Abstract. For accessible set-valued functors it is well known that weak preservation of limits is equivalent to representability, and weak preservation of connected limits to familial representability. In contrast, preservation of weak wide pullbacks is equivalent to being a coproduct of quotients of hom-functors modulo groups of automorphisms. For finitary functors this was proved by André Joyal who called these functors analytic. We introduce a generalization of Joyal’s concept from endofunctors of Set to endofunctors of a symmetric monoidal category. 1.

Weak homomorphisms of coalgebras by

under the supervision of Dr Yde Venema, and submitted to the Board of

ABSTRACT. We survey coalgebras as models of state based systems together with their global and local logics. We convey some useful intuition regarding Set-functors which leads naturally to coalgebraic modal logic where modalities are validity patterns for the successor object of a state. 1.

Information and Computation journal homepage:www.elsevier.com/locate/ic Complete sets of cooperations

The structure map turning a set into the carrier of a final coalgebra is not unique. This fact is well-known but commonly elided. In this paper we argue that any such concrete representation of a set as a final coalgebra is potentially interesting on its own. We discuss several examples, in particular, we consider different coalgebra structures that turn the set of infinite streams into the carrier of a final coalgebra. After that we focus on coalgebra structures that are made up using so-called cooperations. We say that a collection of cooperations is complete for a given set X if it gives rise to a coalgebra structure that turns X into the carrier set of a subcoalgebra of a final coalgebra. Any complete set of cooperations yields a coalgebraic proof and definition principle. We exploit this fact and devise a general definition scheme for constants and functions on a set X that is parametrically in the choice of the complete set of cooperations for X. Key words: Coalgebra, coinduction, infinite data structures, hidden algebra. 1

Abstract not found

In this paper we introduce the notion of an observational coalgebra structure and of a complete set of co-operations. We demonstrate in various example the usefulness of these notions, in particular, we show how they give rise to coalgebraic proof and definition principles. Keywords: Coalgebra, Coinduction, infinite data structures, Hidden Algebra.

We discuss recent work generalising the basic hybrid logic with the difference modality to any reasonable notion of transition. This applies equally to both subrelational transitions such as monotone neighbourhood frames or selection function models as well as those with more structure such as Markov chains and alternating temporal frames. We provide a generic canonical cut-free sequent system and a terminating proof-search strategy for the fragment without the difference modality but including the global modality. Keywords: Global Modality, Difference Modality, Coalgebraic Semantics, Cut-free Sequent System

basic theory of monads and their connection to