Previous |  Up |  Next


variety; Lawvere theory; sifted colimit; filtered colimit
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.
[ALR] Adámek J., Lawvere F.W., Rosický J.: On the duality between varieties and algebraic theories. submitted.
[AP] Adámek J., Porst H.-E.: Algebraic theories of quasivarieties. J. Algebra 208 (1998), 379-398. MR 1655458
[AR] Adámek J., Rosický J.: Locally Presentable and Accessible Categories. Cambridge University Press, 1994. MR 1294136
[Bo] Borceux F.: Handbook of Categorical Algebra. Cambridge University Press, 1994, (in three volumes). Zbl 1143.18003
[GU] Gabriel P., Ulmer F.: Lokal präsentierbare Kategorien. LNM 221, Springer-Verlag, Berlin, 1971. MR 0327863 | Zbl 0225.18004
[L] Lawvere F.W.: Functorial semantics of algebraic theories. Dissertation, Columbia University, 1963. MR 0158921 | Zbl 1062.18004
[S] Street R.: Fibrations in bicategories. Cahiers Topol. Géom. Différentielles Catégoriques XXI (1980), 111-160. MR 0574662 | Zbl 0436.18005
Partner of
EuDML logo