Bases for parametrized iterativity (Q1004388)

Parametrized iterativity of an algebra means the existence of unique solutions of all finitary recursive systems of equations where the recursion is allowed to use only some variables (chosen as a parameter). A generalized definition of iterative recursive system of equations for a general category of algebras is presented. An algebra is iterative if every such system has a unique solution. A functor is called finitary if it preserves all colimits of updirected diagrams. A base for a locally presentable category \(\mathcal A\) is a functor from \(\mathcal A\) into the category of all monadic algebras over finitary functors. It is proved that for every base there exists a free iterative algebra over a finitely presentable object and it can be constructed by colimit construction.