Skip to main content
SpringerLink
Log in

2-Cartesian Fibrations I: A Model for \(\infty \)-Bicategories Fibred in \(\infty \)-Bicategories

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

In this paper, we provide a notion of \(\infty \)-bicategories fibred in \(\infty \)-bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled simplicial sets equipped with an additional collection of triangles containing the scaled 2-simplices, which we call lean triangles, in addition to a collection of edges containing all degenerate 1-simplices. We prove the existence of a left proper combinatorial simplicial model category whose fibrant objects are precisely the 2-Cartesian fibrations over a chosen scaled simplicial set S. Over the terminal scaled simplicial set, this provides a new model structure modeling \(\infty \)-bicategories, which we show is Quillen equivalent to Lurie’s scaled simplicial set model. We conclude by providing a characterization of 2-Cartesian fibrations over an \(\infty \)-bicategory. This characterization then allows us to identify those 2-Cartesian fibrations arising as the coherent nerve of a fibration of \({\text {Set}}^+_{\Delta }\)-enriched categories, thus showing that our definition recovers the preexisting notions of fibred 2-categories.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Availability of data and material

Not applicable.

Notes

  1. Note that this proposition is about the pushout-product of marked biscaled simplicial sets. For readability, we have omitted the marking and biscaling from the notation in the conclusion.

References

  1. Abellán García, F.: Marked colimits and higher cofinality. Homotopy Relat. Struct. 17, 1–22 (2022)

    Article  MathSciNet  Google Scholar 

  2. Abellán García, F., Stern, W.H.: Theorem A for marked 2-categories. J. Pure Appl. Algebra 226(9), 107040 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abellán García, F., Stern, W.H.: Enhanced twisted arrow categories. arXiv:2009.11969

  4. Abellán García, F., Stern, W.H.: 2-Cartesian fibrations II: Higher cofinality. arXiv:2201.09589

  5. Buckley, M.: Fibred 2-categories and bicategories. J. Pure Appl. Algebra. 218(6), 1034–1074 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Adamek, J., Rosicky, J.: Locally presentable and accessible categories. London Mathematical Society Lecture Note Series 189. Cambridge University Press, Cambridge (1994)

  7. Gagna, A., Harpaz, Y., Lanari, E.: On the equivalence of all models for \((\infty,2)\)-categories. Math. Soc, Lond (2022). https://doi.org/10.1112/jlms.12614

    Book  MATH  Google Scholar 

  8. Gagna, A., Harpaz, Y., Lanari, E.: Gray tensor products and lax functors of \((\infty,2)\)-categories. Adv. Math. (2021). https://doi.org/10.1016/j.aim.2021.107986

    Article  MathSciNet  MATH  Google Scholar 

  9. Gagna, A., Harpaz, Y., Lanari, E.: Fibrations and lax limits of \((\infty ,2)\)-categories. arXiv:2012.04537

  10. Gagna, A., Harpaz, Y., Lanari, E.: Cartesian Fibrations of \((\infty ,2)\)-categories arXiv:2107.12356

  11. Lurie, J.: Higher Topos Theory. Princeton University Press, Princeton (2009). the author’s webpage

  12. Lurie, J.: Higher Algebra. (2017). the author’s webpage

  13. Lurie, J.: \((\infty ,2)\)-categories and the Goodwillie calculus. arXiv:0905.0462

  14. Nuiten, J.: On Straightening for Segal Spaces . arXiv:2108.11431

  15. Rasekh, N.: Yoneda Lemma for D-Simplicial Spaces. arXiv:2108.06168

  16. Verity, D.R.B.: Weak complicial sets I. Basic homotopy theory. Adv. Math. 219, 1081–1149 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to an anonymous referee for the very careful reading which helped improve the article. F.A.G. would like to acknowledge the support of the VolkswagenStiftung through the Lichtenberg Professorship Programme while he conducted this research. W.H.S wishes to acknowledge the support of the NSF Research Training Group at the University of Virginia (grant number DMS-1839968) during the preparation of this work.

Funding

The first author acknowledges the support of the VolkswagenStiftung through the Lichtenberg Professorship Programme. The second author acknowledges the support of the NSF Research Training Group at the University of Virginia (Grant Number DMS-1839968) during the preparation of this work.

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    Fernando Abellán García

  2. Department of Mathematics, University of Virginia, 141 Cabell Drive, Kerchof Hall Office 311, Charlottesville, VA, 22904, USA

    Walker H. Stern

Authors
  1. Fernando Abellán García
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Walker H. Stern
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Not applicable.

Corresponding author

Correspondence to Fernando Abellán García.

Ethics declarations

Conflicts of interest

Not applicable.

Code availability

Not applicable.

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

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abellán García, F., Stern, W.H. 2-Cartesian Fibrations I: A Model for \(\infty \)-Bicategories Fibred in \(\infty \)-Bicategories. Appl Categor Struct 30, 1341–1392 (2022). https://doi.org/10.1007/s10485-022-09693-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-022-09693-x

Keywords

Search

Navigation