Homotopical algebra is not concrete


We generalize Freyd’s well-known result that “homotopy is not concrete”, offering a general method to show that under certain assumptions on a model category \(\mathcal {M}\), its homotopy category \(\textsc {ho}(\mathcal {M})\) cannot be concrete. This result is part of an attempt to understand more deeply the relation between set theory and abstract homotopy theory.

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  1. 1.

    As already mentioned, this is a shorthand to refer to the category of small categories with its ‘folk’ model structure having weak equivalences the equivalences of categories, and cofibrations the functors injective on objects.

  2. 2.

    The symbol \(\varpi \) is an alternative glyph for the Greek letter \(\pi \).


  1. 1.

    Baues, H.J.: Algebraic Homotopy. Cambridge University Press, Cambridge (1989)

    Book  Google Scholar 

  2. 2.

    Borceux, F.: Handbook of categorical algebra. 1, Basic category theory. In: Encyclopedia of Mathematics and Its Applications, vol. 50. Cambridge University Press, Cambridge (1994)

  3. 3.

    Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 136, pp. 9–51. Cambridge University Press, Cambridge (2004)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Freyd, P.J.: On the concreteness of certain categories. Symp. Math. 4, 431–456 (1969)

    Google Scholar 

  5. 5.

    Freyd, P.: Homotopy is not concrete. In: The Steenrod Algebra and Its Applications: A Conference to Celebrate NE Steenrod’s Sixtieth Birthday. Springer, Berlin, Heidelberg (1970)

    Google Scholar 

  6. 6.

    Freyd, P.J.: Concreteness. J. Pure Appl. Algebra 3(2), 171–191 (1973)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Fuchs, L.: Abelian Groups. Springer, Berlin (2015)

    Book  Google Scholar 

  8. 8.

    Gabriel, P., Zisman, M.: Calculus of fractions and homotopy theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35. Springer-Verlag New York Inc, New York (1967)

  9. 9.

    Isbell, J.R.: Two set-theoretical theorems in categories. Fundamenta Mathematicae 53(1), 43–49 (1964) (eng)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Jardine, J.F.: Simplicial presheaves. J. Pure Appl. Algebra 47(1), 35–87 (1987)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Graduate Texts in Mathematics, vol. 5. Springer, New York (1998)

  12. 12.

    May, P., Ponto, K.: More Concise Algebraic Topology: Localization, Completion, and Model Categories. University of Chicago Press, Chicago (2011)

    Book  Google Scholar 

  13. 13.

    The \(n\)Lab, Strøm model structure (2017). https://ncatlab.org/nlab/revision/Strom+model+structure/23

  14. 14.

    Strøm, A.: The homotopy category is a homotopy category. Archiv der Mathematik 23(1), 435–441 (1972)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Toën, B.: Simplicial presheaves and derived algebraic geometry. In: Advanced Course on Simplicial Methods in Higher Categories. Notes of the Course, pp. 95–146 (2010)

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The authors would like to thank professor Dan Christensen for a preliminary and attentive reading of the first draft of this paper, professor Jiří Rosický for his support in our investigation, professor Ivo Dell’Ambrogio for persuading us about the relevance of our result to a public of algebraic topologists, and more in general everybody who contributed to the improvement of this work.

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Correspondence to Fosco Loregian.

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F. Loregian is supported by the Grant Agency of the Czech Republic under the Grant P201/12/G028.

Communicated by Jiri Rosicky.

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Di Liberti, I., Loregian, F. Homotopical algebra is not concrete. J. Homotopy Relat. Struct. 13, 673–687 (2018). https://doi.org/10.1007/s40062-018-0197-3

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  • Concrete category
  • Homotopical algebra
  • Model category
  • Faithful functor