Abstract and concrete categories
Abstract and concrete categories
Additions and corrections to “Terminal coalgebras in well-founded set theory”
Theoretical Computer Science
Universal coalgebra: a theory of systems
Theoretical Computer Science - Modern algebra and its applications
Theoretical Computer Science
Category Theory and Computer Science
Infinite trees and completely iterative theories: a coalgebraic view
Theoretical Computer Science
On tree coalgebras and coalgebra presentations
Theoretical Computer Science
Theoretical Computer Science - Logic, semantics and theory of programming
Completely iterative algebras and completely iterative monads
Information and Computation
Electronic Notes in Theoretical Computer Science (ENTCS)
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By the Final Coalgebra Theorem of Aczel and Mendler, every endofunctor of the category of sets has a final coalgebra, which, however, may be a proper class. We generalise this to all ‘well-behaved’ categories ${\frak K}$. The role of the category of classes is played by a free cocompletion ${\frak K}^\infty$ of ${\frak K}$ under transfinite colimits, that is, colimits of ordinal-indexed chains. Every endofunctor $F$ of ${\frak K}$ has a canonical extension to an endofunctor $F^\infty$ of ${\frak K}^\infty$, which is proved to have a final coalgebra (and an initial algebra). Based on this, we prove a general solution theorem: for every endofunctor of a locally presentable category ${\frak K}$ all guarded equation-morphisms have unique solutions. The last result does not need the extension ${\frak K}^\infty$: the solutions are always found within the category ${\frak K}$.