This paper presents an elementary and self-contained proof of an existence theorem of nal coalgebras for endofunctors on the category of sets and functions.

Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic view is also quite effective for studying processes over structured state spaces, e.g. measurable, or continuous. In the present paper we consider coalgebras ...

Given a coalgebra p for HG, an algebra s for GH and the obvious notion of morphism from coalgebra to algebra, we observe that morphisms from Gp to s correspond to morphisms from p to Hs in the manner of an adjunction even when there is no adjunction between G and H. This appears to be one of the basic phenomena underlying the transport of special algebras. A proof of Freyd’s Iterated Square The...

A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a “lax relation lifting”, or “relator” as used by other authors. Simulations appear as coalgebras of this lifted functor, and similarity as greatest simulation. Two-way similarity is then similarity in both directions. In...

A characterization result for equationally de nable classes of certain coalgebras (including basic hidden algebra) shows that a class of coalgebras is de nable by equations if and only if it is closed under coproducts, quotients, sources of morphisms and representative inclusions. The notions of equation and satisfaction are axiomatized in order to include the di erent approaches in the literat...

We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a symmetric coalgebra. A dual version of Brauer's equivalence theorem is presented, allowing a characterization of symmetric coalgebras by comparing certain func...

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 At. 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 At which takes the root of t to s. Moreover, the tree coalge...