nLab distributive law

Distributive laws

Distributive laws


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(b+c)=ab+aca\cdot (b+c) = a\cdot b + a\cdot c and a0=0a\cdot 0 = 0.

Abstractly, there are two monads on the category Set, one (call it T\mathbf{T}) whose algebras are abelian groups, and one (call it S\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 XX the free ring on that set, which consists of formal sums of formal products of elements of XX—in other words, it can be identified with T(S(X))T(S(X)). Thus the question becomes “given two monads T\mathbf{T} and S\mathbf{S}, what further structure is required to make the composite TST S into a monad?”

It is easy to give TST S a unit, as the composite Idη SSη TSTSId \xrightarrow{\eta^S} S \xrightarrow{\eta^T S} T S, but to give it a multiplication we need a transformation from TSTST S T S to TST S. We naturally want to use the multiplications μ T:TTT\mu^T\colon T T \to T and μ S:SSS\mu^S\colon S S \to S, but in order to do this we first need to switch the order of TT and SS. However, if we have a transformation λ:STTS\lambda\colon S T \to T S, then we can define μ TS\mu^{T S} to be the composite TSTSλTTSSμ Tμ STST S T S \xrightarrow{\lambda} T T S S \xrightarrow{\mu^T\mu^S} T S.

Such a transformation, satisfying suitable axioms to make TST S into a monad, is called a distributive law, because of the motivating example relating addition to multiplication in a ring. In that case, STXS T X is a formal product of formal sums such as (x 1+x 2+x 3)(x 4+x 5)(x_1 + x_2 + x_3)\cdot (x_4 + x_5), and the distributive law λ\lambda is given by multiplying out such an expression formally, resulting in a formal sum of formal products such as x 1x 4+x 1x 5+x 2x 4+x 2x 5+x 3x 4+x 3x 5x_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.

Big picture

Monads in any 2-category CC make themselves a 2-category Mnd\mathrm{Mnd} in which 1-morphisms are either lax or colax homomorphisms of monads.

By formal duality the analogue 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 ll from one monad to another enables to define the composite monad T lP\mathbf T\circ_l\mathbf P. This correspondence extends to a 2-functor comp:Mnd(Mnd(C))Mnd(C)\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.

Explicit definition

A distributive law from a monad T=(T,μ T,η T)\mathbf{T} = (T, \mu^T, \eta^T) in AA to an endofunctor PP is a 2-cell l:TPPTl : T P \Rightarrow P T such that l(η T) P=P(η T)l \circ (\eta^T)_P = P(\eta^T) and l(μ T) P=P(μ T)l TT(l)l \circ (\mu^T)_P = P(\mu^T) \circ l_T \circ T(l). In diagrams:

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

A distributive law from a monad T=(T,μ T,η T)\mathbf{T} = (T, \mu^T, \eta^T) to a monad P=(P,μ P,η P)\mathbf{P} = (P, \mu^P, \eta^P) in AA (or of T\mathbf T over P\mathbf P) is a distributive law from T\mathbf T to the endofunctor PP, compatible with μ P,η P\mu^P,\eta^P in the sense that lT(η P)=(η P) Tl \circ T(\eta^P) = (\eta^P)_T and lT(μ P)=(μ P) TP(l)l Pl \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:TPPTl : T P \Rightarrow P T from the monad T\mathbf{T} to the monad P\mathbf{P} are in a canonical bijection with lifts of the monad P\mathbf{P} to a monad P T\mathbf{P}^{\mathbf T} in the Eilenberg-Moore category A TA^{\mathbf T}, such that U T:A TAU^{\mathbf T} : A^{\mathbf T}\to A preserves the monad structure.

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.


Products distributing over coproducts

In a distributive category products distribute over coproducts.

In Cat

  • 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.

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 tensor product of abelian groups and tensor product of modules.

In other 2-categories


On distributive laws for relative monads:

Last revised on August 20, 2023 at 12:50:52. See the history of this page for a list of all contributions to it.