On tree coalgebras and coalgebra presentations

On tree coalgebras and coalgebra presentations

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...

On descent for coalgebras and type transformations

We find a criterion for a morphism of coalgebras over a Barr-exact category to be effective descent and determine (effective) descent morphisms for coalgebras over toposes in some cases. Also, we study some exactness properties of endofunctors of arbitrary categories in connection with natural transformations between them as well as those of functors that these transformations induce between co...

Topology on Coalgebras

Interfaces as functors, programs as coalgebras - A final coalgebra theorem in intensional type theory

In [17,19] Peter Hancock and Anton Setzer introduced rules to extend Martin-Löf’s type theory in order to represent interactive programming. The rules essentially reflect the existence of weakly final coalgebras for a general form of polynomial functor. The standard rules of dependent type theory allow the definition of inductive types, which correspond to initial algebras. Coalgebraic types ar...

On coalgebra based on classes

The category Class of classes and functions is proved to have a number of properties suitable for algebra and coalgebra: every endofunctor has an initial algebra and a terminal coalgebra, the categories of algebras and coalgebras are complete and cocomplete, and every endofunctor generates a free completely iterative monad. A description of a terminal coalgebra for the power-set functor is prov...