Generalizing Substitution

¹ Inst. of Cybernetics, Tallinn Technical Univ., Akadeemia tee 21, EE-12618 Tallinn, Estonia; tarmo@cs.ioc.ee.
² The work was largely done during the author's postdoctoral leave to Dep. de Informática, Univ. do Minho, Braga, Portugal.

Received: 5 December 2002
Accepted: 13 August 2003

Abstract

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 [17] and Aczel, Adámek, Milius and Velebil [12] 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.