topology on coalgebras

Topology on Coalgebras

On Minimal Coalgebras

We define an out-degree for F -coalgebras and show that the coalgebras of outdegree at most κ form a covariety. As a subcategory of all F coalgebras, this class has a terminal object, which for many problems can stand in for the terminal F -coalgebra, which need not exist in general. As examples, we derive structure theoretic results about minimal coalgebras, showing that, for instance minimiza...

Bisimulation Games on Coalgebras

The theory of coalgebras [9] knows several ways to define behavioural equivalence or bisimilarity [11]. Another, complementary, view is to characterize behavioural equivalence via a coalgebraic modal logic [10, 8]. In concurrency theory a third perspective is quite common, where behavioural equivalence is defined via attacker-defender games [12]. Such games are useful both for theoretical reaso...

On coalgebras over algebras

We extend Barr’s well-known characterization of the final coalgebra of a Set-endofunctor H as the completion of its initial algebra to the EilenbergMoore category Alg(M) of algebras associated to a Set-monad M, if H can be lifted to Alg(M). As further analysis, we introduce the notion of commuting pair of endofunctors (T,H) with respect to a monad M and show that under reasonable assumptions, t...

On Coalgebras with Internal Moves

In the first part of the paper we recall the coalgebraic approach to handling the so-called invisible transitions that appear in different state-based systems semantics. We claim that these transitions are always part of the unit of a certain monad. Hence, coalgebras with internal moves are exactly coalgebras over a monadic type. The rest of the paper is devoted to supporting our claim by study...

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...