Inÿnite trees form a free completely iterative theory over any given signature—this fact, proved by Elgot, Bloom and Tindell, turns out to be a special case of a much more general categorical result exhibited in the present paper. We prove that whenever an endofunctor H of a category has ÿnal coalgebras for all functors H () + X , then those coalgebras, TX… (More)
For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra A t. The universal property of these coalgebras, resembling freeness, is that for every state s of every system S there exists a unique coalgebra homomorphism from a unique A t which takes the… (More)
A survey of parts of General Coalgebra is presented with applications to the theory of systems. Stress is laid on terminal coalgebras and coinduction as well as iterative algebras and iterative theories. For my series of lectures on coalgebra on the preconference to CTCS 2002 I prepared a short text called " Introduction to Coalgebra " that was intended to… (More)
A concept of equation morphism is introduced for every endofuctor F of a cocomplete category C. Equationally defined classes of F –algebras for which free algebras exist are called varieties. Every variety is proved to be monadic over C, and conversely, every monadic category is equivalent to a variety. And the Birkhoff Variety Theorem is proved for "… (More)
Eilenberg's variety theorem, a centerpiece of algebraic automata theory, establishes a bijective correspondence between varieties of languages and pseudovarieties of monoids. In the present paper this result is generalized to an abstract pair of algebraic categories: we introduce varieties of languages in a category C, and prove that they correspond to… (More)
Every finitary endofunctor of Set is proved to generate a free iterative theory in the sense of Elgot. This work is based on coalgebras, specifically on parametric corecursion, and the proof is presented for categories more general than just Set.