Skip to main content
Log in

Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript


In this paper we prove that various quasi-categories whose objects are \(\infty \)-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which we characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories arise this way up to equivalence, this analysis covers the general case. Namely, we show that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit cones, whose shape is governed by the weight for pseudo limits over a homotopy coherent diagram but with the defining universal property up to equivalence, rather than isomorphism, of mapping spaces. Our applications follow from the fact that the \((\infty ,1)\)-categorical core of an \(\infty \)-cosmos admits weighted homotopy limits for all flexible weights, which includes in particular the weight for pseudo cones.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Known \(\infty \)-cosmoi of \((\infty ,n)\)-categories include \(\theta _n\)-spaces, iterated complete Segal spaces, n-complicial sets, and n-quasi-categories.

  2. Here we show that the codomain of the comparison map in (4.1.9) is a quasi-category by applying Proposition 4.1.5 in the \(\infty \)-cosmos of quasi-categories.

  3. The simplicial computads are the cofibrant objects [12, §16.2] in the model structure on simplicial categories due to Bergner [2].

  4. For bookkeeping reasons it is convenient to adopt the convention that atomic arrows are not identities, though in a simplicial computad the identities will also admit no non-trivial factorisations. With this convention, an identity arrow factors uniquely as an empty composite of atomic arrows.

  5. For more details about the Leibniz or “pushout-product” construction see [13, §4].


  1. Brown, K.S.: Abstract homotopy theory and generalized sheaf cohomology. Trans. Am. Math. Soc. 186, 419–458 (1973)

    Article  MathSciNet  Google Scholar 

  2. Bergner, J.E.: A model category structure on the category of simplicial categories. Trans. Am. Math. Soc. 359, 2043–2058 (2007)

    Article  MathSciNet  Google Scholar 

  3. Barnea, I., Harpaz, Y., Horel, G.: Pro-categories in homotopy theory. Algebr. Geom. Topol. 17(1), 567–643 (2017)

    Article  MathSciNet  Google Scholar 

  4. Barwick, C., Schommer-Pries, C.: On the unicity of the homotopy theory of higher categories. arXiv:1112.0040 (2011)

  5. Cordier, J.-M.: Sur la notion de diagramme homotopiquement cohérent. In: Proceedings of 3éme Colloque sur les Catégories, Amiens (1980), vol. 23, pp. 93–112 (1982)

  6. Dugger, D., Spivak, D.I.: Mapping spaces in quasi-categories. Algebr. Geom. Topol. 11(1), 263–325 (2011)

    Article  MathSciNet  Google Scholar 

  7. Gambino, N.: Weighted limits in simplicial homotopy theory. J. Pure Appl. Algebra 214(7), 1193–1199 (2010)

    Article  MathSciNet  Google Scholar 

  8. Joyal, A.: Quasi-categories and Kan complexes. J. Pure Appl. Algebra 175, 207–222 (2002)

    Article  MathSciNet  Google Scholar 

  9. Kelly, G.M.: Basic Concepts of Enriched Category Theory. London Mathematical Society. Lecture Notes Series, vol. 64. Cambridge University Press, Cambridge (1982)

    Google Scholar 

  10. Lurie, J.: Higher Topos Theory, Annals of Mathematical Studies, vol. 170. Princeton University Press, Princeton, NJ (2009)

    MATH  Google Scholar 

  11. Riehl, E.: On the structure of simplicial categories associated to quasi-categories. Math. Proc. Camb. Philos. Soc. 150(3), 489–504 (2011)

    Article  MathSciNet  Google Scholar 

  12. Riehl, E.: Categorical Homotopy Theory, New Mathematical Monographs, vol. 24. Cambridge University Press, Cambridge (2014)

    Book  Google Scholar 

  13. Riehl, E., Verity, D.: The theory and practice of Reedy categories. Theory Appl. Categ. 29(9), 256–301 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Riehl, E., Verity, D.: The 2-category theory of quasi-categories. Adv. Math. 280, 549–642 (2015)

    Article  MathSciNet  Google Scholar 

  15. Riehl, E., Verity, D.: Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions. Homol. Homotopy Appl. 17(1), 1–33 (2015)

    Article  MathSciNet  Google Scholar 

  16. Riehl, E., Verity, D.: Homotopy coherent adjunctions and the formal theory of monads. Adv. Math. 286, 802–888 (2016)

    Article  MathSciNet  Google Scholar 

  17. Riehl, E., Verity, D.: Fibrations and Yoneda’s lemma in an \(\infty \)-cosmos. J. Pure Appl. Algebra 221(3), 499–564 (2017)

    Article  MathSciNet  Google Scholar 

  18. Riehl, E., Verity, D.: Kan extensions and the calculus of modules for \(\infty \)-categories. Algebr. Geom. Topol. 17–1, 189–271 (2017)

    Article  MathSciNet  Google Scholar 

  19. Riehl, E., Verity, D.: The comprehension construction. Higher Struct. 2(1), 116–190 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Riehl, E., Verity, D.: On the construction of limits and colimits in \(\infty \)-categories. arXiv:1808.09835 (2018)

  21. Szumiło, K.: Two models for the homotopy theory of cocomplete homotopy theories. arXiv:1411.0303 (2014)

  22. Toën, B.: Vers une axiomatisation de la théorie des catégories supérieures. K-Theory 34(3), 233–263 (2005)

    Article  MathSciNet  Google Scholar 

  23. Verity, D.: Weak complicial sets II, nerves of complicial Gray-categories. In: Davydov, A., et al. (eds.) Categories in Algebra, Geometry and Mathematical Physics (StreetFest), Contemporary Mathematics, vol. 431, pp. 441–467. American Mathematical Society, Providence (2007)

    Chapter  Google Scholar 

  24. Verity, D.: Weak complicial sets I, basic homotopy theory. Adv. Math. 219, 1081–1149 (2008)

    Article  MathSciNet  Google Scholar 

Download references


The authors are grateful for support from the National Science Foundation (DMS-1551129 and DMS-1652600) and from the Australian Research Council (DP160101519). This work was commenced when the second-named author was visiting the first at Harvard and then at Johns Hopkins, continued while the first-named author was visiting the second at Macquarie, and completed after everyone finally made their way home. We thank all three institutions for their assistance in procuring the necessary visas as well as for their hospitality. The final published manuscript benefitted greatly from the astute suggestions of an eagle-eyed referee. We are also grateful to the referee for the sequel [20] who noticed that the proof of the converse result Theorem 6.2.7 originally given there could be simplified enough so that we could include it here.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Emily Riehl.

Additional information

Communicated by Nicola Gambino.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Riehl, E., Verity, D. Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves. Appl Categor Struct 28, 669–716 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification