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Summary: CMCS'01 Preliminary Version
From Varieties of Algebras to Covarieties
of Coalgebras
Jir Adamek 1;2
Department of Theoretical Computer Science
Technical University of Braunschweig
Braunschweig, Germany
Hans{E. Porst 3;4
Department of Mathematics
University of Bremen
Bremen, Germany
Abstract
Varieties of F -algebras with respect to an endofunctor F on an arbitrary cocomplete
category C are de ned as equational classes admitting free algebras. They are
shown to correspond precisely to the monadic categories over C. Under suitable
assumptions satis ed in particular by any endofunctor on Set and Set op the Birkho
Variety Theorem holds. By dualization, covarieties over complete categories C are
introduced, which then correspond to the comonadic categories over C, and allow
for a characterization in dual terms of the Birkho Variety Theorem. Moreover, the
well known conditions of accessibilitly and boundedness for Set-functors F , suÆcient
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