symmetric monoidal (∞,1)-category of spectra
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 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 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 $Id \xrightarrow{\eta^S} S \xrightarrow{\eta^T} T S$, 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 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 $x_1\cdot x_4 + x_1 \cdot x_5 + x_2 \cdot x_4 + x_2 \cdot x_5 + x_3\cdot x_4 + x_3 \cdot x_5$.
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 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 structures), distributive laws between actions of two different monoidal categories on the same category, for PROPs 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 corings.
A 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 $l \circ (\mu^T)_P = P(\mu^T) \circ l_T \circ T(l)$. The latter identitity is the commutativity of the pentagon
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 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 of $\mathbf T$ 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 $l \circ T(\mu^P) = (\mu^P)_T \circ P(l) \circ l_P$. 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, Brzeziński and Majid combined the 4 diagrams into one picture which they call the bow-tie diagram.
Similarly, there are definitions of distributive law of a comonad over a comonad, a monad over a comonad (sometimes called a mixed distributive law), and so on.
In a distributive category products distribute over coproducts.
For many standard choices of tensor products in the presence of direct sums the former distribute over the latter. See at tensor product of abelian groups and tensor product of modules.
strict factorization systems can be identified with distributive laws between categories regarded as monads in Span(Set).
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 this discussion.
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