Models for rational -categories
The study of -categories is intended to understand categories with a notion of homotopy between their morphisms. Namely, an -category is a category enriched in spaces, possibly weakly. This enrichment yields -morphisms for all given by homotopies. However, all morphisms in dimension higher than are invertible up to homotopy because homotopies can be traveled in reverse.
Our understanding of -categories has been advanced thanks to the development of various models for -categories, that is, mathematical objects that exhibit the structure of an -category. Two such models are complete Segal spaces, as introduced by Rezk in [7], and Segal categories, as developed from the homotopical perspective by Bergner in [1].
In [3], we develop an analog of Bergner and Rezk’s work for rational homotopy theory. We introduce rational -categories, which are -categories enriched in spaces whose higher homotopy groups are rational vector spaces. Then, we produce two models for rational -categories, rational complete Segal spaces and rational Segal categories.
Our argument is not exclusive to rational homotopy theory; it works for enrichment in general localizations of spaces. For example, in the chromatic homotopy-theoretic framework developed by Heuts in [4], our work yields two equivalent models for -categories enriched in -periodic spaces for a non-negative integer , with the case being that of rational -categories.
Lastly, we present the future directions of our project on developing rational homotopy-theoretic analogs of other models for -categories. Such models are quasi-categories, as investigated by Joyal in [5] and Lurie in [6], as well as simplicial categories, as studied by Bergner in [2]. We discuss our ongoing work on giving localized analogs of these two models for -categories.
- [1] J. E. Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007), no. 4, 397–436.
- [2] J. E. Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2043–2058.
- [3] E. Chatzitheodoridis, Models for rational -categories, to appear in Homology Homotopy Appl., preprint available at arXiv:2509.22413, 2025.
- [4] G. Heuts, Lie algebras and -periodic spaces, Ann. of Math. 193 (2021), no. 1, 223–301.
- [5] A. Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra. 175 (2002), no. 1, 207–222.
- [6] J. Lurie, Higher Topos Theory, Princeton University Press, 2009.
- [7] C. Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), no. 3, 973–1007.