Abstract. We give a semantic footing to the fold/build syntax of programming with inductive types, covering shortcut deforestation, based on a universal property. Specifically, we give a semantics for inductive types based on limits of algebra structure forgetting functors and show that it is equivalent to the usual initial algebra semantics. We also give a similar semantic account of the augment generalization of build and of the unfold/destroy syntax of coinductive types.
Abstract. We study general structured corecursion, dualizing the work of Osius, Taylor, and others on general structured recursion. We call an algebra of a functor corecursive if it supports general structured corecursion: there is a unique map to it from any coalgebra of the same functor. The concept of antifounded algebra is a statement of the bisimulation principle. We show that it is independent from corecursiveness: Neither condition implies the other. Finally, we call an algebra focusing if its codomain can be reconstructed by iterating structural refinement. This is the strongest condition and implies all the others. IntroductionA line of research started by Osius and Taylor studies the categorical foundations of general structured recursion. A recursive coalgebra (RCA) is a coalgebra of a functor F with a unique coalgebra-to-algebra morphism to any F -algebra. In other words, it is an algebra guaranteeing unique solvability of any structured recursive diagram. The notion was introduced by Osius [Osi74] (it was also of interest to Eppendahl [Epp99]; we studied construction of recursive coalgebras from coalgebras known to be recursive with the help of distributive laws of functors over comonads [CUV06]).Taylor introduced the notion of wellfounded coalgebra (WFCA) and proved that, in a Set-like category, it is equivalent to RCA [Tay96a,Tay96b],[Tay99, Ch. 6]. Formulated in terms of Jacobs's next-time operator [Jac02], it states that any subset of the carrier of the coalgebra containing its next-time subset is isomorphic to the carrier; in other words, the carrier is the least fixed-point of the next-time operator. As this subset is given by those elements of the carrier whose recursive calls tree is wellfounded, the principle really states that the "wellfounded core" of the coalgebra carrier coincides with the whole carrier [BC05]. A closely related characterization consists in the reconstruction of the coalgebra carrier by iterating the next-time operator on the empty set. Under mild assumptions, it is equivalent to both WFCA and RCA.Adámek et al.[ALM07] provided extra characterizations for the important case when the functor has an initial algebra. Backhouse and Doornbos [DB96] studied wellfoundedness in a relational setting.We decided to tackle the dual notions. The importance of this line of research lies in the study of structured recursive definitions which make sense not because of specific properties of the coalgebra marshalling the recursive call arguments but rather thanks to the algebra assembling the recursive call results: here we speak of general structured corecursion. This is typical of definitions that work because of "productivity" rather than "termination", e.g., guardedby-constructors definitions of functions with a coinductive codomain. To our surprise, none of the equivalences established in the dual situation carries over. The duals of the conditions that made those equivalences possible become unreasonable and are false in usual categories, specifically in Set.The dual of RCA is the ...
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