Yoneda lemma for simplicial spaces. (English) Zbl 1516.18018
This paper studies the Yoneda lemma for arbitrary simplicial spaces by introducing left fibrations of simplicial spaces and studying their associated model structure. P. Boavida de Brito [Contemp. Math. 708, 19–44 (2018; Zbl 1405.55023)] developed left fibrations of Segal spaces. The author has written several papers concerned wtih Cartesian fibrations of complete Segal spaces [N. Rasekh, J. Homotopy Relat. Struct. 16, No. 4, 563–604 (2021; Zbl 1492.18022); High. Struct. 7, No. 1, 40–73 (2023); N. Rasekh, Homology Homotopy Appl. 24, No. 2, 135–161 (2022; Zbl 1512.18006)]. The key ideas of this paper have also been generalized in [N. Rasekh, “Yoneda Lemma for \(\mathcal{D}\)-simplicial spaces”, Preprint, arXiv:2108.06168] to the setting of \((\infty,n)\)-categories in the particular model of \(n\)-fold complete Segal spaces. Independently J. Nuiten [“On straightening for Segal spaces”, Preprint, arXiv:2108.11431] has also studied fibrations of \(n\)-fold complete Segal spaces.
The synopsis of the paper goes as follows.
The synopsis of the paper goes as follows.
- § 1
- reviews the classical Yoneda lemma (§1.1), the Grothendieck construction (§1.2) and the fibrational Yoneda lemma for categories (§1.3).
- § 2
- reviews necessary background concepts such as Joyal-Tierney calculus (§2.1), spaces (§2.2), simplicial spaces (§2.3), the Reedy model structure (§2.4) and complete Segal spaces (§2.6).
- § 3
- begins the study of left fibrations. §3.1 introduces left fibrations, giving various alternative characterizations. §3.2 defines a model structure for left fibrations, the covariant model structure, over arbitrary simplicial spaces (Theorem 3.12). §3.3 studies left fibrations over Segal spaces, particularly establishing the Yoneda lemma for Segal spaces (Theorem 3.49).
- § 4
- consists of two subsections. §4.1 focuses on the covariant model structure over nerves of categories, particularly establishing the Grothendieck construction (Theorem 4.18). §4.2 establishes the recognition principle for covariant equivalences (Theorem 4.41).
- § 5
- studies the relation between left fibrations and complete Segal spaces. §5.1 proves the invariance of the covariant model structure (Theorem 5.1) with several significant implications. §5.2 applies these results to the study of colimits in Segal spaces.
- Appendix A
- reviews some key lemmas about model categories.
- Appendix B
- establishes the Quillen equivalence between the covariant model structure for simplicial spaces and that for simplicial sets studied in [J. Lurie, Higher topos theory. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001)].
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18N60 | \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories |
| 18N40 | Homotopical algebra, Quillen model categories, derivators |
| 18N50 | Simplicial sets, simplicial objects |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
Keywords:
higher category theory; simplicial spaces; complete Segal spaces; left fibrations; Yoneda lemmaReferences:
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