$\newcommand{\con}{\mathbf{Set}} \newcommand{\uno}{\mathbf{1}} \newcommand{\Cat}{\mathbf{Cat}}\newcommand{\mon}{\mathbf{Mon}} \newcommand{\ab}{\mathbf{Ab}} \newcommand{\an}{\mathbf{An}}\newcommand {\matcon}{\mathbf{Set\text{-}Mat}} \newcommand{\grp}{\mathbf{Grp}} \newcommand{\ob}{\mathrm{Ob}}$
The Zappa-Szép product, strict factorization systems and distributive laws are related in a certain way. Let us talk first about the Zappa-Szép product (also known as knit product or matched pair of groups).
The Zappa-Szép product is a generalization of the semidirect product for groups in the same way as this product is a generalization of the direct product for groups, and the zappa-szép product is the most general way in which a group is presented as a product of two groups.
The internal Zappa-Szép product is defined as follows. Given a group $G$ and two subgroups $H$ and $K$ of $G$, the following statements are equivalent:
There is an external version of the Zappa-Szép product, which is the one we are interested in, because it's by means of this product that a link between distributive laws and strict factorization systems is established.
Given two groups $K$ and $H$, suppose there are funtions $\alpha:H\times K\rightarrow K$ and $\beta:H\times K\rightarrow H$ such that
Note that if $G$ is an internal Zappa-Szép product of its subgroups $K$ y $H$, then there are funtions $\alpha:H\times K\rightarrow K$ and $\beta:H\times K\rightarrow H$ such that (i), (ii), (iii), (iv), (v) and (vi) in the previous definition are satisfied: their existence follows from the fact that every element $g\in G$ can be written uniquely as a product $kh$, and the fact that if $G$ is an external Zappa-Szép product of the groups $K$ and $H$, then $G$ is an internal Zappa-Szép product of its subgroups $K\times e_H$ and $e_K\times H$.
Let $H,K\in\grp$. Let $S$ and $T$ be the endofunctors $H\times(-)$ and $K\times(-)$ on $\con$, respectively. Then we have the monads $$H\times X,\quad\xymatrix{X\ar[r]^(.4){e_H\times 1_X} & H\times X},\quad\xymatrix{H\times H\times X\ar[r]^(.6){m_H\times 1_X} & H\times X}$$ and $$K\times X,\quad\xymatrix{X\ar[r]^(.4){e_K\times 1_X} & K\times X},\quad\xymatrix{K\times K\times X\ar[r]^(.6){m_K\times 1_X} & K\times X}.$$ Denote their units and multiplicationss as $\eta',\mu'$ y $\eta,\mu$, resp. Let $\gamma$ be a group structure on $K\times H$ such that $K\times e_H,e_K\times H$ are subgroups of $(\gamma,K\times H)$ and $(K\times e_H)\gamma(e_K\times H)=(\gamma,K\times H)$. In other words, such that $(\gamma,K\times H)$ is an internal Zappa-Szép product of $K\times e_H$ and $e_k\times H$, or that $(\gamma, K\times H)$ is an external Zappa-Szép product of $K$ and $H$.
Without loss of generality, we can assume $(k,h)=(k,e_H)\gamma(e_K,h)$ for any $k\in K$ and $h\in H$ (consider the definition of external Zappa-Szép product and the properties of $\alpha$ and $\beta$). Then we have the monad $$(K\times H\times(-),(e_K,e_H)\times 1_{(-)}, \gamma\times 1_{(-)})$$ on $\con$. Since $K\times e_H$ and $e_K\times H$ are subgroups of $(\gamma,K\times H)$, then $\eta S$ and $T\eta'$ are morphisms of monads, and since $(k,h)=(k,e_H)\gamma(e_K,h)$, the previous monad satisfies the middle unitary law, so the monad above induces a distributive law $ST\Rightarrow TS$; namely, $$(\gamma\times 1_{(-)})\cdot\eta ST\eta':H\times K\times(-)\Rightarrow K\times H\times(-);$$ explicitly, $$\xymatrix{(h,k,-)\ar@{|->}[r] & (e_K,h,k,e_H,-)\ar@{|->}[r] & ((e_K,h)\gamma(k,e_H),-)}.$$
Conversely, let $\lambda:ST\Rightarrow TS$ be a distributive law of $S$ over $T$, and consider $\lambda 1:H\times K\times 1\rightarrow K\times H\times 1$. Neglect the singleton, and put $$\lambda 1=(\alpha,\beta),$$ where $\alpha$ and $\beta$ are determined by the following commutative diagram: $$\xymatrix{ & H\times K\ar[d]^{\lambda 1}\ar[dr]^\alpha\ar[dl]_\beta &\\ K & K\times H\ar[l]^{p_K}\ar[r]_{p_H} & H. }$$ Then, by the compatibility of $\lambda$ with the unit of $S$, $$\xymatrix{ & H\times K\times 1\ar[dd]^{\lambda 1}\\ K\times 1\ar[ur]^{e_H\times K\times 1}\ar[dr]_{K\times e_H\times 1} &\\ & K\times H\times 1 }$$ commutes; hence, $\alpha(e_H,k)=k$ y $\beta(e_H,k)=e_H$.
Now, by the compatibility of $\lambda$ with the multiplication of $S$, $$\xymatrix{ H\times H\times K\times 1\ar[rr]^{H\times\lambda 1}\ar[d]_{m_H\times K\times 1} & & H\times K\times H\times 1\ar[rr]^{\lambda_{H\times 1}} & & K\times H\times H\times 1\ar[d]^{K\times m_H\times 1}\\ H\times K\times 1\ar[rrrr]_{\lambda 1} & & & & K\times H\times 1 }$$ commutes; whence, $$\alpha(h_1h_2,k)=\alpha(h_1,\alpha(h_2,k))\quad\text{y}\quad \beta(h_1h_2,k)=\beta(h_1,\alpha(h_2,k))\beta(h_2,k).$$ Similarly, by the compatibility of $\lambda$ with the unit and the multiplication of $T$, $\alpha(h,e_K)=e_K$, $\beta(h,e_K)=h$ and $$\alpha(h,k_1k_2)=\alpha(h,k_1)\alpha(\beta(h,k_1),k_2)\quad\text{y}\quad\beta(h,k_1k_2)=\beta(\beta(h,k_1),k_2).$$
Therefore, we have a Zappa-Szép product of $K$ and $H$.
Thus we have a correspondence between the Zappa-Szép products of $K$ and $H$, and the distributive laws of $H\times(-)$ over $K\times(-)$. Indeed, define $L:\mathbf{Zappa\text{-}Sz\acute{e}p}(K,H)\rightarrow\mathbf{DistLaw}(H,K)$, a function which goes from the Zappa-Szép products of $K$ and $H$ to the distributive laws of $H\times(-)$ ovoer $K\times(-)$ (let $\alpha:H\times K\rightarrow K$ and $\beta:H\times K\rightarrow H$ be the functions which define a Zappa-Szép product of $K$ and $H$): $$\xymatrix{ \mathbf{Zappa\text{-}Sz\acute{e}p}(K,H)\ar[r]^(.55)L & \mathbf{DistLaw}(H,K) }\qquad\qquad\quad$$ $$\xymatrix{ (\alpha,\beta)\ar@{|->}[r] & (\alpha,\beta)\times 1_{(-)}. }$$ Define $N:\mathbf{DistLaw}(H,K)\rightarrow\mathbf{Zappa\text{-}Sz\acute{e}p}(K,H)$ as $$\xymatrix{ \mathbf{DistLaw}(H,K)\ar[r]^(.45)N & \mathbf{Zappa\text{-}Sz\acute{e}p}(K,H) }\qquad\qquad\quad$$ $$\xymatrix{ \lambda\ar@{|->}[r] & (p_K\circ\lambda 1,p_H\circ\lambda 1). }$$ Obviously $NL=1$. Less obvious is that $LN=1$. Since $\con$ is a distributive category, \begin{equation}\label{D:distcatset} Z\times X=\sum\nolimits_{x\in X}Z\times H\times\{x\}. \end{equation}
Consider now the injection $i_x:\{x\}\rightarrow X$; then by naturality of $\lambda$, the following diagram commutes: $$\xymatrix{ H\times K\times\{x\}\ar[rr]^{H\times K\times i_x}\ar[d]_{\lambda_{\{x\}}} & & H\times K\times X\ar[d]^{\lambda_X}\\ K\times H\times\{x\}\ar[rr]_{K\times H\times i_x} & & K\times H\times X. }$$ On the other hand, since $H\times K\times i_x$ is the injection $$\xymatrix{H\times K\times\{x\}\ar[r] & \sum_{x\in X}H\times K\times\{x\}},$$ $K\times H\times i_x$ is the injection $$\xymatrix{K\times H\times\{x\}\ar[r] & \sum_{x\in X}K\times H\times\{x\}}$$ and the equality \eqref{D:distcatset} holds, $\lambda_X=\sum_{x\in X}\lambda_{\{x\}}$. However $\lambda_{\{x\}}=\lambda 1\times 1_{\{x\}}$ for every $x\in X$, so $\lambda_X=\lambda 1\times 1_X$. Hence, $\lambda_X=(p_K\circ\lambda 1,p_H\circ\lambda 1)\times 1_X$.
The Zappa-Szép product is generalized in [3] by showing the equivalence of the concept of distributive law in the bicategory of set-valued matrices and the concept of strict factorization system. Let's see how that is done. First describe the bicategory of set-valued matrices $\matcon$ as follows: the objects (the 0-cells) of $\matcon$ are sets, a 1-cell $M:A\rightarrow B$ is a set-valued matrix, i. e., $M(b,a)\in\con$ for every $a\in A$ and $b\in B$, a 2-cell $\tau:M\Rightarrow N:A\rightarrow B$ is a matrix of functions $\tau(b,a):M(b,a)\rightarrow N(b,a)$. The composite of 1-cells $$\xymatrix{ A\ar[r]^M & B\ar[r]^E & C\ar@{}|{=}[r] & A\ar[r]^{EM} & C }$$ is defined as $$EM(c,a):=\sum_{a\in A}E(c,b)\times M(b,a).$$ Given $A\in\matcon$, we define $$1_A(b,a):= \begin{cases} 1,\text{ the singleton, if $b=a$};\\ \emptyset,\text{ if $b\neq a$}. \end{cases}$$ It is clear that $M 1_A\overset{r}{\cong} M$ and $1_B M\overset{l}{\cong} M$.
A monad $T$ on an object $A$ in this bicategory is precisely a category with set of objects $A$. Let's shed some light on what is happening. Diagrammatically $T$ is determined by the following commutative diagrams: $$\xymatrix{ (TT)T\ar@{}|{\cong}[r]\ar[d]_{\mu T} & T(TT)\ar[r]^(.55){T\mu} & TT\ar[d]^\mu\\ TT\ar[rr]_\mu & & T }$$ and $$\xymatrix{ 1_AT\ar[r]^{\eta T}\ar[dr]_l & TT\ar[d]^\mu & T1_A\ar[l]_{T\eta}\ar[ld]^r\\ & T &, }$$ where $l$ and $r$ are the isomorphisms above induced by the product and the terminal object of $\con$. Now, what $T$ does is to assign to each pair of elements $b,a\in A$ a set $T(b,a)$ of arrows, to give a composite to each composable pair by means of $\mu$, to choose an identity for each element $a\in A$ via $\eta$ and finally, with the previous diagrams, to make composition associative and make composition with an identity a unit law for the arrows of the small category $A$ with objects its elements and its arrows in $T(b,a)$; put differently, a monad in $\matcon$ is a category.
Now let $M$ and $E$ be two categories with set of objects $A$ and let $\lambda:ME\Rightarrow EM$ be a distributive law of $M$ over $E$; $\lambda$ yields a function $$\xymatrix{ ME(a,c)\ar[r]^{\lambda(a,c)} & EM(a,c) }$$ for every pair $(a,c)\in A\times A$. Thus we have a family of functions $$(\xymatrix{ M(a,b)\times E(b,c)\ar[r] & \sum_{i\in A}E(a,i)\times M(i,c) })_{b\in A}.$$ If we write $m:\xymatrix{a\ \ar@{>->}[r] & b}$ for an arrow in $M$ and $e:\xymatrix{b\ar@{->>}[r] & c}$ for an arrow in $E$, then $\lambda$ yields an object $e_\lambda m$ and a composable pair $(e_\alpha m,e_\beta m)$ as shown by: $$\xymatrix{ a\; \ar@{>->}[r]^m\ar@{->>}[d]_{e_\alpha m}\ar@{}|{\triangleleft\scriptscriptstyle{\dashv}}[dr] & b\ar@{->>}[d]^e\\ e_\lambda m\;\ar@{>->}[r]_{e_\beta m} & c. }$$ We call a diagram such as this a $\lambda$-square. Consider the compatibility diagrams for the distributive law $\lambda:ME\Rightarrow EM$: \begin{equation} \vcenter{\xymatrix{ & ME\ar[dd]^\lambda\\ E\ar[ru]^{1E}\ar[dr]_{E1} & \\ & EM, }}\quad\text{compatibility of $\lambda$ with $1$ (CuM)}\notag \end{equation} \begin{equation} \vcenter{\xymatrix{ & ME\ar[dd]^\lambda\\ M\ar[ur]^{M1}\ar[dr]_{1 M} &\\ & EM, }}\quad\text{compatibility of $\lambda$ with $1$ (CuE)}\notag \end{equation} \begin{equation} \vcenter{\xymatrix{ MME\ar[r]^{M\lambda}\ar[d]_{\bullet E} & MEM\ar[r]^{\lambda M} & EMM\ar[d]^{E\bullet}\\ ME\ar[rr]_\lambda & & EM, }}\quad\text{compatibility of $\lambda$ with $\bullet$ (CmM)}\notag \end{equation} \begin{equation} \vcenter{\xymatrix{ MEE\ar[r]^{\lambda E}\ar[d]_{M\bullet} & EME\ar[r]^{E\lambda} & EEM\ar[d]^{\bullet M}\\ ME\ar[rr]_\lambda & & EM, }}\quad\text{compatibility of $\lambda$ with $\bullet$ (CmE)}\notag \end{equation} where we denote by 1 the transformations that provide identities and by $\bullet$ the transformations that provide the composites (the units and the multiplications of the monads $M$ and $E$). Now, in terms of $\lambda$-squares, the compatibility of $\lambda$ with the units is expressed by $$\xymatrix{ b\;\ar@{>->}[r]^{1_b}\ar@{->>}[d]_e\ar@{}|{\triangleleft\scriptscriptstyle{\dashv}}[dr] & b\ar@{->>}[d]^e & & a\;\ar@{>->}[r]^m\ar@{->>}[d]_{1_a}\ar@{}|{\triangleleft\scriptscriptstyle{\dashv}}[dr] & b\ar@{->>}[d]^{1_b}\\ c\;\ar@{>->}[r]_{1_c} & c & & a\;\ar@{>->}[r]_m & b; }$$ i. e., CuM and CuE state that ${1_b}_\lambda e=c,m_\lambda 1_b=a$ and that
From the general theory of distsributive law [1], $\lambda$ induces a composite monad $E_\lambda M$, a category with set of objects $A$ in which an arrow from $a$ to $c$ is given by specifying a third object $b$ and a pair $$\xymatrix{ a\ar@{->>}[r]^e & b\;\ar@{>->}[r]^m & c }$$ with $e$ in $E$ and $m$ in $M$; i. e., the arrows in $E_\lambda M$ are described as a formal composition $e\circ m$. The composition in $E_\lambda M$ is given by the multiplication for the monad $E_\lambda M$; namely, by $$\xymatrix{ & & EEM\ar[dr]^{\bullet M} & \\ EMEM \ar[r]^{E\lambda M} & EEMM\ar[ur]^{EE\bullet}\ar[dr]_{\bullet MM}\ar[rr]^(.55){\bullet\;\bullet} & & EM\\ & & EMM\ar[ur]_{E\bullet} &, }$$ thus the composite of $a\overset{e}{\twoheadrightarrow} b\overset{m}{\rightarrowtail} c$ and $c\overset{f}{\twoheadrightarrow} d\overset{n}{\rightarrowtail} x$ is given by $$\xymatrix{ a\ar@{->>}[r]^e\ar@{->>}[dr]_{e\bullet m_\alpha f} & b\;\ar@{>->}[r]^m\ar@{->>}[d]_(.35){m_\alpha f}\ar@{}|{\triangleleft\scriptscriptstyle{\dashv}}[dr] & c\ar@{->>}[d]^f\\ & m_\lambda f\;\ar@{>->}[r]_(.6){m_\beta f}\ar@{>->}[dr]_{m_\beta f\bullet n} & d\ar@{>->}[d]^n\\ & & x; }$$ i. e., $(e\circ m)\bullet(f\circ n)=(e\bullet m_\alpha f)\circ(m_\beta f\bullet n)$.
It is easy to check that the unit of the composite monad $E_\lambda M$ characterizes the identities in $E_\lambda M$ by $$a\overset{1_a}{\twoheadrightarrow}a\overset{1_a}{\rightarrowtail}a.$$
The morphisms of monads $1M:M\rightarrow E_\lambda M$ and $E1:E\rightarrow E_\lambda M$ are given by $m\mapsto 1\circ m$ and $e\mapsto e\circ 1$, resp. The middle unitary law yields that for every $e\circ m$ in $E_\lambda M$, $$(e\circ 1)\bullet(1\circ m)=e\circ m$$ holds.
Note that if $M$ and $E$ are monoids, the composition in $E_\lambda M$ is the multiplication defined in the case of the Zappa-Szép product for groups.
There is a greater generalization of the Zappa-Szép product in [2]; however, things there are done more à la Ehresmann.
Given a category $C$ with $\ob(C)=:A$, a strict factorization system on $C$ is a pair of subcategories $S:=(E,M)$ of $C$ such that $\ob(M)=\ob(E)=\ob(C)$ and such that for every $f$ in $C$, there is a unique factorization $f=e_fm_f$ with $e_f$ in $E$ and $m_f$ in $M$. Consider $M$ and $E$ as monads on $A$ in $\matcon$. The pair $(E,M)$ induces a distributive law $\lambda_S:ME\Rightarrow EM$; indeed, define $\lambda_S$ by $$\xymatrix{ ME\ar[r]^{\lambda_S} & EM }$$ $$\xymatrix{ a\ar[r]^n & b\ar[r]^f & c\ar@{|->}[r] & a\ar[r]^{e_{n\cdot f}} & i\ar[r]^{m_{n\cdot f}} & c. }\ $$ Its compatibility with the unit of $M$ is obvious, for $m\cdot 1_b=m$. Now, the upper right path of the compatibility diagram of $\lambda_S$ with respect to the multiplication of $M$ (check the diagram above) produces the following diagram: $$\xymatrix{ a\ar[rr]^f\ar[dr]_{e_{f\cdot e_{g\cdot h}}} & & b\ar[r]^g\ar[dr]_(.45){e_{g\cdot h}} & c\ar[r]^h & d\\ & k\ar[rr]_{m_{f\cdot e_{g\cdot h}}} & & j\ar[ur]_{m_{g\cdot h}} &\ ; }$$ so $f\cdot g\cdot h$ has as factorization $e_{f\cdot e_{g\cdot h}}\cdot m_{f\cdot e_{g\cdot h}}\cdot m_{g\cdot h}$, which is unique; whence, the left lower path in the compatibility diagram of $\lambda_S$ with respect to the multiplication of $M$ produces the same result, and in consequence the compatibility of $\lambda_S$ with $M$ is satisfied. Similarly, the compatibility of $\lambda_S$ with $E$ is satisfied.
Conversely, let $\lambda:ME\Rightarrow EM$ be a distributive law in $\matcon$, and consider the subcategories of $E_\lambda M$ defined by $$\lambda E:=\{e\circ 1\mid e\in E\}\quad\text{y}\quad M\lambda:=\{1\circ m\mid m\in M\}.$$ Each of these subcategories contains all the identities of $E_\lambda M$ and thus each contains all the objects of $E_\lambda M$. By the middle unitary law, $e\circ m$ is factorized as $(e\circ 1)\bullet(1\circ m)$. It is clear that this factorization is unique, and therefore $(\lambda E, M\lambda)$ is a strict factorization system $S_\lambda$ for $E_\lambda M$.
We leave this correspondence just right there; i. e., we won't show that $S_{(-)}$ and $\lambda_{(-)}$ are biequivalences inverse of each other, because this is not our objetive right now.
The way a distributive law is induced by the Zappa-Szép product in the group case makes us wonder whether a distributive law in $\matcon$ participates in a correspondence of that kind...
[1] Beck, J. [1969]: Distributive laws, Seminar on Triples and Categorical Homological Theory, ETH 1966/67, 80, 119-140 (1969). ↩
[2] Brin, M. G. [2005]: On the Zappa-Szép Product, Communications in Algebra, 33, 393-424 (2005). ↩
[3] Rosebrugh, R., Wood, R. J. [2002]: Distributive laws and factorization, Journal of Pure and Applied Algebra, 175(1-3), 327-353 (2002). ↩
[4] Takeuchi, M. [1981]: Matched pairs of groups and bismash products of Hopf algebras, Communications in Algebra, 9(8), 841-882 (1981). ↩
The Zappa-Szép product and distributive laws
The Zappa-Szép product, strict factorization systems and distributive laws are related in a certain way. Let us talk first about the Zappa-Szép product (also known as knit product or matched pair of groups).
The Zappa-Szép product is a generalization of the semidirect product for groups in the same way as this product is a generalization of the direct product for groups, and the zappa-szép product is the most general way in which a group is presented as a product of two groups.
The internal Zappa-Szép product is defined as follows. Given a group $G$ and two subgroups $H$ and $K$ of $G$, the following statements are equivalent:
- $G=KH$ y $K\cap H=\{e\}$,
- for every $g\in G$ there is a unique $k\in K$ and a unique $h\in H$ such that $g=kh$.
There is an external version of the Zappa-Szép product, which is the one we are interested in, because it's by means of this product that a link between distributive laws and strict factorization systems is established.
Given two groups $K$ and $H$, suppose there are funtions $\alpha:H\times K\rightarrow K$ and $\beta:H\times K\rightarrow H$ such that
- $\alpha(h_1h_2,k)=\alpha(h_1,\alpha(h_2,k))$,
- $\beta(h_1h_2,k)=\beta(h_1,\alpha(h_2,k))\beta(h_2,k)$,
- $\beta(h,k_1k_2)=\beta(\beta(h,k_1),k_2)$,
- $\alpha(h,k_1k_2)=\alpha(h,k_1)\alpha(\beta(h,k_1),k_2)$,
- $\alpha(e,k)=k$,
- $\beta(h,e)=h$
- $\beta(e,k)=e$ and
- $\alpha(h,e)=e$.
Note that if $G$ is an internal Zappa-Szép product of its subgroups $K$ y $H$, then there are funtions $\alpha:H\times K\rightarrow K$ and $\beta:H\times K\rightarrow H$ such that (i), (ii), (iii), (iv), (v) and (vi) in the previous definition are satisfied: their existence follows from the fact that every element $g\in G$ can be written uniquely as a product $kh$, and the fact that if $G$ is an external Zappa-Szép product of the groups $K$ and $H$, then $G$ is an internal Zappa-Szép product of its subgroups $K\times e_H$ and $e_K\times H$.
Let $H,K\in\grp$. Let $S$ and $T$ be the endofunctors $H\times(-)$ and $K\times(-)$ on $\con$, respectively. Then we have the monads $$H\times X,\quad\xymatrix{X\ar[r]^(.4){e_H\times 1_X} & H\times X},\quad\xymatrix{H\times H\times X\ar[r]^(.6){m_H\times 1_X} & H\times X}$$ and $$K\times X,\quad\xymatrix{X\ar[r]^(.4){e_K\times 1_X} & K\times X},\quad\xymatrix{K\times K\times X\ar[r]^(.6){m_K\times 1_X} & K\times X}.$$ Denote their units and multiplicationss as $\eta',\mu'$ y $\eta,\mu$, resp. Let $\gamma$ be a group structure on $K\times H$ such that $K\times e_H,e_K\times H$ are subgroups of $(\gamma,K\times H)$ and $(K\times e_H)\gamma(e_K\times H)=(\gamma,K\times H)$. In other words, such that $(\gamma,K\times H)$ is an internal Zappa-Szép product of $K\times e_H$ and $e_k\times H$, or that $(\gamma, K\times H)$ is an external Zappa-Szép product of $K$ and $H$.
Without loss of generality, we can assume $(k,h)=(k,e_H)\gamma(e_K,h)$ for any $k\in K$ and $h\in H$ (consider the definition of external Zappa-Szép product and the properties of $\alpha$ and $\beta$). Then we have the monad $$(K\times H\times(-),(e_K,e_H)\times 1_{(-)}, \gamma\times 1_{(-)})$$ on $\con$. Since $K\times e_H$ and $e_K\times H$ are subgroups of $(\gamma,K\times H)$, then $\eta S$ and $T\eta'$ are morphisms of monads, and since $(k,h)=(k,e_H)\gamma(e_K,h)$, the previous monad satisfies the middle unitary law, so the monad above induces a distributive law $ST\Rightarrow TS$; namely, $$(\gamma\times 1_{(-)})\cdot\eta ST\eta':H\times K\times(-)\Rightarrow K\times H\times(-);$$ explicitly, $$\xymatrix{(h,k,-)\ar@{|->}[r] & (e_K,h,k,e_H,-)\ar@{|->}[r] & ((e_K,h)\gamma(k,e_H),-)}.$$
Conversely, let $\lambda:ST\Rightarrow TS$ be a distributive law of $S$ over $T$, and consider $\lambda 1:H\times K\times 1\rightarrow K\times H\times 1$. Neglect the singleton, and put $$\lambda 1=(\alpha,\beta),$$ where $\alpha$ and $\beta$ are determined by the following commutative diagram: $$\xymatrix{ & H\times K\ar[d]^{\lambda 1}\ar[dr]^\alpha\ar[dl]_\beta &\\ K & K\times H\ar[l]^{p_K}\ar[r]_{p_H} & H. }$$ Then, by the compatibility of $\lambda$ with the unit of $S$, $$\xymatrix{ & H\times K\times 1\ar[dd]^{\lambda 1}\\ K\times 1\ar[ur]^{e_H\times K\times 1}\ar[dr]_{K\times e_H\times 1} &\\ & K\times H\times 1 }$$ commutes; hence, $\alpha(e_H,k)=k$ y $\beta(e_H,k)=e_H$.
Now, by the compatibility of $\lambda$ with the multiplication of $S$, $$\xymatrix{ H\times H\times K\times 1\ar[rr]^{H\times\lambda 1}\ar[d]_{m_H\times K\times 1} & & H\times K\times H\times 1\ar[rr]^{\lambda_{H\times 1}} & & K\times H\times H\times 1\ar[d]^{K\times m_H\times 1}\\ H\times K\times 1\ar[rrrr]_{\lambda 1} & & & & K\times H\times 1 }$$ commutes; whence, $$\alpha(h_1h_2,k)=\alpha(h_1,\alpha(h_2,k))\quad\text{y}\quad \beta(h_1h_2,k)=\beta(h_1,\alpha(h_2,k))\beta(h_2,k).$$ Similarly, by the compatibility of $\lambda$ with the unit and the multiplication of $T$, $\alpha(h,e_K)=e_K$, $\beta(h,e_K)=h$ and $$\alpha(h,k_1k_2)=\alpha(h,k_1)\alpha(\beta(h,k_1),k_2)\quad\text{y}\quad\beta(h,k_1k_2)=\beta(\beta(h,k_1),k_2).$$
Therefore, we have a Zappa-Szép product of $K$ and $H$.
Thus we have a correspondence between the Zappa-Szép products of $K$ and $H$, and the distributive laws of $H\times(-)$ over $K\times(-)$. Indeed, define $L:\mathbf{Zappa\text{-}Sz\acute{e}p}(K,H)\rightarrow\mathbf{DistLaw}(H,K)$, a function which goes from the Zappa-Szép products of $K$ and $H$ to the distributive laws of $H\times(-)$ ovoer $K\times(-)$ (let $\alpha:H\times K\rightarrow K$ and $\beta:H\times K\rightarrow H$ be the functions which define a Zappa-Szép product of $K$ and $H$): $$\xymatrix{ \mathbf{Zappa\text{-}Sz\acute{e}p}(K,H)\ar[r]^(.55)L & \mathbf{DistLaw}(H,K) }\qquad\qquad\quad$$ $$\xymatrix{ (\alpha,\beta)\ar@{|->}[r] & (\alpha,\beta)\times 1_{(-)}. }$$ Define $N:\mathbf{DistLaw}(H,K)\rightarrow\mathbf{Zappa\text{-}Sz\acute{e}p}(K,H)$ as $$\xymatrix{ \mathbf{DistLaw}(H,K)\ar[r]^(.45)N & \mathbf{Zappa\text{-}Sz\acute{e}p}(K,H) }\qquad\qquad\quad$$ $$\xymatrix{ \lambda\ar@{|->}[r] & (p_K\circ\lambda 1,p_H\circ\lambda 1). }$$ Obviously $NL=1$. Less obvious is that $LN=1$. Since $\con$ is a distributive category, \begin{equation}\label{D:distcatset} Z\times X=\sum\nolimits_{x\in X}Z\times H\times\{x\}. \end{equation}
Consider now the injection $i_x:\{x\}\rightarrow X$; then by naturality of $\lambda$, the following diagram commutes: $$\xymatrix{ H\times K\times\{x\}\ar[rr]^{H\times K\times i_x}\ar[d]_{\lambda_{\{x\}}} & & H\times K\times X\ar[d]^{\lambda_X}\\ K\times H\times\{x\}\ar[rr]_{K\times H\times i_x} & & K\times H\times X. }$$ On the other hand, since $H\times K\times i_x$ is the injection $$\xymatrix{H\times K\times\{x\}\ar[r] & \sum_{x\in X}H\times K\times\{x\}},$$ $K\times H\times i_x$ is the injection $$\xymatrix{K\times H\times\{x\}\ar[r] & \sum_{x\in X}K\times H\times\{x\}}$$ and the equality \eqref{D:distcatset} holds, $\lambda_X=\sum_{x\in X}\lambda_{\{x\}}$. However $\lambda_{\{x\}}=\lambda 1\times 1_{\{x\}}$ for every $x\in X$, so $\lambda_X=\lambda 1\times 1_X$. Hence, $\lambda_X=(p_K\circ\lambda 1,p_H\circ\lambda 1)\times 1_X$.
The bicategory of set-valued matrices
The Zappa-Szép product is generalized in [3] by showing the equivalence of the concept of distributive law in the bicategory of set-valued matrices and the concept of strict factorization system. Let's see how that is done. First describe the bicategory of set-valued matrices $\matcon$ as follows: the objects (the 0-cells) of $\matcon$ are sets, a 1-cell $M:A\rightarrow B$ is a set-valued matrix, i. e., $M(b,a)\in\con$ for every $a\in A$ and $b\in B$, a 2-cell $\tau:M\Rightarrow N:A\rightarrow B$ is a matrix of functions $\tau(b,a):M(b,a)\rightarrow N(b,a)$. The composite of 1-cells $$\xymatrix{ A\ar[r]^M & B\ar[r]^E & C\ar@{}|{=}[r] & A\ar[r]^{EM} & C }$$ is defined as $$EM(c,a):=\sum_{a\in A}E(c,b)\times M(b,a).$$ Given $A\in\matcon$, we define $$1_A(b,a):= \begin{cases} 1,\text{ the singleton, if $b=a$};\\ \emptyset,\text{ if $b\neq a$}. \end{cases}$$ It is clear that $M 1_A\overset{r}{\cong} M$ and $1_B M\overset{l}{\cong} M$.
A monad $T$ on an object $A$ in this bicategory is precisely a category with set of objects $A$. Let's shed some light on what is happening. Diagrammatically $T$ is determined by the following commutative diagrams: $$\xymatrix{ (TT)T\ar@{}|{\cong}[r]\ar[d]_{\mu T} & T(TT)\ar[r]^(.55){T\mu} & TT\ar[d]^\mu\\ TT\ar[rr]_\mu & & T }$$ and $$\xymatrix{ 1_AT\ar[r]^{\eta T}\ar[dr]_l & TT\ar[d]^\mu & T1_A\ar[l]_{T\eta}\ar[ld]^r\\ & T &, }$$ where $l$ and $r$ are the isomorphisms above induced by the product and the terminal object of $\con$. Now, what $T$ does is to assign to each pair of elements $b,a\in A$ a set $T(b,a)$ of arrows, to give a composite to each composable pair by means of $\mu$, to choose an identity for each element $a\in A$ via $\eta$ and finally, with the previous diagrams, to make composition associative and make composition with an identity a unit law for the arrows of the small category $A$ with objects its elements and its arrows in $T(b,a)$; put differently, a monad in $\matcon$ is a category.
Now let $M$ and $E$ be two categories with set of objects $A$ and let $\lambda:ME\Rightarrow EM$ be a distributive law of $M$ over $E$; $\lambda$ yields a function $$\xymatrix{ ME(a,c)\ar[r]^{\lambda(a,c)} & EM(a,c) }$$ for every pair $(a,c)\in A\times A$. Thus we have a family of functions $$(\xymatrix{ M(a,b)\times E(b,c)\ar[r] & \sum_{i\in A}E(a,i)\times M(i,c) })_{b\in A}.$$ If we write $m:\xymatrix{a\ \ar@{>->}[r] & b}$ for an arrow in $M$ and $e:\xymatrix{b\ar@{->>}[r] & c}$ for an arrow in $E$, then $\lambda$ yields an object $e_\lambda m$ and a composable pair $(e_\alpha m,e_\beta m)$ as shown by: $$\xymatrix{ a\; \ar@{>->}[r]^m\ar@{->>}[d]_{e_\alpha m}\ar@{}|{\triangleleft\scriptscriptstyle{\dashv}}[dr] & b\ar@{->>}[d]^e\\ e_\lambda m\;\ar@{>->}[r]_{e_\beta m} & c. }$$ We call a diagram such as this a $\lambda$-square. Consider the compatibility diagrams for the distributive law $\lambda:ME\Rightarrow EM$: \begin{equation} \vcenter{\xymatrix{ & ME\ar[dd]^\lambda\\ E\ar[ru]^{1E}\ar[dr]_{E1} & \\ & EM, }}\quad\text{compatibility of $\lambda$ with $1$ (CuM)}\notag \end{equation} \begin{equation} \vcenter{\xymatrix{ & ME\ar[dd]^\lambda\\ M\ar[ur]^{M1}\ar[dr]_{1 M} &\\ & EM, }}\quad\text{compatibility of $\lambda$ with $1$ (CuE)}\notag \end{equation} \begin{equation} \vcenter{\xymatrix{ MME\ar[r]^{M\lambda}\ar[d]_{\bullet E} & MEM\ar[r]^{\lambda M} & EMM\ar[d]^{E\bullet}\\ ME\ar[rr]_\lambda & & EM, }}\quad\text{compatibility of $\lambda$ with $\bullet$ (CmM)}\notag \end{equation} \begin{equation} \vcenter{\xymatrix{ MEE\ar[r]^{\lambda E}\ar[d]_{M\bullet} & EME\ar[r]^{E\lambda} & EEM\ar[d]^{\bullet M}\\ ME\ar[rr]_\lambda & & EM, }}\quad\text{compatibility of $\lambda$ with $\bullet$ (CmE)}\notag \end{equation} where we denote by 1 the transformations that provide identities and by $\bullet$ the transformations that provide the composites (the units and the multiplications of the monads $M$ and $E$). Now, in terms of $\lambda$-squares, the compatibility of $\lambda$ with the units is expressed by $$\xymatrix{ b\;\ar@{>->}[r]^{1_b}\ar@{->>}[d]_e\ar@{}|{\triangleleft\scriptscriptstyle{\dashv}}[dr] & b\ar@{->>}[d]^e & & a\;\ar@{>->}[r]^m\ar@{->>}[d]_{1_a}\ar@{}|{\triangleleft\scriptscriptstyle{\dashv}}[dr] & b\ar@{->>}[d]^{1_b}\\ c\;\ar@{>->}[r]_{1_c} & c & & a\;\ar@{>->}[r]_m & b; }$$ i. e., CuM and CuE state that ${1_b}_\lambda e=c,m_\lambda 1_b=a$ and that
- ${1_b}_\alpha e=e$,
- ${1_b}_\beta e=1_c$,
- $m_\alpha 1_b=1_a$,
- $m_\beta 1_b=m$.
- $(mn)_\alpha e=m_\alpha(n_\alpha e)$,
- $(mn)_\beta e=m_\beta(n_\alpha e)\bullet n_\beta e$.
- $m_\alpha(ef)=m_\alpha e\bullet(m_\beta e)_\alpha f$,
- $m_\beta(ef)=(m_\beta e)_\beta f$.
From the general theory of distsributive law [1], $\lambda$ induces a composite monad $E_\lambda M$, a category with set of objects $A$ in which an arrow from $a$ to $c$ is given by specifying a third object $b$ and a pair $$\xymatrix{ a\ar@{->>}[r]^e & b\;\ar@{>->}[r]^m & c }$$ with $e$ in $E$ and $m$ in $M$; i. e., the arrows in $E_\lambda M$ are described as a formal composition $e\circ m$. The composition in $E_\lambda M$ is given by the multiplication for the monad $E_\lambda M$; namely, by $$\xymatrix{ & & EEM\ar[dr]^{\bullet M} & \\ EMEM \ar[r]^{E\lambda M} & EEMM\ar[ur]^{EE\bullet}\ar[dr]_{\bullet MM}\ar[rr]^(.55){\bullet\;\bullet} & & EM\\ & & EMM\ar[ur]_{E\bullet} &, }$$ thus the composite of $a\overset{e}{\twoheadrightarrow} b\overset{m}{\rightarrowtail} c$ and $c\overset{f}{\twoheadrightarrow} d\overset{n}{\rightarrowtail} x$ is given by $$\xymatrix{ a\ar@{->>}[r]^e\ar@{->>}[dr]_{e\bullet m_\alpha f} & b\;\ar@{>->}[r]^m\ar@{->>}[d]_(.35){m_\alpha f}\ar@{}|{\triangleleft\scriptscriptstyle{\dashv}}[dr] & c\ar@{->>}[d]^f\\ & m_\lambda f\;\ar@{>->}[r]_(.6){m_\beta f}\ar@{>->}[dr]_{m_\beta f\bullet n} & d\ar@{>->}[d]^n\\ & & x; }$$ i. e., $(e\circ m)\bullet(f\circ n)=(e\bullet m_\alpha f)\circ(m_\beta f\bullet n)$.
It is easy to check that the unit of the composite monad $E_\lambda M$ characterizes the identities in $E_\lambda M$ by $$a\overset{1_a}{\twoheadrightarrow}a\overset{1_a}{\rightarrowtail}a.$$
The morphisms of monads $1M:M\rightarrow E_\lambda M$ and $E1:E\rightarrow E_\lambda M$ are given by $m\mapsto 1\circ m$ and $e\mapsto e\circ 1$, resp. The middle unitary law yields that for every $e\circ m$ in $E_\lambda M$, $$(e\circ 1)\bullet(1\circ m)=e\circ m$$ holds.
Note that if $M$ and $E$ are monoids, the composition in $E_\lambda M$ is the multiplication defined in the case of the Zappa-Szép product for groups.
There is a greater generalization of the Zappa-Szép product in [2]; however, things there are done more à la Ehresmann.
Strict factorization systems
Given a category $C$ with $\ob(C)=:A$, a strict factorization system on $C$ is a pair of subcategories $S:=(E,M)$ of $C$ such that $\ob(M)=\ob(E)=\ob(C)$ and such that for every $f$ in $C$, there is a unique factorization $f=e_fm_f$ with $e_f$ in $E$ and $m_f$ in $M$. Consider $M$ and $E$ as monads on $A$ in $\matcon$. The pair $(E,M)$ induces a distributive law $\lambda_S:ME\Rightarrow EM$; indeed, define $\lambda_S$ by $$\xymatrix{ ME\ar[r]^{\lambda_S} & EM }$$ $$\xymatrix{ a\ar[r]^n & b\ar[r]^f & c\ar@{|->}[r] & a\ar[r]^{e_{n\cdot f}} & i\ar[r]^{m_{n\cdot f}} & c. }\ $$ Its compatibility with the unit of $M$ is obvious, for $m\cdot 1_b=m$. Now, the upper right path of the compatibility diagram of $\lambda_S$ with respect to the multiplication of $M$ (check the diagram above) produces the following diagram: $$\xymatrix{ a\ar[rr]^f\ar[dr]_{e_{f\cdot e_{g\cdot h}}} & & b\ar[r]^g\ar[dr]_(.45){e_{g\cdot h}} & c\ar[r]^h & d\\ & k\ar[rr]_{m_{f\cdot e_{g\cdot h}}} & & j\ar[ur]_{m_{g\cdot h}} &\ ; }$$ so $f\cdot g\cdot h$ has as factorization $e_{f\cdot e_{g\cdot h}}\cdot m_{f\cdot e_{g\cdot h}}\cdot m_{g\cdot h}$, which is unique; whence, the left lower path in the compatibility diagram of $\lambda_S$ with respect to the multiplication of $M$ produces the same result, and in consequence the compatibility of $\lambda_S$ with $M$ is satisfied. Similarly, the compatibility of $\lambda_S$ with $E$ is satisfied.
Conversely, let $\lambda:ME\Rightarrow EM$ be a distributive law in $\matcon$, and consider the subcategories of $E_\lambda M$ defined by $$\lambda E:=\{e\circ 1\mid e\in E\}\quad\text{y}\quad M\lambda:=\{1\circ m\mid m\in M\}.$$ Each of these subcategories contains all the identities of $E_\lambda M$ and thus each contains all the objects of $E_\lambda M$. By the middle unitary law, $e\circ m$ is factorized as $(e\circ 1)\bullet(1\circ m)$. It is clear that this factorization is unique, and therefore $(\lambda E, M\lambda)$ is a strict factorization system $S_\lambda$ for $E_\lambda M$.
We leave this correspondence just right there; i. e., we won't show that $S_{(-)}$ and $\lambda_{(-)}$ are biequivalences inverse of each other, because this is not our objetive right now.
The way a distributive law is induced by the Zappa-Szép product in the group case makes us wonder whether a distributive law in $\matcon$ participates in a correspondence of that kind...
References
[1] Beck, J. [1969]: Distributive laws, Seminar on Triples and Categorical Homological Theory, ETH 1966/67, 80, 119-140 (1969). ↩
[2] Brin, M. G. [2005]: On the Zappa-Szép Product, Communications in Algebra, 33, 393-424 (2005). ↩
[3] Rosebrugh, R., Wood, R. J. [2002]: Distributive laws and factorization, Journal of Pure and Applied Algebra, 175(1-3), 327-353 (2002). ↩
[4] Takeuchi, M. [1981]: Matched pairs of groups and bismash products of Hopf algebras, Communications in Algebra, 9(8), 841-882 (1981). ↩
