Abstract: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.
Keywords: variety, Lawvere theory, sifted colimit, filtered colimit
AMS Subject Classification: 18C10, 08B99, 18A30