To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is first-order acyclic sharing graphs represented by let-syntax, and others are extensions with higher-order constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced

The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal

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. 1

We instantiate the general comonad-based construction of recursion schemes for the initial algebra of a functor F to the cofree recursive comonad on F. Differently from the scheme based on the cofree comonad on F in a similar fashion, this scheme allows not only recursive calls on elements structurally smaller than the given argument, but also subsidiary recursions. We develop a Mendler formulation of the scheme via a generalized Yoneda lemma for initial algebras involving strong dinaturality and hint a relation to circular proofs à la Cockett, Santocanale.

A note on strong dinaturality, initial algebras