Abstract and concrete categories
Abstract and concrete categories
Terminal coalgebras in well-founded set theory
Theoretical Computer Science
Universal coalgebra: a theory of systems
Theoretical Computer Science - Modern algebra and its applications
Category Theory and Computer Science
The temporal logic of coalgebras via Galois algebras
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science
Implementing collection classes with monads
Mathematical Structures in Computer Science
On the fusion of coalgebraic logics
CALCO'11 Proceedings of the 4th international conference on Algebra and coalgebra in computer science
Coalgebraic correspondence theory
FOSSACS'10 Proceedings of the 13th international conference on Foundations of Software Science and Computational Structures
A coalgebraic perspective on minimization and determinization
FOSSACS'12 Proceedings of the 15th international conference on Foundations of Software Science and Computational Structures
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For any set-endofunctor $T : {\mathcal S}et \rightarrow {\mathcal S}et$ there exists a largest sub-cartesian transformation μ to the filter functor ${\mathbb F}: {\mathcal S}et \rightarrow {\mathcal S}et$. Thus we can associate with every T-coalgebra A a certain filter-coalgebra $A_{\mathbb F}$. Precisely, when T (weakly) preserves preimages, μ is natural, and when T (weakly) preserves intersections, μ factors through the covariant powerset functor ${\mathbb P}$, thus providing for every T-coalgebra A a Kripke structure $A_{\mathbb P}$. We characterize preservation of preimages, preservation of intersections, and preservation of both preimages and intersections via the existence of natural, sub-cartesian or cartesian transformations from T to either ${\mathbb F}$ or ${\mathbb P}$. Moreover, we define for arbitrary T-coalgebras ${\mathcal A}$ a next-time operator $\bigcirc_{\mathcal A}$ with associated modal operators □ and $\lozenge$ and relate their properties to weak limit preservation properties of T. In particular, for any T-coalgebra ${\mathcal A}$ there is a transition system ${\mathcal K}$ with $\bigcirc_{A} = \bigcirc_{K}$ if and only if T preserves intersections.