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    ×
  • Bulletin of the Australian Mathematical Society, Volume 17, Issue 3
  • December 1977, pp. 433-450

Colimits of algebras revisited

  • Jiří Adámek (a1)
  • DOI: http://dx.doi.org/10.1017/S0004972700010704
  • Published online: 01 April 2009
Abstract

It has “been open for some time whether, given an algebraic theory (triple, monad) Π in a cocomplete category K, also the category KΠ of Π-algebras must be cocomplete. We solve this in the negative by exhibiting a free algebraic theory Π in the category Gra of graphs such that GraΠ is not cocomplete. Further, we improve somewhat the well-known colimit theorem of Barr and Linton by showing that the base category need not be complete.

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[3]Michael A. Arbib and Ernest G. Manes , “Machines in a category: an expository introduction”, SIAM Rev. 16 (1974), 163192.

[4]Michael Barr , “Coequalizers and free triples”, Math. Z. 116 (1970), 307322.

[7]F.E.J. Linton , “Coequalizers in categories of algebras”, Seminar on triples and categorical homology theory, 7590 (Lecture Notes in Mathematics, 80. Springer-Verlag, Berlin, Heidelberg, New York, 1969).

[8]S. Mac Lane , Categories for the working mathematician (Graduate Texts in Mathematics, 5. Springer-Verlag, New York, Heidelberg, Berlin, 1971).

[9]Ernest G. Manes , Algebraic theories (Graduate Texts in Mathematics, 26. Springer-Verlag, New York, Heidelberg, Berlin, 1976).

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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