# A duality between infinitary varieties and algebraic theories

Jiří Adámek; Václav Koubek; Jiří Velebil

Commentationes Mathematicae Universitatis Carolinae (2000)

- Volume: 41, Issue: 3, page 529-541
- ISSN: 0010-2628

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topAdámek, Jiří, Koubek, Václav, and Velebil, Jiří. "A duality between infinitary varieties and algebraic theories." Commentationes Mathematicae Universitatis Carolinae 41.3 (2000): 529-541. <http://eudml.org/doc/248619>.

@article{Adámek2000,

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 $\mathcal \{D\}$-colimits in $\text\{Set\}$, then $\mathcal \{D\}$ 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.},

author = {Adámek, Jiří, Koubek, Václav, Velebil, Jiří},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {variety; Lawvere theory; sifted colimit; filtered colimit; varieties; diagrams; limits},

language = {eng},

number = {3},

pages = {529-541},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {A duality between infinitary varieties and algebraic theories},

url = {http://eudml.org/doc/248619},

volume = {41},

year = {2000},

}

TY - JOUR

AU - Adámek, Jiří

AU - Koubek, Václav

AU - Velebil, Jiří

TI - A duality between infinitary varieties and algebraic theories

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2000

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 41

IS - 3

SP - 529

EP - 541

AB - 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 $\mathcal {D}$-colimits in $\text{Set}$, then $\mathcal {D}$ 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.

LA - eng

KW - variety; Lawvere theory; sifted colimit; filtered colimit; varieties; diagrams; limits

UR - http://eudml.org/doc/248619

ER -

## References

top- Adámek J., Lawvere F.W., Rosický J., On the duality between varieties and algebraic theories, submitted.
- Adámek J., Porst H.-E., Algebraic theories of quasivarieties, J. Algebra 208 (1998), 379-398. (1998) MR1655458
- Adámek J., Rosický J., Locally Presentable and Accessible Categories, Cambridge University Press, 1994. MR1294136
- Borceux F., Handbook of Categorical Algebra, Cambridge University Press, 1994, (in three volumes). Zbl1143.18003
- Gabriel P., Ulmer F., Lokal präsentierbare Kategorien, LNM 221, Springer-Verlag, Berlin, 1971. Zbl0225.18004MR0327863
- Lawvere F.W., Functorial semantics of algebraic theories, Dissertation, Columbia University, 1963. Zbl1062.18004MR0158921
- Street R., Fibrations in bicategories, Cahiers Topol. Géom. Différentielles Catégoriques XXI (1980), 111-160. (1980) Zbl0436.18005MR0574662

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