Geodesic
A duality between infinitary varieties and algebraic theories
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 3, pp. 529-541

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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.
Classification : 08B99, 18A30, 18C10
Keywords: variety; Lawvere theory; sifted colimit; filtered colimit 
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Adámek, Jiří; Koubek, Václav; Velebil, Jiří. A duality between infinitary varieties and algebraic theories. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 3, pp. 529-541. http://geodesic.mathdoc.fr/item/CMUC_2000__41_3_a9/