 
Summary: The Intersection of Algebra and Coalgebra
J. Ad’amek
Technical University of Braunschweig, PO Box 3329
38023 Braunschweig, Germany
Abstract
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
quasicovarieties.
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
