--- title: Walking Distributive Law url: https://www.emergentmind.com/topics/walking-distributive-law type: topic --- # Walking Distributive Law Searching arXiv for the cited papers and closely related work on walking distributive laws and distributive laws of monads. The walking distributive law is the universal strict 2-categorical object encoding two monads on a common object together with a distributive law between them. In the semi-strict formulation, it is defined by $$ \mathrm{Dist} := B\Delta_+ \otimes_l B\Delta_+, $$ where \(B\Delta_+\) is the walking monad and \(\otimes_l\) is the lax Gray tensor product; strict 2-functors out of \(\mathrm{Dist}\) are equivalently distributive laws in the target 2-category [2509.22208]. A parallel presentation describes a single-object 2-category \(\mathbb W\) generated by two endo-1-cells, their monad structures, and a 2-cell \(\lambda: S\circ T \Rightarrow T\circ S\), subject to the monad axioms and Beck coherences, thereby exhibiting the same universal role in a more explicitly algebraic form [2404.00581]. ## 1. Monad-theoretic precursor: the walking monad The construction begins with the observation that a monad in a 2-category \(\mathcal B\) is a lax 2-functor $$ M:\mathbf 1 \to \mathcal B $$ from the terminal strict 2-category \(\mathbf 1\). Such a lax 2-functor sends the unique object to an object \(X\in\mathcal B\), the unique 1-cell to an endo-1-cell \(t:X\to X\), and carries structure 2-cells $$ M_u:\mathrm{Id}_X \Rightarrow t, \qquad M_c:t\cdot t \Rightarrow t $$ satisfying the usual unit and associativity axioms [2509.22208]. This factorization yields a fully faithful embedding $$ \mathrm{Lax}[\mathbf 1,\mathcal B] \simeq \mathrm{Mnd}(\mathcal B). $$ Concretely, the strict 2-category \(B\Delta_+\), described as the delooping of the monoidal category \((\Delta_+,\oplus,[-1])\), is the free 2-category on one object \(*\) with one 1-cell \([0]\) and unary operations \(\delta,\sigma\). Its universal property is that strict 2-functors $$ B\Delta_+ \to \mathcal B $$ correspond to monads in \(\mathcal B\). For this reason \(B\Delta_+\) is called the walking monad [2509.22208]. The walking monad is the basic “free” carrier of monad structure. This suggests that any universal object encoding interactions between monads should arise by combining copies of \(B\Delta_+\) in a monoidal structure on \(2\)-Cat. ## 2. Gray tensor as the mechanism of interaction The relevant monoidal structure is the lax Gray tensor product \(\otimes_l\) on \(2\)-Cat, characterized by the adjunction $$ 2\mathrm{Cat}[\mathcal B,\mathrm{Lax}[\mathcal C,\mathcal D]] \;\simeq\; 2\mathrm{Cat}[\mathcal B\otimes_l \mathcal C,\mathcal D]. $$ For strict 2-categories \(\mathcal B,\mathcal C\), the tensor \(\mathcal B\otimes_l\mathcal C\) has 0-cells given by pairs \((b,c)\), 1-cells generated under composition by horizontal steps \((f,\mathrm{id})\) and \((\mathrm{id},g)\), and 2-cells generated by \((\phi,\mathrm{id})\), \((\mathrm{id},\psi)\), and the interchanger $$ \gamma_{f,g}:(f,\mathrm{id})\cdot(\mathrm{id},g)\Rightarrow(\mathrm{id},g)\cdot(f,\mathrm{id}), $$ subject to coherence relations expressing compatibility with vertical and horizontal composition, functoriality in \(f\) and \(g\), and the pentagon–triangle axioms making \(\gamma\) into a strength [2509.22208]. In the summary formulation, the interchanger additionally satisfies identities such as $$ \gamma_{f,\mathrm{id}}=\mathrm{id}, \qquad \gamma_{\mathrm{id},g}=\mathrm{id}, $$ together with composition laws including $$ \gamma_{f_1\cdot f_2,g} = (\gamma_{f_1,g}\cdot(f_2,\mathrm{id})) \circ ((f_1,\mathrm{id})\cdot \gamma_{f_2,g}), $$ and analogous relations [2509.22208]. The Gray tensor is therefore not merely a product of 2-categories. Its interchanger generates exactly the kind of nontrivial comparison 2-cell needed to express a distributive law between two endo-1-cells. ## 3. The semi-strict walking distributive law The walking distributive law is defined by Gray-tensoring the walking monad with itself: $$ \mathrm{Dist}:=B\Delta_+\otimes_l B\Delta_+. $$ Concretely, \(\mathrm{Dist}\) has one object \(\star\) and two generating 1-cells $$ t=([0],\mathrm{id}), \qquad s=(\mathrm{id},[0]). $$ Each of \(t\) and \(s\) inherits a monad structure, written \((\eta^t,\mu^t)\) and \((\eta^s,\mu^s)\), from the respective copies of \(B\Delta_+\). In addition, there is a distinguished 2-cell $$ \sigma:s\cdot t \Rightarrow t\cdot s $$ coming from the interchanger \(\gamma_{[0],[0]}\) [2509.22208]. The defining relations are exactly the usual four Beck distributive-law axioms. In the ordinary categorical notation of two monads \((S,\eta^S,\mu^S)\) and \((T,\eta^T,\mu^T)\) on the same category \(\mathcal C\), a natural transformation $$ \lambda:ST\Rightarrow TS $$ is a distributive law precisely when the following equations hold: $$ \lambda\circ(S\eta^T)=\eta^T S, \qquad \lambda\circ(\eta^S T)=T\eta^S, $$ $$ \lambda\circ(\mu^S T)=(T\mu^S)\circ(\lambda S)\circ(S\lambda), $$ $$ \lambda\circ(S\mu^T)=(\mu^T S)\circ(T\lambda)\circ(\lambda T). $$ These are the unit and multiplication coherences that the interchanger-generated 2-cell must satisfy [2404.00581]. A common misconception is that a walking distributive law must already include higher braid-like coherence. In the two-monad case, no higher coherence is needed because there are only two monads [2509.22208]. The nontrivial higher coherence appears only in iterated variants. ## 4. Universal property and algebraic presentation The universal property of \(\mathrm{Dist}\) states that a strict 2-functor $$ \mathrm{Dist}\to \mathcal B $$ is equivalently the datum of an object \(X\in\mathcal B\), two monads \((t,\eta^t,\mu^t)\) and \((s,\eta^s,\mu^s)\) on \(X\), and a 2-cell \(\sigma:s\cdot t\Rightarrow t\cdot s\) satisfying the Beck axioms. Equivalently, $$ 2\mathrm{Cat}[\mathrm{Dist},\mathcal B]\simeq \mathrm{DistLaw}(\mathcal B), $$ where the right-hand side is the ordinary category of distributive laws in \(\mathcal B\) [2509.22208]. An explicitly presented version of the same idea is the strictly presented 2-category \(\mathbb W\). It has a single object \(\star\), generating 1-cells \(S:\star\to\star\) and \(T:\star\to\star\), generating 2-cells \(\eta^S\), \(\mu^S\), \(\eta^T\), \(\mu^T\), and \(\lambda:S\circ T\Rightarrow T\circ S\), and relations given by the usual monad axioms for \((S,\eta^S,\mu^S)\) and \((T,\eta^T,\mu^T)\) together with the Beck coherence diagrams for \(\lambda\) [2404.00581]. Given any 2-category \(\mathcal K\) and such data in \(\mathcal K\), there is a unique strict 2-functor $$ F:\mathbb W\to \mathcal K $$ sending the generators to the chosen monads and distributive law [2404.00581]. This universal property explains the adjective “walking.” The object walks exactly the structure of a distributive law into any target 2-category, and nothing beyond that structure. ## 5. Relation to composite theories and rewriting Distributive laws admit an algebraic interpretation through composite theories. If \(S\) and \(T\) are algebraic theories with signatures \(\Sigma_S,\Sigma_T\) and equational axioms \(E_S,E_T\) presenting finitary monads \(S,T\), then from a distributive law \(\lambda\) one forms a new theory $$ U^\lambda=(\Sigma_U,E_U), \qquad \Sigma_U=\Sigma_S\uplus\Sigma_T, \qquad E_U=E_S\cup E_T\cup E_\lambda, $$ where \(E_\lambda\) consists of exactly the mixed equations forced by \(\lambda\) [2404.00581]. The paper states two structural properties. First, every \(U^\lambda\)-term can be rewritten, using \(E_S\cup E_T\cup E_\lambda\), to a normal form in \(\Sigma_T^*\Sigma_S^*\), that is, a \(T\)-term whose variables are \(S\)-terms. Second, any two such separations represent the same element of \(TSV\), so they coincide modulo the \(S\)- and \(T\)-axioms. Hence \(U^\lambda\) is precisely the composite theory of \(T\) after \(S\) [2404.00581]. A term-rewriting system \(R\) is obtained by orienting each distribution equation as a left-to-right rewrite rule $$ s(t_1,\dots,t_n)\longrightarrow t(s_1,\dots,s_m), $$ under the assumption that each left-hand side has layer \((1,1)\). If \(R\) is terminating, for example by a suitable multiset path order or a polynomial interpretation sending \(S\)-symbols to large polynomials and \(T\)-symbols to smaller ones, then the smaller subset \(E'\subset E_\lambda\) of all \((1,1)\)-layer equations suffices to derive all of \(E_\lambda\) [2404.00581]. The prototypical “ring” example takes \(S=\) monoids and \(T=\) Abelian groups, with key rewrite rules $$ x\,(y+z)\to (x\,y)+(x\,z), \qquad (y+z)\,x\to (y\,x)+(z\,x). $$ By polynomial interpretation these rules terminate, and Prover9 or similar tools quickly show that the monoid and Abelian-group axioms plus these two rules derive other mixed equations such as \(x\cdot 0=0\), \(0\cdot x=0\), and \(({-}x)\,y=-(x\,y)\) [2404.00581]. This correspondence situates the walking distributive law simultaneously in 2-category theory and in algebraic syntax. A plausible implication is that the universal 2-categorical presentation and the composite-theory presentation should be viewed as equivalent organizational perspectives on the same coherence data. ## 6. Parametric and iterated variants The parametric walking distributive law is obtained by adjoining an additional Gray factor. If one wishes to parameterize by a 2-category \(\mathcal P\), the definition is $$ \mathrm{PDist}(\mathcal P,\mathcal B) := 2\mathrm{Cat}[\mathcal P,\mathrm{Dist}(\mathcal B)] \simeq 2\mathrm{Cat}[\mathcal P,2\mathrm{Cat}[B\Delta_+\otimes_l B\Delta_+,\mathcal B]] \leftrightarrow 2\mathrm{Cat}[\mathcal P\otimes_l B\Delta_+\otimes_l B\Delta_+,\mathcal B]. $$ In particular, when \(\mathcal P=B\Gamma\) for a monoidal category \(\Gamma\), parameters enter by an extra Gray factor \(\mathcal P\otimes_l B\Delta_+\otimes_l B\Delta_+\) [2509.22208]. The same source demonstrates applicability by providing two concrete examples involving the Writer and Either monads, explicitly describing morphisms of such parametric distributive laws [2509.22208]. Iterated variants are formed by repeated Gray tensoring. The 3-fold semi-strict distributive law is $$ B\Delta_+\otimes_l B\Delta_+\otimes_l B\Delta_+, $$ with three monads \(t_1,t_2,t_3\) and three interchangers \(\sigma_{i,j}\) for \(i