On the generic solution to \(P(X)\cong X\) in distributive categories
A category \({\mathcal C}\) is distributive if it has finite products and sums such that the canonical morphisms \(X\times Y+ X\times Z\to X \times (Y+Z)\) are isomorphisms. The free distributive category on one generator is given. A polynomial \(P\) in one variable with natural number coefficients can be evaluated at any object \(X\) of the distributive category \({\mathcal C}\) and this value is denoted by \(P(X)\). A \(P\)-algebra in \({\mathcal C}\) is a pair \((X,s)\) of an object \(X\) in \({\mathcal C}\) and a morphism \(s: P(X) \to X\). It is rigid if \(s\) is an isomorphism. A classifying distributive category for \(P\)-algebras, and another one for rigid \(P\)-algebras are build up. For a class of polynomials, concrete descriptions of these categories are given, together with a concrete description of isomorphism classes of objects.
scientific article; zbMATH DE number 1121682