On an algebraic description of the category of \Sigma-coalgebras
by
Christian Dzierzon
University of Bremen, Department of Mathematics

Let \Sigma be a signature of operation symbols \sigma with arities |\sigma| bounded by a cardinal \lambda. The category CoAlg\Sigma of the corresponding polynomial functor H_\Sigma:Set --> Set, with H_\SigmaX=\coprod_{\sigma in \Sigma}{\sigma}×X^|\sigma| has been proven to be locally finitely presentable by [1], showing that the \Sigma-labelled trees, considered as coalgebras, form a strong generator of finitely presentable objects in CoAlg\Sigma. Thus, CoAlg\Sigma is equivalent to the category of models of an essentially algebraic theory (see e.g. [2]). We strengthen this result by showing that CoAlg\Sigma in fact is a many-sorted variety of unary algebras.

[1]     J. Adámek, H.-E. Porst: On Tree Coalgebras and Coalgebra Presentations, to appear in Theoretical Computer Science.

[2]     J. Adámek, J. Rosický: Locally Presentable and Accessible Categories, Cambridge University Press, Cambridge (1994).

Date received: January 30, 2003