A duality between $\lambda$-ary varieties and $\lambda$-ary algebraic theories is proved as a direct generalization of the finitary case studied by the first author, F.W. Lawvere and J. Rosick'y. We also prove that for every uncountable cardinal $\lambda $, whenever $\lambda $-small products commute with $\Cal D$-colimits in $\text{Set}$, then $\Cal D$ must be a $\lambda $-filtered category. We nevertheless introduce the concept of $\lambda$-sifted colimits so that morphisms between $\lambda$-ary varieties (defined to be $\lambda$-ary, regular right adjoints) are precisely the functors preserving limits and $\lambda$-sifted colimits.
@article{119187,
author = {Ji\v r\'\i\ Ad\'amek and V\'aclav Koubek and Ji\v r\'\i\ Velebil},
title = {A duality between infinitary varieties and algebraic theories},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {41},
year = {2000},
pages = {529-541},
zbl = {1035.08004},
mrnumber = {1795083},
language = {en},
url = {http://dml.mathdoc.fr/item/119187}
}
Adámek, Jiří; Koubek, Václav; Velebil, Jiří. A duality between infinitary varieties and algebraic theories. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 529-541. http://gdmltest.u-ga.fr/item/119187/
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