Abstract. We show that, for functors with suitable mild restrictions, the initial algebra in the category of sets and functions gives rise to the final coalgebra in the (Kleisli) category of sets and relations. The finality principle thus obtained leads to the finite trace semantics of nondeterministic systems, which extends the trace semantics for coalgebras previously introduced by the second author. We demonstrate the use of our technical result by giving the first coalgebraic account on contextfree grammars, where we obtain generated context-free languages via the finite trace semantics. Additionally, the constructions of both finite and possibly infinite parse trees are shown to be monads. Hence our extension of the application domain of coalgebras identifies several new mathematical constructions and structures. 1

Abstract. We propose a novel, comonadic approach to dataflow (streambased) computation. This is based on the observation that both general and causal stream functions can be characterized as coKleisli arrows of comonads and on the intuition that comonads in general must be a good means to structure context-dependent computation. In particular, we develop a generic comonadic interpreter of languages for context-dependent computation and instantiate it for stream-based computation. We also discuss distributive laws of a comonad over a monad as a means to structure combinations of effectful and context-dependent computation. We apply the latter to analyse clocked dataflow (partial stream based) computation. 1

This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributive laws. 1

The concept of recursive coalgebra of a functor was introduced in the 1970s by Osius in his work on categorical set theory to discuss the relationship between wellfounded induction and recursively specified functions. In this paper, we motivate the use of recursive coalgebras as a paradigm of structured recursion in programming semantics, list some basic facts about recursive coalgebras and, centrally, give new conditions for the recursiveness of a coalgebra based on comonads, comonadcoalgebras and distributive laws of functors over comonads. We also present an alternative construction using countable products instead of cofree comonads.

Abstract: We propose a novel discipline for programming stream functions and for the semantic description of stream manipulation languages based on the observation that both general and causal stream functions can be characterized as coKleisli arrows of comonads. This seems to be a promising application for the old, but very little exploited idea that if monads abstract notions of computation of a value, comonads ought to be useable as an abstraction of notions of value in a context. We also show that causal partial-stream functions can be described in terms of a combination of a comonad and a monad.

Neil Ghani Dept. of Math. and Comp. Sci.

Abstract. The category Class of classes and functions is proved to have a number of properties suitable for algebra and coalgebra: every endofunctor has an initial algebra and a terminal coalgebra, the categories of algebras and coalgebras are complete and cocomplete, and every endofunctor generates a free completely iterative monad. A description of a terminal coalgebra for the power-set functor is provided.

This paper shows that the approach of [2,12] for obtaining coinductive solutions of equations on infinite terms is a special case of a more general recent approach of [4] using distributive laws.

Completely iterative monads of Elgot et al. are the monads such that every guarded iterative equation has a unique solution. Free completely iterative monads are known to exist on every iteratable endofunctor H, i. e., one with final coalgebras of all functors H ( ) + X. We show that conversely, if H generates a free completely iterative monad, then it is iteratable. Key words: monad, completely iterative, iterable 1

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.