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\section*{distributive law}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{distributive_laws}{}\section*{{Distributive laws}}\label{distributive_laws}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{big_picture}{Big picture}\dotfill \pageref*{big_picture} \linebreak
\noindent\hyperlink{explicit_definition}{Explicit definition}\dotfill \pageref*{explicit_definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{products_distributing_over_coproducts}{Products distributing over coproducts}\dotfill \pageref*{products_distributing_over_coproducts} \linebreak
\noindent\hyperlink{in_cat}{In Cat}\dotfill \pageref*{in_cat} \linebreak
\noindent\hyperlink{tensor_products_distributing_over_direct_sums}{Tensor products distributing over direct sums}\dotfill \pageref*{tensor_products_distributing_over_direct_sums} \linebreak
\noindent\hyperlink{in_other_2categories}{In other 2-categories}\dotfill \pageref*{in_other_2categories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Sometimes in [[mathematics]] we want to consider objects equipped with two different types of [[extra structure]] which interact in a suitable way. For instance, a [[ring]] is a [[set]] equipped with both (1) the structure of an (additive) [[abelian group]] and (2) the structure of a (multiplicative) [[monoid]], which satisfy the distributive laws $a\cdot (b+c) = a\cdot b + a\cdot c$ and $a\cdot 0 = 0$.

Abstractly, there are two [[monads]] on the [[category]] [[Set]], one (call it $\mathbf{T}$) whose [[algebra over a monad|algebras]] are abelian groups, and one (call it $\mathbf{S}$) whose algebras are monoids, and so we might ask ``can we construct, from these two monads, a third monad whose algebras are rings?'' Such a monad would assign to each set $X$ the [[free object|free]] ring on that set, which consists of formal sums of formal products of elements of $X$---in other words, it can be identified with $T(S(X))$. Thus the question becomes ``given two monads $\mathbf{T}$ and $\mathbf{S}$, what further structure is required to make the composite $T S$ into a monad?''

It is easy to give $T S$ a unit, as the composite $Id \xrightarrow{\eta^S} S \xrightarrow{\eta^T S} T S$, but to give it a multiplication we need a transformation from $T S T S$ to $T S$. We naturally want to use the multiplications $\mu^T\colon T T \to T$ and $\mu^S\colon S S \to S$, but in order to do this we first need to switch the order of $T$ and $S$. However, if we have a transformation $\lambda\colon S T \to T S$, then we can define $\mu^{T S}$ to be the composite $T S T S \xrightarrow{\lambda} T T S S \xrightarrow{\mu^T\mu^S} T S$.

Such a transformation, satisfying suitable axioms to make $T S$ into a monad, is called a \emph{distributive law}, because of the motivating example relating addition to multiplication in a ring. In that case, $S T X$ is a formal product of formal sums such as $(x_1 + x_2 + x_3)\cdot (x_4 + x_5)$, and the distributive law $\lambda$ is given by multiplying out such an expression formally, resulting in a formal sum of formal products such as $x_1\cdot x_4 + x_1 \cdot x_5 + x_2 \cdot x_4 + x_2 \cdot x_5 + x_3\cdot x_4 + x_3 \cdot x_5$.

\hypertarget{big_picture}{}\subsection*{{Big picture}}\label{big_picture}

[[monad|Monads]] in any [[2-category]] $C$ make themselves a 2-category $\mathrm{Mnd}$ in which 1-cells are either lax or colax morphisms of monads; by dualization the same is true for [[comonads]]. Monads internal to the 2-category of monads are called \emph{distributive laws}. In particular, distributive laws themselves make a 2-category. There are other variants like distributive laws between a monad and an [[endofunctor]], ``mixed'' distributive laws between a monad and a comonad (the variants for algebras and coalgebras called [[entwining structure]]s), distributive laws between actions of two different monoidal categories on the same category, for [[PROP]]s and so on. Having a distributive law $l$ from one monad to another enables to define the composite monad $\mathbf T\circ_l\mathbf P$. This correspondence extends to a 2-functor $\mathrm{comp}:\mathrm{Mnd}(\mathrm{Mnd}(C))\to\mathrm{Mnd}(C)$. An analogue of this 2-functor in the mixed setup is a homomorphism of bicategories from the bicategory of entwinings to a bicategory of [[coring]]s.

\hypertarget{explicit_definition}{}\subsection*{{Explicit definition}}\label{explicit_definition}

A \textbf{distributive law} from a monad $\mathbf{T} = (T, \mu^T, \eta^T)$ in $A$ to an endofunctor $P$ is a 2-cell $l : T P \Rightarrow P T$ such that $l \circ (\eta^T)_P = P(\eta^T)$ and $l \circ (\mu^T)_P = P(\mu^T) \circ l_T \circ T(l)$. In diagrams: $\backslash$begin\{tikzcd\} \& P $\backslash$ar\{dl\}swap\{$\backslash$eta{\tt \symbol{94}}T P\} $\backslash$ar\{dr\}\{P $\backslash$eta{\tt \symbol{94}}T\} $\backslash$ T P $\backslash$ar\{rr\}\{l\} \&\& P T $\backslash$end\{tikzcd\} $\backslash$begin\{tikzcd\} T T P $\backslash$ar\{d\}\{$\backslash$mu{\tt \symbol{94}}T P\} $\backslash$ar\{r\}\{T l\} \& T P T $\backslash$ar\{r\}\{l T\} \& P T T $\backslash$ar\{d\}\{P $\backslash$mu{\tt \symbol{94}}T\} $\backslash$ T P $\backslash$ar\{rr\}\{l\} \&\& P T $\backslash$end\{tikzcd\}

Distributive laws from the monad $\mathbf{T}$ to the endofunctor $P$ are in a canonical bijection with lifts of $P$ to an endofunctor $P^{\mathbf T}$ in the [[Eilenberg-Moore category]] $A^{\mathbf T}$, satisfying $U^{\mathbf T} P^{\mathbf T} = P U^{\mathbf T}$. Indeed, the endofunctor $P^{\mathbf T}$ is given by $(M,\nu) \mapsto (P M,P(\nu)\circ l_M)$.

A \textbf{distributive law} from a monad $\mathbf{T} = (T, \mu^T, \eta^T)$ to a monad $\mathbf{P} = (P, \mu^P, \eta^P)$ in $A$ (or \emph{of} $\mathbf T$ \emph{over} $\mathbf P$) is a distributive law from $\mathbf T$ to the endofunctor $P$, compatible with $\mu^P,\eta^P$ in the sense that $l \circ T(\eta^P) = (\eta^P)_T$ and $l \circ T(\mu^P) = (\mu^P)_T \circ P(l) \circ l_P$. In diagrams: $\backslash$begin\{tikzcd\} \& T $\backslash$ar\{dl\}swap\{T $\backslash$eta{\tt \symbol{94}}P\} $\backslash$ar\{dr\}\{$\backslash$eta{\tt \symbol{94}}P T\} $\backslash$ T P $\backslash$ar\{rr\}\{l\} \&\& P T $\backslash$end\{tikzcd\} $\backslash$begin\{tikzcd\} T P P $\backslash$ar\{d\}\{T $\backslash$mu{\tt \symbol{94}}P\} $\backslash$ar\{r\}\{l P\} \& P T P $\backslash$ar\{r\}\{P l\} \& P P T $\backslash$ar\{d\}\{$\backslash$mu{\tt \symbol{94}}P T\} $\backslash$ T P $\backslash$ar\{rr\}\{l\} \&\& P T $\backslash$end\{tikzcd\}

Thus all together a distributive law from a monad to a monad is a 2-cell for which 2 triangles and 2 pentagons commute. In the entwining case, Brzeziski and Majid combined the 4 diagrams into one picture which they call the \emph{bow-tie diagram}.

Similarly, there are definitions of distributive law of a comonad over a comonad, a monad over a comonad (sometimes called a \textbf{mixed} distributive law), and so on.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{products_distributing_over_coproducts}{}\subsubsection*{{Products distributing over coproducts}}\label{products_distributing_over_coproducts}

In a \emph{[[distributive category]]} [[products]] distribute over [[coproducts]].

\hypertarget{in_cat}{}\subsubsection*{{In Cat}}\label{in_cat}

\begin{itemize}%
\item There is a distributive law of the monad (on [[Set]]) for [[monoids]] over the monad for [[abelian groups]], whose composite is the monad for [[rings]]. This is the canonical example which gives the name to the whole concept.

\end{itemize}
\hypertarget{tensor_products_distributing_over_direct_sums}{}\paragraph*{{Tensor products distributing over direct sums}}\label{tensor_products_distributing_over_direct_sums}

For many standard choices of [[tensor products]] in the presence of [[direct sums]] the former distribute over the latter. See at \emph{[[tensor product of abelian groups]]} and \emph{[[tensor product of modules]]}.

\hypertarget{in_other_2categories}{}\subsubsection*{{In other 2-categories}}\label{in_other_2categories}

\begin{itemize}%
\item [[strict factorization systems]] can be identified with distributive laws between categories regarded as monads in [[span|Span(Set)]].


\item More generally, [[factorization systems over a subcategory]] can be identified with distributive laws in [[Prof]]. Ordinary [[orthogonal factorization systems]] are a special case. The latter can also be obtained by other weakenings; see for instance \href{http://golem.ph.utexas.edu/category/2010/07/ternary_factorization_systems.html}{this discussion}.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[distributivity for monoidal structures]]
\item [[monad transformer]]

\end{itemize}
\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

\begin{itemize}%
\item H. Appelgate, [[Michael Barr]], J. Beck, [[Bill Lawvere]], [[Fred Linton]], E, Manes, [[Myles Tierney]], F. Ulmer, \emph{Seminar on triples and categorical homology theory}, ETH 1966/67, edited by B. Eckmann, LNM 80, Springer 1969. (includes article Jon Beck, Distributive laws, pages 119--140). Republished in \emph{Reprints in Theory and Applications of Categories}, No. 18 (2008) pp. 1-303. (\href{http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html}{web})


\item [[Michael Barr]], [[Charles Wells]], \emph{Toposes, triples and theories}, Springer-Verlag 1985; \href{http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html}{web remake at TAC}


\item [[Gabi Böhm]], \emph{Internal bialgebroids, entwining structures and corings}, AMS Contemp. Math. 376 (2005) 207--226; \href{http://front.math.ucdavis.edu/math.QA/0311244}{arXiv:math.QA/0311244}


\item [[T. Brzeziński]], [[S. Majid]], \emph{Coalgebra bundles}, Comm. Math. Phys. 191 (1998), no. 2, 467--492 (\href{http://arxiv.org/abs/q-alg/9602022}{arXiv version}).


\item T. Brzeziski, R. Wisbauer, \emph{Corings and comodules}, London Math. Soc. Lec. Note Series 309, Cambridge 2003.


\item T. F. Fox, [[Martin Markl]], \emph{Distributive laws, bialgebras, and cohomology}, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), 167--205, Contemp. Math. 202, AMS 1997.


\item [[Steve Lack]], \emph{Composing PROPS}, \href{http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html}{Theory Appl. Categ.} 13 (2004), No. 9, 147--163.


\item [[Steve Lack]], [[Ross Street]], \emph{The formal theory of monads II}, Special volume celebrating the 70th birthday of Professor Max Kelly. J. Pure Appl. Algebra 175 (2002), no. 1-3, 243--265.


\item [[Martin Markl]], \emph{Distributive laws and Koszulness}, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 307--323 (\href{http://www.numdam.org/numdam-bin/fitem?id=AIF_1996__46_2_307_0}{numdam})


\item [[Ross Street|R. Street]], \emph{The formal theory of monads}, J. Pure Appl. Alg. 2, 149--168 (1972)


\item [[Z. Škoda]], \emph{Distributive laws for monoidal categories} (\href{http://front.math.ucdavis.edu/math.CT/0406310}{arXiv:0406310}); \emph{Equivariant monads and equivariant lifts versus a 2-category of distributive laws} (\href{http://front.math.ucdavis.edu/0707.1609}{arXiv:0707.1609}); \emph{Bicategory of entwinings} (\href{http://arxiv.org/abs/0805.4611}{arXiv:0805.4611})


\item [[Z. Škoda]], \emph{Some equivariant constructions in noncommutative geometry}, Georgian Math. J. 16 (2009) 1; 183--202 (\href{http://front.math.ucdavis.edu/0811.4770}{arXiv:0811.4770})


\item R. Wisbauer, \emph{Algebras versus coalgebras}, Appl. Categ. Structures \textbf{16} (2008), no. 1-2, 255--295.


\item [[Francisco Marmolejo]] and [[Adrian Vazquez-Marquez]], \emph{No-iteration mixed distributive laws}, \href{https://doi.org/10.1017/S0960129514000656}{doi}


\item Enrique Ruiz Hernández, \emph{Another characterization of no-iteration distributive laws}, \href{https://arxiv.org/abs/1910.06531}{arxiv}



\end{itemize}
[[!redirects distributive laws]]

[[!redirects distributivity law]]

[[!redirects distributivity]]



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