Order-enriched solid functors

Lurdes Sousa; Walter Tholen

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 4, page 553-580
  • ISSN: 0010-2628

Abstract

top
Order-enriched solid functors, as presented in this paper in two versions, enjoy many of the strong properties of their ordinary counterparts, including the transfer of the existence of weighted (co)limits from their codomains to their domains. The ordinary version of the notion first appeared in Trnková's work on automata theory of the 1970s and was subsequently studied by others under various names, before being put into a general enriched context by C. Anghel. Our focus in this paper is on differentiating the order-enriched notion from the ordinary one, mostly in terms of the functor's behaviour with respect to specific weighted (co)limits, and on the presentation of examples, which include functors of general varieties of ordered algebras and special ones, such as ordered vector spaces.

How to cite

top

Sousa, Lurdes, and Tholen, Walter. "Order-enriched solid functors." Commentationes Mathematicae Universitatis Carolinae 60.4 (2019): 553-580. <http://eudml.org/doc/295079>.

@article{Sousa2019,
abstract = {Order-enriched solid functors, as presented in this paper in two versions, enjoy many of the strong properties of their ordinary counterparts, including the transfer of the existence of weighted (co)limits from their codomains to their domains. The ordinary version of the notion first appeared in Trnková's work on automata theory of the 1970s and was subsequently studied by others under various names, before being put into a general enriched context by C. Anghel. Our focus in this paper is on differentiating the order-enriched notion from the ordinary one, mostly in terms of the functor's behaviour with respect to specific weighted (co)limits, and on the presentation of examples, which include functors of general varieties of ordered algebras and special ones, such as ordered vector spaces.},
author = {Sousa, Lurdes, Tholen, Walter},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {ordered category; (strongly) order-solid functor; weighted (co)limit; ordered algebra},
language = {eng},
number = {4},
pages = {553-580},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Order-enriched solid functors},
url = {http://eudml.org/doc/295079},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Sousa, Lurdes
AU - Tholen, Walter
TI - Order-enriched solid functors
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 4
SP - 553
EP - 580
AB - Order-enriched solid functors, as presented in this paper in two versions, enjoy many of the strong properties of their ordinary counterparts, including the transfer of the existence of weighted (co)limits from their codomains to their domains. The ordinary version of the notion first appeared in Trnková's work on automata theory of the 1970s and was subsequently studied by others under various names, before being put into a general enriched context by C. Anghel. Our focus in this paper is on differentiating the order-enriched notion from the ordinary one, mostly in terms of the functor's behaviour with respect to specific weighted (co)limits, and on the presentation of examples, which include functors of general varieties of ordered algebras and special ones, such as ordered vector spaces.
LA - eng
KW - ordered category; (strongly) order-solid functor; weighted (co)limit; ordered algebra
UR - http://eudml.org/doc/295079
ER -

References

top
  1. Adámek J., Herrlich H., Strecker G. E., Abstract and Concrete Categories, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, New York, 1990. MR1051419
  2. Adámek J., Rosický J., Vitale E. M., Algebraic Theories, Cambridge Tracts in Mathematics, 184, Cambridge University Press, Cambridge, 2011. MR2757312
  3. Adámek J., Sousa L., KZ-monadic categories and their logic, Theory Appl. Categ. 32 (2017), paper no. 10, 338–379. MR3633707
  4. Adámek J., Sousa L., Velebil J., 10.1017/S0960129514000024, Math. Structures Comput. Sci. 25 (2015), no. 1, 6–45. MR3283679DOI10.1017/S0960129514000024
  5. Anghel C., Factorizations and Initiality in Enriched Categories, Doctoral Dissertation, Fernuniversität, Hagen, 1987. 
  6. Anghel C., 10.1080/00927879008823905, Comm. Algebra 18 (1990), no. 1, 135–181. MR1037900DOI10.1080/00927879008823905
  7. Anghel C., 10.1080/00927879008823906, Comm. Algebra 18 (1990), no. 1, 183–192. MR1037901DOI10.1080/00927879008823906
  8. Antoine P., Étude élémentaire des catégories d'ensembles structurés, Bull. Soc. Math. Belg. 18 (1966), 142–164, 387–414 (French). MR0200321
  9. Börger R., Tholen W., 10.1007/BF01214264, Math. Z. 160 (1978), no. 2, 135–138 (German). MR0498772DOI10.1007/BF01214264
  10. Börger R., Tholen W., 10.4153/CJM-1990-012-x, Canad. J. Math. 42 (1990), no. 2, 213–229. MR1051726DOI10.4153/CJM-1990-012-x
  11. Brümmer G. C. L., A Categorical Study of Initiality in Uniform Topology, Ph.D. Thesis, University of Cape Town, Cape Town, 1971. 
  12. Carvalho M., Sousa L., 10.1016/j.topol.2010.12.016, Topology Appl. 158 (2011), no. 17, 2408–2422. MR2838390DOI10.1016/j.topol.2010.12.016
  13. Carvalho M., Sousa L., 10.1007/s10485-015-9413-z, Appl. Categorical Structures 25 (2017), no. 1, 83–104. MR3606496DOI10.1007/s10485-015-9413-z
  14. Čech E., Topological Spaces, Academia, Prague, 1966. 
  15. Dubuc E., Adjoint triangles, Reports of the Midwest Category Seminar, Springer, Berlin, 1968, pages 69–91. MR0233864
  16. Fritz T., 10.1017/S0960129515000444, Math. Structures Comput. Sci. 27 (2017), no. 6, 850–938. MR3683631DOI10.1017/S0960129515000444
  17. Herrlich H., 10.1016/0016-660X(74)90016-6, General Topology and Appl. 4 (1974), 125–142. MR0343226DOI10.1016/0016-660X(74)90016-6
  18. Hoffmann R.-E., Die kategorielle Auffassung der Initial- und Finaltopologie, Doctoral Dissertation, Ruhr-Universität, Bochum, 1972 (German). 
  19. Hoffmann R.-E., 10.1002/mana.3210740124, Math. Nachr. 74 (1976), 295–307. MR0428256DOI10.1002/mana.3210740124
  20. Hofmann D. (ed.), Seal G. J. (ed.), Tholen W. (ed.), Monoidal Topology, Encyclopedia of Mathematics and Its Applications, 153, Cambridge University Press, Cambridge, 2014. MR3307673
  21. Hofmann D., Sousa L., Aspects of algebraic algebras, Log. Methods Comput. Sci. 13 (2017), no. 3, paper no. 4, 25 pages. MR3673245
  22. Hong Y. H., Studies on Categories of Universal Topological Algebras, Ph.D. Thesis, McMaster University, Hamilton, 1974. MR2702868
  23. Hušek M., S -categories, Comment. Math. Univ. Carolinae 5 (1964), 37–46. MR0174027
  24. Jameson G., Ordered Linear Spaces, Lecture Notes in Mathematics, 141, Springer, Berlin, 1970. MR0438077
  25. Kelly G. M., Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Note Series, 64, Cambridge University Press, Cambridge, 1982. Zbl1086.18001MR0651714
  26. Kurz A., Velebil J., 10.1017/S096012951500050X, Math. Structures Comput. Sci. 27 (2017), no. 7, 1153–1194. MR3705669DOI10.1017/S096012951500050X
  27. Lawvere F. W., Functorial Semantics of Algebraic Theories, Ph.D. Thesis, Columbia University, New York, 1963. MR2939398
  28. Linton F. E. J., Some aspects of equational categories, Proc. of the Conf. Categorical Algebra, La Jolla, 1965, Springer, New York, 1966, pages 84–94. MR0209335
  29. MacLane S., Categories for the Working Mathematician, Graduate Texts in Mathematics, 5, Springer, Berlin, 1971. Zbl0705.18001MR0354798
  30. Manes E. G., 10.1007/BF02945002, Algebra Universalis 2 (1972), 7–17. MR0311741DOI10.1007/BF02945002
  31. Picado J., Pultr A., Frames and Locales: Topology without Points, Frontiers in Mathematics, Birkhäuser/Springer Basel AG, Basel, 2012. MR2868166
  32. Roberts J. E., 10.1016/0021-8693(68)90044-6, J. Algebra 8 (1968), 181–193. MR0224674DOI10.1016/0021-8693(68)90044-6
  33. Schaefer H. H., Topological Vector Spaces, Graduate Texts in Mathematics, 3, Springer, New York, 1971. Zbl0983.46002MR0342978
  34. Shukla W., On Top Categories, Ph.D. Thesis, Indian Institute of Technology, Kanpur, 1971. 
  35. Sousa L., 10.1007/BF00877631, Appl. Categ. Structures 3 (1995), no. 2, 105–118. MR1329186DOI10.1007/BF00877631
  36. Sousa L., 10.1016/j.jpaa.2016.07.002, J. Pure Appl. Algebra 221 (2017), no. 2, 422–448. MR3545270DOI10.1016/j.jpaa.2016.07.002
  37. Street R., Tholen W., Wischnewsky M., Wolff H., 10.1016/0022-4049(80)90035-3, J. Pure Appl. Algebra 16 (1980), no. 3, 299–314. MR0558494DOI10.1016/0022-4049(80)90035-3
  38. Taylor J. C., 10.4153/CMB-1965-057-7, Canad. Math. Bull. 8 (1965), 771–781. MR0212069DOI10.4153/CMB-1965-057-7
  39. Tholen W., M -functors, Nordwestdeutsches Kategorienseminar, Tagung, Bremen, 1976, Math.-Arbeitspapiere, No. 7, Teil A: Math. Forschungspapiere, Universität Bremen, Bremen, 1976, pages 178–185. MR0486037
  40. Tholen W., 10.1016/0016-660X(78)90050-8, General Topology and Appl. 8 (1978), no. 2, 197–206. MR0480669DOI10.1016/0016-660X(78)90050-8
  41. Tholen W., 10.1016/0022-4049(79)90040-9, J. Pure Appl. Algebra 15 (1979), no. 1, 53–73. MR0532963DOI10.1016/0022-4049(79)90040-9
  42. Tholen W., 10.1017/S0004972700005992, Bull. Austral. Math. Soc. 21 (1980), no. 2, 169–173. MR0574836DOI10.1017/S0004972700005992
  43. Tholen W., Wischnewsky M. B., 10.1016/0022-4049(79)90041-0, J. Pure Appl. Algebra 15 (1979), no. 1, 75–92. MR0532964DOI10.1016/0022-4049(79)90041-0
  44. Trnková V., 10.1007/3-540-07389-2_188, Mathematical Foundations of Computer Science 1975, Fourth Sympos., Mariánské Lázně, 1975, Lecture Notes in Computer Science, 32, Springer, Berlin, 1975, 138–152. MR0393175DOI10.1007/3-540-07389-2_188
  45. Wischnewsky M. B., 10.1007/BF01110107, Math. Z. 127 (1972), 83–91 (German). MR0308238DOI10.1007/BF01110107
  46. Wischnewsky M. B., A lifting theorem for right adjoints, Cahiers Topologie Géom. Differentielle 19 (1978), no. 2, 155–168. MR0528344
  47. Wyler O., 10.1007/BF01222531, Arch. Math. (Basel) 22 (1971), 7–17. MR0287563DOI10.1007/BF01222531
  48. Wyler O., 10.1016/0016-660X(71)90106-1, General Topology and Appl. 1 (1971), no. 1, 17–28. MR0282324DOI10.1016/0016-660X(71)90106-1

NotesEmbed ?

top

You must be logged in to post comments.