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Generating the algebraic theory of C(X): the case of partially ordered compact spaces

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Dirk Hofmann, Renato Neves, and Pedro Nora

It is known since the late 1960's that the dual of the category of compact
Hausdorff spaces and continuous maps is a variety - not finitary, but
bounded by $\aleph_1$. In this note we show that the dual of the category
of partially ordered compact spaces and monotone continuous maps is an
$\aleph_1$-ary quasivariety, and describe partially its algebraic theory.
Based on this description, we extend these results to categories of
Vietoris coalgebras and homomorphisms on ordered compact spaces. We also
characterise the $\aleph_1$-copresentable partially ordered compact
spaces.

Keywords:
Ordered compact space, quasivariety, duality, coalgebra, Vietoris functor, copresentable object, metrisable

2010 MSC:
18B30, 18D20, 18C35, 4A05, 54F05

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 12, pp 276-295.

Published 2018-04-26.

http://www.tac.mta.ca/tac/volumes/33/12/33-12.pdf

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