GENERALIZING SUBSTITUTION

摘要

It is well known that, given an endofunctor H on a category C, the initial (A + H ―)-algebras (if existing), i.e., the algebras of (wellfounded) H-terms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Moss and Aczel, Adamek, Milius and Velebil have shown that a similar monad, which even enjoys the additional special property of having iterations for all guarded substitution rules (complete iterativeness), arises from the inverses of the final (A + H ―)-coalgebras (if existing), i.e., the algebras of non-wellfounded H-terms. We show that, upon an appropriate generalization of the notion of substitution, the same can more generally be said about the initial T'(A, ― )-algebras resp. the inverses of the final T'(A,―)-coalgebras for any endobifunctor T' on any category C such that the functors T'(―,X) uniformly carry a monad structure.