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 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.