Equipping weak equivalences with algebraic structure

  • John BourkeEmail author


We investigate the extent to which the weak equivalences in a model category can be equipped with algebraic structure. We prove, for instance, that there exists a monad T such that a morphism of topological spaces admits T-algebra structure if and only it is a weak homotopy equivalence. Likewise for quasi-isomorphisms and many other examples. The basic trick is to consider injectivity in arrow categories. Using algebraic injectivity and cone injectivity we obtain general results about the extent to which the weak equivalences in a combinatorial model category can be equipped with algebraic structure.


Monads Algebraic injectives Weak equivalences 

Mathematics Subject Classification

Primary 55U35 Secondary 18C35 



The author gratefully acknowledges the support of an Australian Research Council Discovery Grant DP160101519 and the support of the Grant Agency of the Czech Republic under the grant 19-00902S. Particular thanks are due to Emily Riehl whose interest in an algebraic version of Smith’s theorem got me thinking about this topic and to Lukáš Vokřínek who helped me to see the connection between \(Ex_{\infty }\) and the generating cones for simplicial sets. Thanks also to the organisers of the PSSL101 in Leeds for providing the opportunity to present this work, and to the members of the Australian Category Seminar for listening to me speak about it.


  1. 1.
    Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, vol. 189. Cambridge University Press, Cambridge (1994)Google Scholar
  2. 2.
    Ara, D.: On the homotopy theory of Grothendieck \(\omega \)-groupoids. J. Pure Appl. Algebra 217, 1237–1278 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ara, D., Métayer, F.: The Brown–Golasiński model structure on strict \(\infty \)-groupoids revisited. Homol. Homot. Appl. 13(1), 121–142 (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Awodey, S.: A cubical model of homotopy type theory. Ann. Pure Appl. Log. 169(12), 1270–1294 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beke, T.: Sheafifiable homotopy model categories. Math. Proc. Camb. Philos. Soc. 129(3), 447–475 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourke, J., Garner, R.: Algebraic weak factorisation systems I: accessible awfs. J. Pure Appl. Algebra 220, 108–147 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bourke, J., Garner, R.: Algebraic weak factorisation systems 2: categories of weak maps. J. Pure Appl. Algebra 220, 148–174 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Diers, Y.: Catégories localement multiprésentables. Archiv der Mathematik 34(1), 153–170 (1980)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Diers, Y.: Multimonads and multimonadic categories. J. Pure Appl. Algebra 17, 153–170 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Diers, Y.: Some spectra relative to functors. J. Pure Appl. Algebra 22, 57–74 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dugger, D., Isaksen, D.: Weak equivalences of simplicial presheaves. In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory, 97–113, Contemp. Mathematics, vol. 346. American Mathematical Society, Providence (2004)Google Scholar
  12. 12.
    Gabriel, P., Ulmer, F.: Lokal präsentierbare Kategorien, Lecture Notes in Mathematics, vol. 221. Springer, Berlin (1971)Google Scholar
  13. 13.
    Gambino, N., Sattler, C.: The Frobenius condition, right properness, and uniform fibrations. J. Pure Appl. Algebra 221, 3027–3068 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Garner, R.: Understanding the small object argument. Appl. Categ. Struct. 17(3), 247–285 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Goerss, P., Jardine, J.: Simplicial Homotopy Theory, Progress in Mathematics, vol. 174. Birkhäuser, Boston (1999)Google Scholar
  16. 16.
    Hess, K., Kedziorek, M., Riehl, E., Shipley, B.: A necessary and sufficient condition for induced model structures. J. Topol. 10, 324–369 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    John, R.: A note on implicational subcategories. In: Proc. Colloq. Szeged, Coll. Math. J. Bolyai, vol. 17, pp. 213–222, North-Holland, Amsterdam (1975)Google Scholar
  18. 18.
    Joyal, A., Street, R.: Pullbacks equivalent to pseudopullbacks. Cahiers de Topologie et Geométrie Différentielle Catégoriques 34(2), 153–156 (1993)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kan, D.: On C.S.S. complexes. Am. J. Math. 79(3), 449–476 (1957)Google Scholar
  20. 20.
    Kelly, G.M.: A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bull. Aust. Math. Soc. 22(1), 1–83 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Koubek, V., Reiterman, J.: Categorical constructions of free algebras, colimits and completions of partial algebras. J. Pure Appl. Algebra 14, 195–231 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lack, S.: A Quillen model structure for 2-categories. K-Theory 26, 171–205 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lack, S.: A Quillen model structure for bicategories. K-Theory 33, 185–197 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lack, S.: Homotopy-theoretic aspects of 2-monads. J. Homotopy Relat. Struct. 7(2), 229–260 (2007)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lack, S.: A Quillen model structure for gray-categories. J. K-Theory 8(2), 183–221 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lafont, Y., Métayer, F., Worytkiewicz, K.: A folk model structure on omega-cat. Adv. Math. 224(3), 1183–1231 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Makkai, M., Paré, R.: Accessible categories: the foundations of categorical model theory. In: Contemporary Mathematics, vol. 104. American Mathematical Society, Providence (1989)Google Scholar
  28. 28.
    Nikolaus, T.: Algebraic models for higher categories. Indag. Math. (N.S.) 21(1–2), 52–75 (2011)Google Scholar
  29. 29.
    Quillen, D.: Homotopical algebra. In: Lecture Notes in Mathematics, , vol. 43. Springer, Berlin (1967)Google Scholar
  30. 30.
    Riehl, E.: Algebraic model structures. N. Y. J. Math. 17, 173–231 (2011)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Rosický, J.: On combinatorial model categories. Appl. Categ. Struct. 17, 303–316 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Swan, A.: Identity types in algebraic model structures and cubical sets. (2018). Arxiv Preprint arXiv:1808.00915
  33. 33.
    van den Berg, B., Garner, R.: Topological and simplicial models of identity types. Trans. ACM Comput. Log. 13(1), 3:1–3:44 (2012)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMasaryk UniversityBrnoCzech Republic

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