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On nal coalgebras of continuous functors Ji Admek 1

Summary: On nal coalgebras of continuous functors
Ji°í Adámek 1
Technical University of Braunschweig, Postfach 3329
38106 Braunschweig, Germany
email: J.Adamek@tu-bs.de
Abstract. Continuous endofunctors F of locally nitely presentable cate-
gories carry a natural metric on their nal coalgebra. Whenever F (0) has an
element, this metric is proved to be a Cauchy completion of the initial algebra
of F . This is illustrated on the poset of real numbers represented as a nal
coalgebra of an endofunctor of Pos by Pavlovi˘ and Pratt. Under additional
assumptions on the locally nitely presentable category, all nitary endofunctors
are proved to have a nal coalgebra constructed in ! + ! steps of the natural
iteration construction.
I Introduction
Important data types are dened via initial algebras (typical for nite data
types) or nal coalgebras (typical for potentially innite data types) of suitable
functors F : K ! K. Here K is a category of data types and structure-preserving
maps under investigation. Typically K is a locally presentable category in the
sense of Gabriel and Ulmer [GU]. It is well-known that, depending on the inter-
nal structure of the category K, there is a connection between a nal coalgebra


Source: Adámek, Jiri - Institut für Theoretische Informatik, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig


Collections: Computer Technologies and Information Sciences