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On varieties and covarieties in a category

Published online by Cambridge University Press:  02 April 2003

JIR˘Í ADÁMEK  and
HANS-E. PORST
JIR˘Í ADÁMEK
Affiliation:
Department of Theoretical Computer Science, Technical University of Braunschweig, 38023 Braunschweig, Germany
HANS-E. PORST
Affiliation:
Department of Mathematics, University of Bremen, 28344 Bremen, Germany
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Abstract

A concept of equation morphism is introduced for every endofuctor $F$ of a cocomplete category $\Ce$. Equationally defined classes of $F$-algebras for which free algebras exist are called varieties. Every variety is proved to be monadic over $\Ce$, and, conversely, every monadic category is equivalent to a variety. The Birkhoff Variety Theorem is also proved for $`{\sf Set}\hbox{-like}'$ categories.

By dualising, we arrive at a concept of coequation such that covarieties, that is, coequationally specified classes of coalgebras with cofree objects, correspond precisely to comonadic categories. Natural examples of covarieties are presented.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 13 , Issue 2 , April 2003 , pp. 201 - 232
Copyright
2003 Cambridge University Press

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