},\underline{q}}.

In Cartesian coordinates, the vertices of an n-cube can be labelled by the elements \underline{p} = (p_{1}, …, p_{n}) of the set \{0, 1\}^{n}. Sometimes we represent \underline{p} ∈ \{0, 1\}^{n} by listing those values of i for which p_{i} = 1. For example, 12 = (1, 1, 0, …, 0), 3 = (0, 0, 1, 0, …, 0), etc.. The n-cube has an edge \underline{p} → \underline{q} if and only if there is some integer 1 ≤ i ≤ n such that p_{j} = q_{j} for all j ≠ i, p_{i} = 0 and q_{i} = 1. We denote this situation by \underline{q} = \underline{p} + i. For \underline{p}, \underline{q} ∈ \{0, 1\}^{n}, we say that \underline{p} < \underline{q} if, for any 1 ≤ i, j ≤ n, the equality p_{i}q_{j} = 1 implies i < j. For \underline{p} < \underline{q} we define \underline{p} + \underline{q} ∈ \{0, 1\}^{n} putting (\underline{p} + \underline{q})_{i} := p_{i} + q_{i}. We denote by \underline{0} := (0, 0, …, 0) and \underline{1} := (1, 1, …, 1) the constant elements of \{0, 1\}^{n}.

The construction of the promised 2-functor Wdl^{(n)}(\overline{K}) → Mnd(\overline{K})^{2^{n+1}} relies on a few lemmas below. A routine computation proves the first one:

4.2. Lemma. For any object \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{0≤i<j≤2} of Wdl^{(2)}(\overline{K}), there is a homomorphism of monads in \overline{K},

φ_{0,2}^{1} := \overline{λ}_{012}.s_{0}η_{1}s_{2} : (s_{0}s_{2}, \overline{λ}_{02}) → (s_{0}s_{1}s_{2}, \overline{λ}_{012}).

This means that there is a 1-cell ((A, A), φ_{0,2}^{1}):(A, (s_{0}s_{1}s_{2}, \overline{λ}_{012})) → (A, (s_{0}s_{2}, \overline{λ}_{02})) in Mnd(\overline{K}) (where A is the object underlying the monads (A, s_{i})).

4.3. Lemma. For any object \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{0≤i<j≤4} of Wdl^{(4)}(\overline{K}), the morphisms as in Lemma 4.2 constitute a commutative diagram in \overline{K}:

\xymatrix{(s_{0}s_{2}s_{4}, \overline{λ}_{024}) \ar[r]^{φ_{0,24}^{1}} \ar[d]_{φ_{02,4}^{3}} & (s_{0}s_{1}s_{2}s_{4}, \overline{λ}_{0124}) \ar[d]^{φ_{012,4}^{3}} \\ (s_{0}s_{2}s_{3}s_{4}, \overline{λ}_{0234}) \ar[r]_{φ_{0,234}^{1}} & (s_{0}s_{1}s_{2}s_{3}s_{4}, \overline{λ}_{01234}) .}

Proof. In view of Lemma 4.2, both paths around the diagram are equal to \overline{λ}_{01234}. s_{0}η_{1}s_{2}η_{3}s_{4}.

4.4. Lemma. For any 1-cell \{ξ_{i} : s′_{i}v → vs_{i}\}_{0≤i≤2} in Wdl^{(2)}(\overline{K}), the morphisms in Lemma 4.2 induce a commutative square

\xymatrix{(A, (s_{0}s_{1}s_{2}, \overline{λ}_{012})) \ar[r]^{((v,\overline{v}),ξ_{012})} \ar[d]_{((A,A),φ_{0,2}^{1})} & (A′, (s′_{0}s′_{1}s′_{2}, \overline{λ′}_{012})) \ar[d]^{((A′,A′),φ′_{0,2}^{1})} \\ (A, (s_{0}s_{2}, \overline{λ}_{02})) \ar[r]_{((v,\overline{v}),ξ_{02})} & (A′, (s′_{0}s′_{2}, \overline{λ′}_{02}))}

in Mnd(\overline{K}).

Proof. Both paths around the diagram are computed to be equal to

v\overline{λ}_{012}.vs_{0}η_{1}s_{2}.ξ_{0}s_{2}.s′_{0}ξ_{2} = v\overline{λ}_{012}.ξ_{0}s_{1}s_{2}.s′_{0}ξ_{1}s_{2}.s′_{0}s′_{1}ξ_{2}.\overline{λ′}_{012}v.s′_{0}η′_{1}s′_{2}v,

where the last equality follows by the unitality of ξ_{i} and the normalization property ξ_{012}.\overline{λ′}_{012}v = ξ_{012}.

4.5. A 2-functor F : Wdl^{(n)}(\overline{K}) → Mnd(\overline{K})^{2^{n+1}} For any non-negative integer n, and any \underline{0} ≠ \underline{p} = (p_{0}, …, p_{n}) ∈ \{0, 1\}^{n+1}, taking those values of 0 ≤ i ≤ n for that p_{i} = 1, defines a 2-functor Wdl^{(n)}(\overline{K}) → Wdl^{(−1+∑_{i} p_{i})}(\overline{K}). Composing it with the unique iterated weak wreath product 2-functor Wdl^{(−1+∑_{i} p_{i})}(\overline{K}) → Mnd(\overline{K}) in Section 2, yields a 2-functor Wdl^{(n)}(\overline{K}) → Mnd(\overline{K}). Denote the image of

\xymatrix{\{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{0≤i<j≤n} \ar[rr]^{\{ξ_{i}:s′_{i}v→vs_{i}\}_{0≤i≤n}} \ar[rr]_{\{ξ′_{i}:s′_{i}v′→v′s_{i}\}_{0≤i≤n}} & ⇓ ω & \{λ′_{i,j} : s′_{j}s′_{i} → s′_{i}s′_{j}\}_{0≤i<j≤n}}

under it by

\xymatrix{(A, (s\underline{p}, \overline{λ}\underline{p})) \ar[rr]^{ξ\underline{p}:s′ ; \underline{p}v→vs\underline{p}} \ar[rr]_{ξ′ ; \underline{p}:s′ ; \underline{p}v′→v′s\underline{p}} & ⇓ ω & (A′, (s′ ; \underline{p}, \overline{λ′}\underline{p})) .}

Then (A, s\underline{p}) is the weak wreath product of those demimonads (A, s_{i}) for which p_{i} = 1. For \underline{0} ∈ \{0, 1\}^{n+1}, put (A, s\underline{0}) := (A, A) and ξ\underline{0} := v.

By Lemmas 4.2 and 4.3, for any object \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{0≤i<j≤n} of Wdl^{(n)}(\overline{K}), for any \underline{p} ∈ \{0, 1\}^{n+1}, and for any 0 ≤ i, j ≤ n such that p_{i} = p_{j} = 0, there is a commutative square

\xymatrix{(A, (s_{\underline{p}+i+j}, \overline{λ}_{\underline{p}+i+j})) \ar[r]^{((A,A),φ_{\underline{p}+j}^{i})} \ar[d]_{((A,A),φ_{\underline{p}+i}^{j})} & (A, (s_{\underline{p}+j}, \overline{λ}_{\underline{p}+j})) \ar[d]^{((A,A),φj ; \underline{p})} \\ (A, (s_{\underline{p}+i}, \overline{λ}_{\underline{p}+i})) \ar[r]_{((A,A),φi ; \underline{p})} & (A, (s\underline{p}, \overline{λ}\underline{p}))}

in Mnd(\overline{K}). Such squares constitute a commutative n + 1-cube in Mnd(\overline{K}).

By Lemma 4.4, for any 1-cell \{ξ_{i} : s′_{i}v → vs_{i}\}_{0≤i≤n} of Wdl^{(n)}(\overline{K}), there is a commutative square

\xymatrix{(A, (s_{\underline{p}+i}, \overline{λ}_{\underline{p}+i})) \ar[r]^{((v,\overline{v}),ξ_{\underline{p}+i})} \ar[d]_{((A,A),φi ; \underline{p})} & (A′, (s′_{\underline{p}+i}, \overline{λ′}_{\underline{p}+i})) \ar[d]^{((A′,A′),φ′i ; \underline{p})} \\ (A, (s\underline{p}, \overline{λ}\underline{p})) \ar[r]_{((v,\overline{v}),ξ\underline{p})} & (A′, (s′ ; \underline{p}, \overline{λ′}\underline{p}))}

in Mnd(\overline{K}). Hence the 1-cells ((v, \overline{v}), ξ\underline{p}) : (A, (s\underline{p}, \overline{λ}\underline{p})) → (A′, (s′ ; \underline{p}, \overline{λ′}\underline{p})) constitute a 1-cell in Mnd(\overline{K})^{2^{n+1}}.

Finally, for a 2-cell ω : \{ξ_{i} : s′_{i}v → vs_{i}\}_{0≤i≤n} → \{ξ′_{i} : s′_{i}v′ → v′s_{i}\}_{0≤i≤n} in Wdl^{(n)}(\overline{K}), ω is a 2-cell ((v, \overline{v}), ξ\underline{p}) → ((v′, \overline{v′}), ξ′ ; \underline{p}) in Mnd(\overline{K}), for any \underline{p} ∈ \{0, 1\}^{n+1}, which constitutes a 2-cell in Mnd(\overline{K})^{2^{n+1}}.

The above maps define the stated 2-functor F : Wdl^{(n)}(\overline{K}) → Mnd(\overline{K})^{2^{n+1}}.

4.6. Theorem. For any 2-category K, and any non-negative integer n, the 2-functor Wdl^{(n)}(\overline{K}) → Mnd(\overline{K})^{2^{n+1}} in Paragraph 4.5 is fully faithful.

Proof. Faithfulness is obvious. In order to prove fullness on the 1-cells, take a 1cell \{ζ\underline{p} : s′ ; \underline{p}v → vs\underline{p}\}_{\underline{p}∈\{0,1\}^{n+1}} in Mnd(\overline{K})^{2^{n+1}} between objects arising from 0-cells \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{0≤i<j≤n} and \{λ′_{i,j} : s′_{j}s′_{i} → s′_{i}s′_{j}\}_{0≤i<j≤n} of Wdl^{(n)}(\overline{K}). This includes in particular 1-cells ξ_{i} := ζ_{i} : s′_{i}v → vs_{i} in Mnd(\overline{K}). We claim that for i ∈ \{0, …, n\} they constitute a 1-cell in Wdl^{(n)}(\overline{K}) and each ζ\underline{p} is equal to their weak wreath product. By commutativity of the squares

\xymatrix{(A, (s_{i}s_{j}, \overline{λ}_{ij})) \ar[r]^{((v,\overline{v}),ζ_{ij})} \ar[d]_{((A,A),φ_{i}^{j})} & (A′, (s′_{i}s′_{j}, \overline{λ′}_{ij})) \ar[d]^{((A′,A′),φ′_{i}^{j})} & (A, (s_{i}s_{j}, \overline{λ}_{ij})) \ar[r]^{((v,\overline{v}),ζ_{ij})} \ar[d]_{((A,A),φ_{j}^{i})} & (A′, (s′_{i}s′_{j}, \overline{λ′}_{ij})) \ar[d]^{((A′,A′),φ′_{j}^{i})} \\ (A, (s_{i}, \overline{s}_{i})) \ar[r]_{((v,\overline{v}),ξ_{i})} & (A′, (s′_{i}, \overline{s}′_{i})) & (A, (s_{j}, \overline{s}_{j})) \ar[r]_{((v,\overline{v}),ξ_{j})} & (A′, (s′_{j}, s′_{j}))}

in Mnd(\overline{K}), we conclude the commutativity of the diagrams

\xymatrix{s′_{i}v \ar[d]_{ξ_{i}} \ar[r]^{s′_{i}η′_{j}v} & s′_{i}s′_{j}v \ar[r]^{\overline{λ′}_{ij}v} \ar[dr]^{ζ_{ij}} & s′_{i}s′_{j}v \ar[d]^{ζ_{ij}} & s′_{j}v \ar[d]_{ξ_{j}} \ar[r]^{η′_{i}s′_{j}v} & s′_{i}s′_{j}v \ar[r]^{\overline{λ′}_{ij}v} \ar[dr]^{ζ_{ij}} & s′_{i}s′_{j}v \ar[d]^{ζ_{ij}} \\ vs_{i} \ar[r]_{vs_{i}η_{j}} & vs_{i}s_{j} \ar[r]_{v\overline{λ}_{ij}} & vs_{i}s_{j} & vs_{j} \ar[r]_{vη_{i}s_{j}} & vs_{i}s_{j} \ar[r]_{v\overline{λ}_{ij}} & vs_{i}s_{j}}  (29)

in K. Since ((v, \overline{v}), ζ_{ij}) is a 1-cell in Mnd(\overline{K}), the following diagram commutes

[figure: s′_{i}s′_{j}v; s′_{i}ξ_{j}; s′_{i}η′_{i}s′_{j}v; \overline{λ′}_{ij}v; (29); s′_{i}vs_{j}; ξ_{i}s_{j}; s′_{i}vη_{i}s_{j}; vs_{i}s_{j}; vs_{i}η_{i}s_{j}; v\overline{λ}_{ij}; s′_{i}vs_{i}s_{j}; ξ_{i}s_{i}s_{j}; s′_{i}η′_{j}vs_{i}s_{j}; s′_{i}v\overline{λ}_{ij}; (29); vs_{i}^{2}s_{j}; vs_{i}η_{j}s_{i}s_{j}; v(s_{i}s_{j})^{2}; v\overline{λ}_{ij}s_{i}s_{j}; s_{i}^{′2}s′_{j}v; s′_{i}ζ_{ij}; s′_{i}η′_{j}s′_{i}s′_{j}v; s′_{i}vs_{i}s_{j}; s′_{i}η′_{j}vs_{i}s_{j}; s′_{i}s′_{j}vs_{i}s_{j}; ζ_{ij}s_{i}s_{j}; v(s_{i}s_{j})^{2}; vμ_{ij}; (s′_{i}s′_{j})^{2}v; s′_{i}s′_{j}ζ_{ij}; μ′_{ij}v; s′_{i}s′_{j}vs_{i}s_{j}; ζ_{ij}s_{i}s_{j}; v(s_{i}s_{j})^{2}; vμ_{ij}; s′_{i}s′_{j}v; ζ_{ij}; vs_{i}s_{j} ,] (30)

where we denoted μ_{ij} = μ_{i}μ_{j}.s_{i}λ_{i,j}s_{j}. Since ζ_{ij}.\overline{λ′}_{ij}v = ζ_{ij}, this says that ζ_{ij} is equal to v\overline{λ}_{ij}.ξ_{i}s_{j}.s′_{i}ξ_{j}; that is, ζ_{ij} is the weak wreath product of ξ_{i} and ξ_{j}. With this expression of ζ_{ij} at hand, also the following diagram commutes.

[figure: s′_{j}s′_{i}v; s′_{j}s′_{i}η′_{j}v; s′_{j}ξ_{i}; λ′_{i,j}v; (29); s′_{j}s′_{i}s′_{j}v; η′_{i}s′_{j}s′_{i}s′_{j}v; s′_{j}ζ_{ij}; (s′_{i}s′_{j})^{2}v; μ′_{ij}v; s′_{i}s′_{j}ζ_{ij}; s′_{i}s′_{j}v; ζ_{ij}; s′_{i}ξ_{j}; s′_{j}vs_{i}; s′_{j}vs_{i}η_{j}; ξ_{j}s_{i}; s′_{j}vs_{i}s_{j}; s′_{j}v\overline{λ}_{ij}; s′_{j}vs_{i}s_{j}; η′_{i}s′_{j}vs_{i}s_{j}; ξ_{j}s_{i}s_{j}; (29); s′_{i}s′_{j}vs_{i}s_{j}; ζ_{ij}s_{i}s_{j}; s′_{i}vs_{j}; ξ_{i}s_{j}; vs_{j}s_{i}; vs_{j}s_{i}η_{j}; vs_{j}s_{i}s_{j}; vs_{j} \overline{λ}_{ij}; vη_{i}s_{j}s_{i}s_{j}; vs_{j}s_{i}s_{j}; vη_{i}s_{j}s_{i}s_{j}; v(s_{i}s_{j})^{2}; vs_{i}s_{j} \overline{λ}_{ij}; vμ_{ij}; v(s_{i}s_{j})^{2}; v\overline{λ}_{ij}s_{i}s_{j}; v(s_{i}s_{j})^{2}; vμ_{ij}; vs_{i}s_{j}; v\overline{λ}_{ij}; vs_{j}s_{i}; vλ_{i,j}; vs_{j}s_{i}; vs_{i}s_{j}]

That is, (v, ξ_{i}, ξ_{j}) is a 1-cell in Wdl(\overline{K}). Thus the collection \{ξ_{i} : s′_{i}v → vs_{i}\}_{0≤i≤n} is a 1-cell in Wdl^{(n)}(\overline{K}). The same reasoning as in (30) shows that its image under the 2-functor in the claim is the 1-cell \{ζ\underline{p} : s\underline{p}v → vs\underline{p}\}_{\underline{p}∈\{0,1\}^{n+1}} in Mnd(\overline{K})^{2^{n+1}} that we started with. Fullness on the 2-cells is evident.

4.7. If idempotent 2-cells split Let K be a 2-category in which idempotent 2-cells split; that is, biequivalent to \overline{K}, and consider the 2-functor

\xymatrix{Wdl^{(n)}(K) \ar[r]^{≃} & Wdl^{(n)}(\overline{K}) \ar[r]^{F} & Mnd(\overline{K})^{2^{n+1}} ,}  (31)

where F is the 2-functor introduced in Paragraph 4.5. By Paragraph 4.5, it takes a 1-cell \{ξ_{i} : s′_{i}v → vs_{i}\}_{0≤i≤n} to an n + 2-cube in Mnd(\overline{K}), with faces of the form in the first diagram in

\xymatrix{(A, (s_{\underline{p}+i}, \overline{λ}_{\underline{p}+i})) \ar[r]^{((v,v),ξ_{\underline{p}+i})} \ar[d]_{((A,A),φi ; \underline{p})} & (A′, (s′_{\underline{p}+i}, \overline{λ′}_{\underline{p}+i})) \ar[d]^{((A′,A′),φ′i ; \underline{p})} & (A, z_{\underline{p}+i}) \ar[r]^{(v,ζ_{\underline{p}+i})} \ar[d]_{(A,ψi ; \underline{p})} & (A′, z′_{\underline{p}+i}) \ar[d]^{(A′,ψ′ ; \underline{p}^{i})} \\ (A, (s\underline{p}, \overline{λ}\underline{p})) \ar[r]_{((v,v),ξ\underline{p})} & (A′, (s′ ; \underline{p}, \overline{λ′}\underline{p})) & (A, z\underline{p}) \ar[r]_{(v,ζ\underline{p})} & (A′, z′ ; \underline{p}).}

(32)

Let us choose a biequivalence pseudofunctor \overline{K} → K adopting the convention that we split any identity 2-cell trivially; that is, via identity 2-cells. Then the induced biequivalence Mnd(\overline{K}) → Mnd(K) takes the first square in (32) to a commutative square in Mnd(K) of the form in the second diagram. Mapping {ξ_{i} : s′_{i}v → vs_{i}}_{0≤i≤n} to the n + 2-cube in Mnd(K) formed by these faces; and mapping a 2-cell ω to the 2-cell in Mnd(K)^{2^{n+1}} whose value at each \underline{p} ∈ \{0, 1\}^{n+1} is given by ω; we obtain a fully faithful 2-functor Wdl^{(n)}(K) → Mnd(K)^{2^{n+1}}. Its composition with the biequivalence Mnd(K)^{2^{n+1}} →^{≃} Mnd(\overline{K})^{2^{n+1}} is 2-naturally isomorphic to (31).

4.8. If Eilenberg-Moore objects exist Recall (from [Street (1972)]) that a 2-category K is said to admit Eilenberg-Moore objects provided that the evident inclusion I : K → Mnd(K) possesses a right 2-adjoint J. Whenever J exists, it induces a fully faithful 2-functor Mnd(K) → K^{2} as follows. It takes a monad (A, t) to the 1-cell part of the counit of the 2-adjunction I ⊣ J evaluated at (A, t); that is, the so-called “forgetful morphism” J(A, t) → A in K. (The terminology certainly comes from its form in K = Cat.) It takes a 1-cell (v, ψ) to the pair (v, J(v, ψ)) and it takes a 2-cell ω to the pair (ω, Jω).

4.9. Corollary. Let K be a 2-category in which idempotent 2-cells split and which admits Eilenberg-Moore objects. Then composing the fully faithful 2-functor Wdl^{(n)}(K) → Mnd(K)^{2^{n+1}} in Paragraph 4.7 with J^{2^{n+1}} : Mnd(K)^{2^{n+1}} → K^{2^{n+1}}, we obtain a fully faithful embedding.

5. The n-ary factorization problem

The aim of this section is to find sufficient and necessary conditions on a demimonad (that is, a monad in the local idempotent closure of a 2-category) to be isomorphic to a weak wreath product of n demimonads. Some facts about the n = 2 case are recalled in Paragraph 1.6.

In the next theorem we shall use the notation introduced after Paragraph 4.1.

5.1. Theorem. For any demimonad (A, s) in an arbitrary 2-category K, the following assertions are equivalent.

(i) There is an object \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{1≤i<j≤n} of Wdl^{(n−1)}(\overline{K}) such that the corresponding n-ary weak wreath product (that is, its image under the 2-functor in Theorem 2.10) is isomorphic to (A, s).

(ii) There is an n dimensional cube whose 2-faces are commutative squares of monad morphisms in \overline{K} of the form (A, φi ; \underline{p}) : (A, s_{\underline{p}+i}) → (A, s\underline{p}), such that the following hold. For \underline{p} < \underline{q} ∈ \{0, 1\}^{n}, denote by φ\underline{p}\underline{q} and by φ\underline{p}\underline{q} the (unique) morphisms composed along any path to \underline{p} + \underline{q} from \underline{p} and from \underline{q}, respectively. Then

(a) (A, s\underline{0}) is the trivial monad (A, A) and (A, s\underline{1}) is isomorphic to (A, s).

(b) For all \underline{p} < \underline{q} ∈ \{0, 1\}^{n}, the 2-cell

π_{\underline{p},\underline{q}} := ( \xymatrix{s\underline{p}s\underline{q} \ar[r]^{^{φ\underline{p}\underline{q} φ\underline{p}\underline{q}}} & s_{\underline{p}+\underline{q}}s_{\underline{p}+\underline{q}} \ar[r]^{μ_{\underline{p}+\underline{q}}} & s_{\underline{p}+\underline{q}}} )

possesses an s\underline{p}-s\underline{q} bimodule section ι_{\underline{p},\underline{q}} in \overline{K}.

(c) For all \underline{p} < \underline{q} < \underline{r} ∈ \{0, 1\}^{n}, the morphisms ι in part (b) render commutative the following diagrams.

\xymatrix{s_{\underline{p}+\underline{q}+\underline{r}} \ar[r]^{ι_{\underline{p},\underline{q}+\underline{r}}} \ar[d]_{ι_{\underline{p}+\underline{q},\underline{r}}} & s\underline{p}s_{\underline{q}+\underline{r}} \ar[d]^{s\underline{p}ι_{\underline{q},\underline{r}}} \\ s_{\underline{p}+\underline{q}}s\underline{r} \ar[r]_{ι_{\underline{p},\underline{q}}s\underline{r}} & s\underline{p}s\underline{q}s\underline{r}}  (33)

\xymatrix{s_{\underline{q}+\underline{r}}s\underline{p} \ar[r]^{ι_{\underline{q},\underline{r}}s\underline{p}} \ar[dd]_{_{φ\underline{p}_{\underline{q}+\underline{r}} φ\underline{p}_{\underline{q}+\underline{r}}}} & s\underline{q}s\underline{r}s\underline{p} \ar[d]^{s\underline{q}φ\underline{p}\underline{r} φ\underline{p}\underline{r}} & s\underline{r}s_{\underline{p}+\underline{q}} \ar[r]^{s\underline{r}ι_{\underline{p},\underline{q}}} \ar[dd]_{_{φ^{\underline{p}+\underline{q}}\underline{r} φ_{\underline{p}+\underline{q}}\underline{r}}} & s\underline{r}s\underline{p}s\underline{q} \ar[d]^{φ\underline{p}\underline{r} φ\underline{p}\underline{r}s\underline{q}} \\  & s\underline{q}s_{\underline{p}+\underline{r}}s_{\underline{p}+\underline{r}} \ar[d]^{s\underline{q}μ_{\underline{p}+\underline{r}}} &  & s_{\underline{p}+\underline{r}}s_{\underline{p}+\underline{r}}s\underline{q} \ar[d]^{μ_{\underline{p}+\underline{r}}s\underline{q}} \\ s_{\underline{p}+\underline{q}+\underline{r}}s_{\underline{p}+\underline{q}+\underline{r}} \ar[dd]_{μ_{\underline{p}+\underline{q}+\underline{r}}} & s\underline{q}s_{\underline{p}+\underline{r}} \ar[d]^{s\underline{q}ι_{\underline{p},\underline{r}}} & s_{\underline{p}+\underline{q}+\underline{r}}s_{\underline{p}+\underline{q}+\underline{r}} \ar[dd]_{μ_{\underline{p}+\underline{q}+\underline{r}}} & s_{\underline{p}+\underline{r}}s\underline{q} \ar[d]^{ι_{\underline{p},\underline{r}}s\underline{q}} \\  & s\underline{q}s\underline{p}s\underline{r} \ar[d]^{φ\underline{p}\underline{q} φ\underline{p}\underline{q}s\underline{r}} &  & s\underline{p}s\underline{r}s\underline{q} \ar[d]^{s\underline{p}φ\underline{q}\underline{r} φ\underline{q}\underline{r}} \\ s_{\underline{p}+\underline{q}+\underline{r}} \ar[dd]_{ι_{\underline{p},\underline{q}+\underline{r}}} & s_{\underline{p}+\underline{q}}s_{\underline{p}+\underline{q}}s\underline{r} \ar[d]^{μ_{\underline{p}+\underline{q}}s\underline{r}} & s_{\underline{p}+\underline{q}+\underline{r}} \ar[dd]_{ι_{\underline{p}+\underline{q},\underline{r}}} & s\underline{p}s_{\underline{q}+\underline{r}}s_{\underline{q}+\underline{r}} \ar[d]^{s\underline{p}μ_{\underline{q}+\underline{r}}} \\  & s_{\underline{p}+\underline{q}}s\underline{r} \ar[d]^{ι_{\underline{p},\underline{q}}s\underline{r}} &  & s\underline{p}s_{\underline{q}+\underline{r}} \ar[d]^{s\underline{p}ι_{\underline{q},\underline{r}}} \\ s\underline{p}s_{\underline{q}+\underline{r}} \ar[r]_{s\underline{p}ι_{\underline{q},\underline{r}}} & s\underline{p}s\underline{q}s\underline{r} & s_{\underline{p}+\underline{q}}s\underline{r} \ar[r]_{ι_{\underline{p},\underline{q}}s\underline{r}} & s\underline{p}s\underline{q}s\underline{r}}

(34)

Proof. \underline{(i)⇒(ii).} The 2-functor in Paragraph 4.5 takes \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{1≤i<j≤n} to a commutative n-cube in Mnd(\overline{K}) with edges (A, φi ; \underline{p}) : (A, s_{\underline{p}+i}) → (A, s\underline{p}) of the form in Lemma 4.2. In this cube (A, s\underline{0}) is the trivial monad (A, A) and (A, s\underline{1}) is the n-ary weak wreath product which is isomorphic to (A, s) by assumption. Thus property (a) holds. By construction of the 2-functor in Paragraph 4.5, (A, s_{\underline{p}+\underline{q}}) is the weak wreath product of (A, s\underline{p}) and (A, s\underline{q}), for all \underline{p} < \underline{q} ∈ \{0, 1\}^{n}. Hence the 2-cell π_{\underline{p},\underline{q}} in part (b) possesses a bilinear section ι_{\underline{p},\underline{q}} by Paragraph 1.6. It remains to show that the diagrams in part (c) commute.

The monic 2-cell ι_{\underline{p},\underline{q}} is given by \overline{λ}_{\underline{p}+\underline{q}} : (s\underline{p}s\underline{q}, \overline{λ}_{\underline{p}+\underline{q}}) → (s\underline{p}s\underline{q}, \overline{s}\underline{p}\overline{s}\underline{q}); ι_{\underline{p},\underline{q}+\underline{r}} is equal to \overline{λ}_{\underline{p}+\underline{q}+\underline{r}} : (s\underline{p}s\underline{q}s\underline{r}, \overline{λ}_{\underline{p}+\underline{q}+\underline{r}}) → (s\underline{p}s\underline{q}s\underline{r}, \overline{s}\underline{p}\overline{λ}_{\underline{q}+\underline{r}}) and so on. Hence (33) takes the form

\xymatrix{(s\underline{p}s\underline{q}s\underline{r}, \overline{λ}_{\underline{p}+\underline{q}+\underline{r}}) \ar[r]^{^{\overline{λ}^{\underline{p}+\underline{q}+\underline{r}}}} \ar[d]_{_{\overline{λ}_{\underline{p}+\underline{q}+\underline{r}}}} & (s\underline{p}s\underline{q}s\underline{r}, \overline{s}\underline{p}\overline{λ}_{\underline{q}+\underline{r}}) \ar[d]^{s\underline{p}\overline{λ}_{\underline{q}+\underline{r}}} \\ (s\underline{p}s\underline{q}s\underline{r}, \overline{λ}_{\underline{p}+\underline{q}}s\underline{r}) \ar[r]_{_{\overline{λ}_{\underline{p}+\underline{q}}\overline{s}\underline{r}}} & (s\underline{p}s\underline{q}s\underline{r}, s\underline{p}s\underline{q}s\underline{r})}

which commutes by (7) and (10). In the vertical paths of the diagrams in (34), note the occurrence of the weak distributive laws λ_{\underline{p},\underline{q}}, λ_{\underline{p},\underline{q}+\underline{r}}, etc. Thus the first diagram in (34) takes the form

\xymatrix{(s\underline{q}s\underline{r}s\underline{p}, \overline{λ}_{\underline{q}+\underline{r}}\overline{s}\underline{p}) \ar[d]_{\overline{λ}_{\underline{q}+\underline{r}}s\underline{p}} \ar[rr]^{λ_{\underline{p},\underline{q}+\underline{r}}} &  & (s\underline{p}s\underline{q}s\underline{r}, \overline{s}\underline{p}\overline{λ}_{\underline{q}+\underline{r}}) \ar[d]^{s\underline{p}\overline{λ}_{\underline{q}+\underline{r}}} \\ (s\underline{q}s\underline{r}s\underline{p}, \overline{s}\underline{q}\overline{s}\underline{r}\overline{s}\underline{p}) \ar[r]_{s\underline{q}λ_{\underline{p},\underline{r}}} & (s\underline{q}s\underline{p}s\underline{r}, \overline{s}\underline{q}\overline{s}\underline{p}\overline{s}\underline{r}) \ar[r]_{λ_{\underline{p},\underline{q}}s\underline{r}} & (s\underline{p}s\underline{q}s\underline{r}, \overline{s}\underline{p}\overline{s}\underline{q}\overline{s}\underline{r})}

which is evidently commutative in view of (17). Commutativity of the second diagram in (34) follows symmetrically.

\underline{(ii)⇒(i).} By assumption, for all 1 ≤ i < j ≤ n, the 2-cell

π_{i,j} := ( \xymatrix{s_{i}s_{j} \ar[r]^{^{φ_{i}^{j} φ^{i}^{j}}} & s_{ij}s_{ij} \ar[r]^{μ_{ij}} & s_{ij}} )

possesses an s_{i}-s_{j} bimodule section ι_{i,j}. Hence by Paragraph 1.6, there is a weak distributive law λ_{i,j} := ι_{i,j}.μ_{ij}.φ^{i}_{j} φ_{i}^{j} : s_{j}s_{i} → s_{i}s_{j} in \overline{K} such that the corresponding weak wreath product is isomorphic to s_{ij}. Let us prove that the collection {λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}}_{1≤i<j≤n} obeys the Yang-Baxter conditions, for all 1 ≤ i < j < k ≤ n. This follows by commutativity of the following diagram.

[figure: s_{k}s_{j}s_{i}; s_{k}φ^{i}_{j} φ_{i}^{j}; φ^{j}_{k} φ_{j}^{k}s_{i}; s_{k}s_{ij}s_{ij}; s_{k}μ_{ij}; _{φ^{ij}_{k} φ_{ij}_{k} φ_{ij}_{k}}; s_{k}s_{ij}; s_{k}ι_{i,j}; _{φ^{ij}_{k} φ_{ij}_{k}}; (34); s_{k}s_{i}s_{j}; φ^{i}_{k} φ_{i}^{k}s_{j}; s_{ik}s_{ik}s_{j}; μ_{ik}s_{j}; s_{jk}s_{jk}s_{i}; _{φ^{i}_{jk} φ^{i}_{jk} φ_{i}_{jk}}; μ_{jk}s_{i}; s_{ijk}s_{ijk}s_{ijk}; s_{ijk}μ_{ijk}; μ_{ijk}s_{ijk}; s_{ijk}s_{ijk}; μ_{ijk}; s_{ik}s_{j}; ι_{i,k}s_{j}; s_{i}s_{k}s_{j}; s_{i}φ^{j}_{k} φ_{j}^{k}; s_{jk}s_{i}; _{φ^{i}_{jk} φ_{i}_{jk}}; ι_{j,k}s_{i}; (34); s_{ijk}s_{ijk}; μ_{ijk}; s_{ijk}; ι_{ij,k}; ι_{i,jk}; (33); s_{ij}s_{k}; ι_{i,j}s_{k}; s_{i}s_{jk}s_{jk}; s_{i}μ_{jk}; s_{j}s_{k}s_{i}; s_{j}φ^{i}_{k} φ_{i}^{k}; s_{i}s_{jk}; s_{i}ι_{j,k}; s_{i}s_{jk}; s_{i}ι_{j,k}; s_{j}s_{ik}s_{ik}; s_{j}μ_{ik}; s_{j}s_{ik}; s_{j}ι_{i,k}; s_{j}s_{i}s_{k}; φ^{i}_{j} φ_{i}^{j}s_{k}; s_{ij}s_{ij}s_{k}; μ_{ij}s_{k}; s_{ij}s_{k}; ι_{i,j}s_{k}; s_{i}s_{j}s_{k}]

Thus \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{1≤i<j≤n} is an object of Wdl^{(n−1)}(\overline{K}). It remains to show that the corresponding n-ary weak wreath product is isomorphic to (A, s\underline{1}) ≅ (A, s).

As observed above, (A, s_{12}) is isomorphic to the weak wreath product of (A, s_{1}) and (A, s_{2}) with respect to the weak distributive law λ_{1,2} := ι_{1,2}.μ_{12}.φ^{1}_{2} φ_{1}^{2} : s_{2}s_{1} → s_{1}s_{2}. Similarly, (A, s_{123}) is isomorphic to the weak wreath product of (A, s_{12}) and (A, s_{3}) with respect to λ_{12,3} := ι_{12,3}.μ_{123}.φ^{12}_{3} φ_{12}^{3} : s_{3}s_{12} → s_{12}s_{3}. Moreover, precomposing both paths around the second diagram in (34) (for \underline{p} = 1, \underline{q} = 2 and \underline{r} = 3) by s_{3}π_{1,2} = s_{3}μ_{12}.s_{3}φ_{1}^{2}φ^{1}_{2}, we obtain that the weak distributive law λ_{12,3} : (s_{3}s_{12}, \overline{s}_{3}\overline{s}_{12}) → (s_{12}s_{3}, \overline{s}_{12}\overline{s}_{3}) differs by the isomorphisms \xymatrix{ι_{1,2} : (s_{12}, \overline{s}_{12}) \ar[r] & (s_{1}s_{2}, \overline{λ}_{12}) : π_{1,2} \ar[l]} from the image s_{1}λ_{2,3}.λ_{1,3}s_{2}.s_{3}\overline{λ}_{12} : (s_{3}s_{1}s_{2}, s_{3}\overline{λ}_{12}) → (s_{1}s_{2}s_{3}, \overline{λ}_{12}s_{3}) of \{λ_{i,j} : s_{j}s_{i} → s_{i}s_{j}\}_{1≤i<j≤3} under the 2-functor C_{1} : Wdl^{(2)}(\overline{K}) → Wdl^{(1)}(\overline{K}) in Theorem 2.6. Hence (A, s_{123}) is isomorphic to the ternary weak wreath product of {(A, s_{i})}_{1≤i≤3}. Iterating this reasoning we conclude that (A, s\underline{1}) is isomorphic to the n-ary weak wreath product of {(A, s_{i})}_{1≤i≤n}.

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Research Institute for Particle and Nuclear Physics, Budapest, H-1525 Budapest 114, P.O.B. 49, Hungary

Email: G.Bohm@rmki.kfki.hu

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