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PreprintCC BY-NC-SA 4.0arXiv

A model structure for cartesian 2-fibrations

  • Cesar Bardomiano Martinez
  • Jana K. Nickel
  • Maru Sarazola
  • Daniel Teixeira
  • Santiago Toro Oquendo
  • Paula Verdugo
Open at sourcePDFDOICite

Abstract

Cartesian 2-fibrations provide a way to understand indexed categories, but their classical ``straightening'' construction requires several layers of weak coherence data. This paper develops a homotopical framework that replaces much of this bookkeeping with a fully strict model. By using marked 2-categories to record the cartesian morphisms and 2-cells, we construct a model structure whose fibrant objects are precisely the cartesian 2-fibrations over a fixed 2-category π’ž. We then show that the marked Grothendieck construction identifies these 2-fibrations, up to weak equivalence, with strict 2-functors from π’ž into 2Cat. As an additional contribution, we construct localizations of 2-categories that simultaneously invert selected morphisms and 2-cells.

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Cite

BibTeXRIS
BibTeX
@misc{martinez2026model,
  title = {{A model structure for cartesian 2-fibrations}},
  author = {Cesar Bardomiano Martinez and Jana K. Nickel and Maru Sarazola and Daniel Teixeira and Santiago Toro Oquendo and Paula Verdugo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11759},
  eprint = {2609.11759},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11759},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - A model structure for cartesian 2-fibrations
AU  - Cesar Bardomiano Martinez
AU  - Jana K. Nickel
AU  - Maru Sarazola
AU  - Daniel Teixeira
AU  - Santiago Toro Oquendo
AU  - Paula Verdugo
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11759
UR  - https://arxiv.org/abs/2609.11759
AB  - Cartesian 2-fibrations provide a way to understand indexed categories, but their classical ``straightening'' construction requires several layers of weak coherence data. This paper develops a homotopical framework that replaces much of this bookkeeping with a fully strict model. By using marked 2-categories to record the cartesian morphisms and 2-cells, we construct a model structure whose fibrant objects are precisely the cartesian 2-fibrations over a fixed 2-category π’ž. We then show that the marked Grothendieck construction identifies these 2-fibrations, up to weak equivalence, with strict 2-functors from π’ž into 2Cat. As an additional contribution, we construct localizations of 2-categories that simultaneously invert selected morphisms and 2-cells.
ER  -