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Keywords:
variety; Lawvere theory; sifted colimit; filtered colimit
variety; Lawvere theory; sifted colimit; filtered colimit
Summary:
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.
References:
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