Final Coalgebras And a Solution Theorem for Arbitrary Endofunctors

✍ Scribed by Jiří Adámek; Stefan Milius; Jiří Velebil

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 411 KB
• Volume: 65
• Category: Article
• ISSN: 1571-0661
• DOI: 10.1016/s1571-0661(04)80357-8

✦ Synopsis

Every endofunctor (F) of Set has an initial algebra and a final coalgebra, but they are classes in general. Consequently, the endofunctor (F^{\infty}) of the category of classes that (F) induces generates a completely iterative monad (T). And solutions of arbitrary guarded systems of iterative equations w.r.t. (F) exist, and can be found in naturally defined subsets of the classes (T Y).

More generally, starting from any category (\mathcal{K}), we can form a free cocompletion (\mathcal{K}^{\infty}) of (\mathcal{K}) under small-filtered colimits (e.g., (\operatorname{Set}^{\infty}) is the category of classes), and we give sufficient conditions to obtain analogous results for arbitrary endofunctors of (\mathcal{K}).