A calculus of transition systems (towards universal coalgebra)

By representing transition systems as coalgebras, the three main ingredients of their theory: coalgebra, homomorphism, and bisimulation, can be seen to be in a precise correspondence to the basic notions of universal algebra: Sigma-algebra, homomorphism, and substitutive relation (or congruence). In this paper, some standard results from universal algebra (such as the three isomorphism theorems and facts on the lattices of subalgebras and congruences) are reformulated (using the afore mentioned correspondence) and proved for transition systems

Weighted colimits and formal balls in generalized metric spaces

(a) Limits of Cauchy sequences in a (possibly non-symmetric) metric space are shown to be weighted colimits (a notion introduced by Borceux and Kelly, 1975). As a consequence, further insights from enriched category theory are applicable to the theory of metric spaces, thus continuing Lawvere's (1973) approach. Many of the recently proposed d

Weighted colimits and formal balls in generalized metric spaces

(a) Limits of Cauchy sequences in a (possibly non-symmetric) metric space are shown to be weighted colimits (a notion introduced by Borceux and Kelly, 1975). As a consequence, further insights from enriched category theory are applicable to the theory of metric spaces, thus continuing Lawvere's (1973) approach. Many of the recently proposed d

Coalgebra, concurrency and control

Coalgebra is used to generalize notions and techniques from concurrency theory, in order to apply them to problems concerning the supervisory control of discrete event systems. The main ingredients of this approach are the characterization of controllability in terms of (a variant of) the notion of bisimulation, and the observation that the fa

Elements of generalized ultrametric domain theory

Generalized ultrametric spaces are a common generalization of preorders and ordinary ultra-metric spaces, as was observed by Lawvere (1973). Guided by his enriched-categorical view on (ultra)metric spaces, we generalize the standard notions of Cau

Coinductive counting : bisimulation in enumerative combinatorics

The recently developed coinductive calculus of streams finds here a further application in enumerative combinatorics. A general methodology is developed to solve a wide variety of basic counting problems in a uniform way: (1) the objects to be counted are enumerated by means of an infinite (weighted) automaton; (2) the automaton is minimized b