Coherence for product monoids and their actions (Q2432580)

Let \((\mathcal{V},\square,e)\) be a monoidal category and let \((A,\mu^A,\eta^A)\), \((B,\mu^B,\eta^B)\) be two monoids in \(\mathcal{V}\). Given a Beck distributive law [\textit{J. Beck}, Lect. Notes Math. 80, 119--140 (1969; Zbl 0186.02902)], \(\iota\colon B\square A\to A\square B\), there is a natural monoid structure on \(A\square B\), \((A\square B,\mu,\eta)\), where \(\mu\colon(A\square B)\square (A\square B)\to A\square B\) is the composite \(\mu^A\square\mu^B\circ 1_A\square\iota\square 1_B\) and \(\eta\colon e\to A\square B\) is the composite of \(\eta^A\square\eta^B\) with the identification \(e\equiv e\square e\). And vice versa: A morphism \(\iota\colon B\square A\to A\square B\) is a Beck distributive law if and only if this construction gives a monoid structure on \(A\square B\). Given a Beck distributive law, and a formal word \(W\) in the symbols \(A,B,e\). Then all morphisms \(W\to A\square B\) which are compositions of \(\square\) products of the symbols \(1_e, 1_A,1_B, \mu^A, \mu^B, \eta^A,\eta^B,\iota\) coincide. And again vice versa: If this holds for a morphism \(\iota\), then \(\iota\) is a Beck distributive law. It is crucial that \(W\) is a formal word so that \(eA\) is not identified with \(A\) for instance. Suppose \((A\square B)\) acts on \(X\) via \(\nu\colon(A\square B)\square X\to X\), then one can define an \(A\) action \(\nu^A\) and a \(B\) action \(\nu^B\) on \(X\) s.t. \(\nu\) is the composition. And \(\nu^A,\nu^B\) are unique with this property. Given \(A\) and \(B\) actions \(\nu^A,\nu^B\) on \(X\) and a Beck distributive law the composite is an \((A\square B)\) action provided certain morphisms of formal words in \(A,B, X,e\) to \(X\) are equivalent. This is Theorem 2.1 in the paper.