internalization and categorical algebra
algebra object (associative, Lie, …)
internal category ($\to$ more)
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Sometimes in mathematics one considers 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 $Id \xrightarrow{\eta^S} S \xrightarrow{\eta^T S} T S$, 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 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 $x_1\cdot x_4 + x_1 \cdot x_5 + x_2 \cdot x_4 + x_2 \cdot x_5 + x_3\cdot x_4 + x_3 \cdot x_5$.
(terminology – what distributes over what)
The eponymous example of distributivity in arithmetic
hence
(where, of course, the equality as such works in both directions, but the distribution of factors over summands is the step from left to right) suggests that a suitable transformation of (co)monads of the form
should be referred to as $T_1$ distributing over $T_2$ instead of the other way around.
However, already the original reference Beck 1969 §1 uses the opposite terminology.
Authors sticking to this original but arguably reverse terminological convention include Brookes & Van Stone 1993, while other authors tacitly switch to the other terminological convention (eg. Barr & Wells 1985 §9 2.1, Power & Watanabe 2002 p. 138).
Monads in any 2-category $C$ make themselves a 2-category $\mathrm{Mnd}$ in which 1-morphisms are either lax or colax homomorphisms of monads (cf. monad transformations). By formal duality the analogue is true for comonads.
Distributivity laws may be understood as monads internal to this 2-category of monads.
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 $\mathrm{comp} \,\colon\, \mathrm{Mnd}(\mathrm{Mnd}(C))\to\mathrm{Mnd}(C)$. An analogue of this 2-functor in the mixed setup is a 2-functor from the bicategory of entwinings to a bicategory of corings.
A distributive law for a monad $\mathbf{T} = (T, \mu^T, \eta^T)$ in $A$ over an endofunctor $P$ is a 2-morphism $l : T P \Rightarrow P T$ 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:
Distributive laws for the monad $\mathbf{T}$ over 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 for a monad $\mathbf{T} = (T, \mu^T, \eta^T)$ over a monad $\mathbf{P} = (P, \mu^P, \eta^P)$ in $A$ is a distributive law for $\mathbf T$ over 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:
The correspondence between distributive laws and endofunctor liftings extends to a correspondence between distributive laws and monad liftings. That is, distributive laws $l \colon T P \Rightarrow P T$ from the monad $\mathbf{T}$ to the monad $\mathbf{P}$ are in a canonical bijection with lifts of the monad $\mathbf{P}$ to a monad $\mathbf{P}^{\mathbf T}$ in the Eilenberg-Moore category $A^{\mathbf T}$, such that $U^{\mathbf T} \colon A^{\mathbf T}\to A$ preserves the monad structure.
Thus all together a distributive law for a monad over 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.
(distributivity as monads in monads)
As mentioned earlier, one can understand a distributivity law of a monad $(a,s)$ over another monad $(a,t)$ as displaying $s \colon a \to a$ as a monad on $(a,t)$ in the 2-category $\mathbf{Mnd}(A)$ of monads in a 2-category $A$.
Specifically, a monad in $\mathbf{Mnd}(A)$ over $(a,t)$ (which is a monad in $A$!) is comprised of the following data:
A 1-morphism $s \colon a \to a$ in $A$, together with an intertwiner $\lambda \colon t s \to s t$ satisfying the equations here
Two 2-morphisms (in $\mathbf{Mnd}(A)$) $\sigma \colon 1 \Rightarrow (s,\lambda)$ and $\nu \colon (s,\lambda)(s,\lambda) \Rightarrow (s,\lambda)$ which correspond to two 2-morpisms in $A$ $\sigma \colon 1 \Rightarrow s$, $\nu \colon s s \Rightarrow s$ commuting with the intertwiners of $1$, $s$ and $ss$.
Then $\lambda$ is the distributive law sought, and the laws $\lambda$, $\sigma$ and $\nu$ satisfy correspond to those of a distributive law.
(…)
The distributivity law of
a comonad $\mathcal{C}$ over
a monad $\mathcal{E}$
on the same category $\mathbf{C}$
is as follows (Brookes & Van Stone 1993 Def. 3, Power & Watanabe 2002):
such that the following diagrams commute for all objects $D$:
Given this distributivity structure, there is a two-sided (“double”) Kleisli category (Brookes & Van Stone 1993 Thm. 2, Power & Watanabe 2002, Prop. 7.4) whose objects are those of $\mathbf{C}$, and whose morphisms $D_1 \to D_2$ are morphisms in $\mathbf{C}$ of the form
with two-sided Kelisli composition
given by the (co-)bind-operation on the factors connected by the distributivity transformation:
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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On distributive laws for relative monads:
Invertible distributive laws are considered in Lemma 4.12 of:
Last revised on June 2, 2024 at 12:20:21. See the history of this page for a list of all contributions to it.