Generalizing Substitution

Abstract

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 [2] 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.Mathematics Subject Classification. 08B20, 18C15, 18C50.

Build, Augment and Destroy, Universally

“…A similar combinator is possible for any monad obtained from a parameterized monad via initial algebras as described in [32]. Some parameterized inductive types covered by this more general construction are, e.g., finitely branching node labelled trees and inductive hyperfunctions [17].…”

Substitution in non-wellfounded syntax with variable binding

“…Another goal will be to check how well the various versions of typed lambda calculus with monotone inductive and coinductive constructors of rank 2 introduced in [1,2,25] are suited for coding our constructions for non-wellfounded syntax with variable binding (from the point of view of achieving the intuitively desirable reduction behaviours). Some other possible directions for continuing this research are: to identify more examples of heterogeneous signatures and substitution systems, to see if a further generalization of the results of [38] along the lines of the present paper is possible.…”

Iterative Algebras for a Base

“…The notion of a base (under the name parametrized monad) has been introduced by Tarmo Uustalu [18] who generalized some results of our paper [1]. In the present paper we continue in the same vein by generalizing results of [5] from H-algebras to base algebras.…”

“…Thus our original hope that we will just indicate how to modify the previous proof ideas to the present generality failed, and we feel obliged to present detailed proofs again. The concrete examples of bases below (see 2.5) and their free iterative algebras (see 3.3-3.6) were already considered in [18].…”