Totality of product completions

Jiří Adámek; Lurdes Sousa; Walter Tholen

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 1, page 9-24
  • ISSN: 0010-2628

Abstract

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Categories whose Yoneda embedding has a left adjoint are known as total categories and are characterized by a strong cocompleteness property. We introduce the notion of multitotal category 𝒜 by asking the Yoneda embedding 𝒜 → [ 𝒜 o p , 𝒮 e t ] to be right multiadjoint and prove that this property is equivalent to totality of the formal product completion Π 𝒜 of 𝒜 . We also characterize multitotal categories with various types of generators; in particular, the existence of dense generators is inherited by the formal product completion iff measurable cardinals cannot be arbitrarily large.

How to cite

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Adámek, Jiří, Sousa, Lurdes, and Tholen, Walter. "Totality of product completions." Commentationes Mathematicae Universitatis Carolinae 41.1 (2000): 9-24. <http://eudml.org/doc/248624>.

@article{Adámek2000,
abstract = {Categories whose Yoneda embedding has a left adjoint are known as total categories and are characterized by a strong cocompleteness property. We introduce the notion of multitotal category $\mathcal \{A\}$ by asking the Yoneda embedding $\mathcal \{A\} \rightarrow [\mathcal \{A\}^\{op\},\mathcal \{S\}et]$ to be right multiadjoint and prove that this property is equivalent to totality of the formal product completion $\Pi \mathcal \{A\}$ of $\mathcal \{A\}$. We also characterize multitotal categories with various types of generators; in particular, the existence of dense generators is inherited by the formal product completion iff measurable cardinals cannot be arbitrarily large.},
author = {Adámek, Jiří, Sousa, Lurdes, Tholen, Walter},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {multitotal category; multisolid functor; formal product completion; multitotal category; multisolid functor; formal product completion},
language = {eng},
number = {1},
pages = {9-24},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Totality of product completions},
url = {http://eudml.org/doc/248624},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Adámek, Jiří
AU - Sousa, Lurdes
AU - Tholen, Walter
TI - Totality of product completions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 1
SP - 9
EP - 24
AB - Categories whose Yoneda embedding has a left adjoint are known as total categories and are characterized by a strong cocompleteness property. We introduce the notion of multitotal category $\mathcal {A}$ by asking the Yoneda embedding $\mathcal {A} \rightarrow [\mathcal {A}^{op},\mathcal {S}et]$ to be right multiadjoint and prove that this property is equivalent to totality of the formal product completion $\Pi \mathcal {A}$ of $\mathcal {A}$. We also characterize multitotal categories with various types of generators; in particular, the existence of dense generators is inherited by the formal product completion iff measurable cardinals cannot be arbitrarily large.
LA - eng
KW - multitotal category; multisolid functor; formal product completion; multitotal category; multisolid functor; formal product completion
UR - http://eudml.org/doc/248624
ER -

References

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  1. Adámek J., Herrlich H., Strecker G.E., Abstract and Concrete Categories, John Wiley and Sons, New York, 1990. MR1051419
  2. Adámek J., Rosický J., Accessible and Locally Presentable Categories, Cambridge University Press, Cambridge, 1995. 
  3. Adámek J., Tholen W., Total categories with generators, J. Algebra 133 (1990), 63-78. (1990) MR1063381
  4. Börger R., Tholen W., Total categories and solid functors, Canad. J. Math. 42.1 (1990), 213-229. (1990) MR1051726
  5. Börger R., Tholen W., Wischnewsky M.B., Wolff H., Compact and hypercomplete categories, J. Pure Appl. Algebra 21 (1981), 120-140. (1981) MR0614376
  6. Carboni A., Johnstone P.T., Connected limits, familial representability and Artin glueing, Math. Struct. in Comp. Science 5 (1995), 1-19. (1995) Zbl0849.18002MR1377312
  7. Diers Y., Catègories localisables, These de doctorat d'état, Université Pierre et Marie Curie - Paris 6, 1977. 
  8. Diers Y., Catègories localement multiprésentables, Arch. Math. 34 (1980), 344-356. (1980) Zbl0453.18002MR0593951
  9. Gabriel P., Ulmer F., Lokal präsentierbare Kategorien, Lecture Notes in Math. 221, Springer, Berlin, 1971. Zbl0225.18004MR0327863
  10. Isbell J.R., Adequate subcategories, Illinois J. Math. 4 (1960), 541-552. (1960) Zbl0104.01704MR0175954
  11. Kelly M., A survey of totality for enriched and ordinary categories, Cahiers Topologie Géom. Différentielle Catégoriques 27 (1986), 109-131. (1986) Zbl0593.18007MR0850527
  12. Rosický J., Tholen W., Accessibility and the solution set condition, J. Pure Appl. Algebra 98 (1995), 189-208. (1995) MR1319969
  13. Sousa L., Note on multisolid categories, J. Pure Appl. Algebra 129 (1998), 201-205. (1998) Zbl0939.18003MR1624462
  14. Street R., The family approach to total cocompleteness and toposes, Trans. Amer. Math. Soc. 284 (1984), 355-369. (1984) Zbl0512.18001MR0742429
  15. Street R., Walters R.F.C., Yoneda structures on 2 -categories, J. Algebra 50 (1978), 350-379. (1978) Zbl0401.18004MR0463261
  16. Tholen W., Semi-topological functors I, J. Pure Appl. Algebra 15 (1979), 53-73. (1979) Zbl0413.18001
  17. Tholen W., Note on total categories, Bull. Austral. Math. Soc. 21 (1980), 169-173. (1980) Zbl0431.18002MR0574836
  18. Tholen W., MacNeille completions of concrete categories with local properties, Comment. Math., Univ. St. Pauli 28 (1979), 179-202. (1979) MR0578672
  19. Wood R.J., Some remarks on total categories, J. Algebra 75 (1982), 538-545. (1982) Zbl0504.18001MR0653907

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