Applied Categorical Structures

, Volume 22, Issue 1, pp 13–28 | Cite as

A Colimit Decomposition for Homotopy Algebras in Cat

Article

Abstract

Badzioch showed that in the category of simplicial sets each homotopy algebra of a Lawvere theory is weakly equivalent to a strict algebra. In seeking to extend this result to other contexts Rosický observed a key point to be that each homotopy colimit in SSet admits a decomposition into a homotopy sifted colimit of finite coproducts, and asked the author whether a similar decomposition holds in the 2-category of categories Cat. Our purpose in the present paper is to show that this is the case.

Keywords

Homotopy algebra Flexibility Codescent object 

Mathematics Subject Classifications (2010)

18D05 55U35 

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References

  1. 1.
    Albert, M.H., Kelly, G.M.: The closure of a class of colimits. J. Pure Appl. Algebra 51(1–2), 1–17 (1988)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Badzioch, B.: Algebraic theories in homotopy theory. Ann. Math. 155(3), 895–913 (2002)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bergner, J.E.: Rigidification of algebras over multi-sorted theories. Algebr. Geom. Topol. 6, 1925–1955 (2005)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bird, G.J., Kelly, G.M., Power, A.J., Street, R.: Flexible limits for 2-categories. J. Pure Appl. Algebra 61(1), 1–27 (1989)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Blackwell, R., Kelly, G.M., Power, A.J.: Two-dimensional monad theory. J. Pure Appl. Algebra 59(1), 1–41 (1989)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bourke, J., Garner, R.: On semiflexible, flexible and pie algebras. J. Pure Appl. Algebra (2012). doi:10.1016/j.jpaa.2012.06.002
  7. 7.
    Dugger, D.: Combinatorial model categories have presentations. Adv. Math. 164(1), 177–201 (2001)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Gambino, N.: Homotopy limits for 2-categories. Math. Proc. Camb. Philos. Soc. 145(1), 43–63 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Hovey, M.: Model categories. In: Mathematical Surveys and Monographs, vol. 63. American Mathematical Society (1999)Google Scholar
  10. 10.
    Kelly, G.M.: Doctrinal adjunction. In: Category Seminar (Sydney, 1972/1973). Lecture Notes in Mathematics, vol. 420, pp. 257–280. Springer (1974)Google Scholar
  11. 11.
    Kelly, G.M.: Basic concepts of enriched category theory. In: London Mathematical Society Lecture Note Series, vol. 64. Cambridge University Press, Cambridge (1982)Google Scholar
  12. 12.
    Kelly, G.M.: Elementary observations on 2-categorical limits. Bull. Aust. Math. Soc. 39(2), 301–317 (1989)CrossRefMATHGoogle Scholar
  13. 13.
    Kelly, G.M., Schmitt, V.: Notes on enriched categories with colimits of some class. Theory Appl. Categ. 14, 399–423 (2005)MATHMathSciNetGoogle Scholar
  14. 14.
    Kelly, G.M., Lack, S., Walters, R.F.C.: Coinverters and categories of fractions for categories with structure. Appl. Categ. Struct. 1(1), 95–102 (1993)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Lack, S.: Codescent objects and coherence. J. Pure Appl. Algebra 175(1–3), 223–241 (2002)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Lack, S.: Homotopy-theoretic aspects of 2-monads. J. Homotopy Relat. Struct. 7(2), 229–260 (2007)MathSciNetGoogle Scholar
  17. 17.
    Lack, S.: A 2-categories companion. In: Towards Higher Categories, IMA Vol. Math. Appl., vol. 152, pp. 105–191. Springer (2010)Google Scholar
  18. 18.
    Lack, S., Shulman, M.: Enhanced 2-categories and limits for lax morphisms. Adv. Math. 229(1), 294–356 (2011)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Makkai, M., Paré, R.: Accessible categories: the foundations of categorical model theory. In: Contemporary Mathematics, vol. 104. American Mathematical Society (1989)Google Scholar
  20. 20.
    Power, J., Robinson, E.: A characterization of pie limits. Math. Proc. Camb. Philos. Soc. 110(1), 33–47 (1991)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Riehl, E.: Algebraic model structures. N.Y. J. Math. 17, 173–231 (2011)MATHMathSciNetGoogle Scholar
  22. 22.
    Rosický, J.: Rigidification of algebras over essentially algebraic theories. arXiv:1206.0422v1 (2012, preprint)
  23. 23.
    Street, R.: Categorical and combinatorial aspects of descent theory. Appl. Categ. Struct. 12(5–6), 537–576 (2004)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMasaryk UniversityBrnoCzech Republic

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