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The Intersection of Algebra and Coalgebra Technical University of Braunschweig, PO Box 3329

Summary: The Intersection of Algebra and Coalgebra
J. Ad’amek
Technical University of Braunschweig, PO Box 3329
38023 Braunschweig, Germany
Presheaf categories are well­known to be varieties of algebras and covarieties of
coalgebras. We prove the converse: if a category is a variety as well as a covariety,
then it is a presheaf category. The result remains true for quasivarieties which are
Key words: variety, covariety, presheaf category
1991 MSC: 18C05, 18B20, 08B99,18C20
1 Introduction
The aim of the paper is to prove that the equation
algebra # coalgebra = presheaves
holds for the category of many­sorted sets. In the more restrictive case of
algebra and coalgebra on (single­sorted) sets the equation is
algebra # coalgebra = M­sets (M a monoid).
This shows that here, essentially, just sequential automata form the intersec­
tion of algebra and coalgebra. In fact, a sequential automaton can be viewed
as an algebra, or as a coalgebra: the main ingredient, the next­state function


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


Collections: Computer Technologies and Information Sciences