Two-variable fibrations, factorisation systems and \(\infty\)-categories of spans. (English) Zbl 1529.18019
This paper establishes a universal property for \(\infty\)-categories of spans in the generality of Barwick’s adequate triples, explicitly describing the cocartesian fibration corresponding to the span functor and showing that the latter restricts to a self-equivalence on the class of orthogonal adequate triples.
The synopsis of the paper goes as follows.
The synopsis of the paper goes as follows.
- § 2
- reviews the theory of adequate triples and their associated span \(\infty\)-categories [C. Barwick, Adv. Math. 304, 646–727 (2017; Zbl 1348.18020); C. Barwick et al., Tunis. J. Math. 2, No. 1, 97–146 (2020; Zbl 1461.18009)] under the name effective Burnside \(\infty\)-categories. An alternative viewpoint on the material is presented by translating the assertions along the equivalence between \(\infty \)-categories and complete Segal \(\infty\)-groupoids.
- § 3
- aims to describe the cocartesian fibration associated to the functor \[ \mathrm{Span}:\mathrm{AdTrip}\rightarrow\mathrm{Cat} \] which is itself given by a certain span \(\infty\)-category of adequate triples. As a direct application, the authors obtain a new and fairly direct proof of the main result of [C. Barwick et al., Theory Appl. Categ. 33, 67–94 (2018; Zbl 1423.18025)], which identifies the dual cartesian fibration of a cocartesian fibration in terms of a span construction.
- § 4
- describes a different perspective on the equivalence between cartesian fibrations and cocartesian fibrations of \(\infty\)-categories established in [E. Lanari, High. Struct. 5, No. 1, 1–17 (2021; Zbl 1481.18031)], establishing Theorem 4.12.
Theorem. Taking span \(\infty\)-categories gives rise to a \(C_{2}\)-action on the full subcategory of orthogonal adequate triples \[ \mathrm{Span}^{\bot}:\mathrm{AdTrip}^{\bot}\rightarrow\mathrm{AdTrip}^{\bot} \] - § 5
- aims to analyze the interaction between orthogonal adequate triples and cartesian fibrations, particularly showing that orthogonal adequate triples \(\left( X,X_{\mathrm{in}},X^{\mathrm{eg}}\right) \)uniquely correspond to cartesian fibrations with contractible fibers by taking \(X\)to \[ X\rightarrow X\left[ \left( X^{\mathrm{eg}}\right) ^{-1}\right] \] generalizing results of E. Lanari [High. Struct. 5, No. 1, 1–17 (2021; Zbl 1481.18031)]. The authors study various kinds of fibrations between adequate triples that are preserved by the dualization equivalence from Theorem 4.12.
- § 6
- purposes to use the results of the previous section to dualize and straighten various kinds of fibrations over a product of \(\infty\)-categories. The various fibrations defined over \(\left( A,B\right) ^{\bot}\)in the previous section recover a subset of the fibrations considered in [R. Haugseng et al., Proc. Lond. Math. Soc. (3) 127, No. 4, 889–957 (2023; Zbl 1528.18022)]. In particular, the authors extend the explicit description of dual (co)cartesian fibrations to the situation of curved, ortho- and Gray fibrations. As an application, an explicit description of parametrized adjoints from [R. Haugseng et al., Proc. Lond. Math. Soc. (3) 127, No. 4, 889–957 (2023; Zbl 1528.18022)], extending previous work of T. Torii [Springer Proc. Math. Stat. 309, 325–380 (2020; Zbl 1462.55003)], is obtained.
- § 7
- uses the dualities between various types of two-variable fibrations considered in the previous section to identify oplax arrows and twisted arrow \(\infty\)-categories as duals.
- § 8
- deduces the following theorem.
Theorem. The Yoneda embedding \[ A\rightarrow\mathcal{P}\left( A\right) \] canonically extends to a natural transformation of functors \[ \mathrm{Cat}\rightarrow\mathrm{CAT} \] from the inclusion to the composite \[ \mathrm{Cat}\overset{\mathrm{Fun}\left( -^{\mathrm{op}},\mathrm{Grp}\right) }{\rightarrow}\left( \mathrm{CAT}^{\mathrm{R}}\right) ^{\mathrm{op}} \simeq\mathrm{CAT}^{\bot}\subseteq\mathrm{CAT} \] - Appendix A
- establishes a rigidity result implying that any two ways of straightening or dualizing a two-variable fibration are naturally equivalent.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18N70 | \(\infty\)-operads and higher algebra |
| 18N60 | \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories |
| 18N65 | \((\infty, n)\)-categories and \((\infty,\infty)\)-categories |
Citations:
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