Enhanced twisted arrow categories. (English) Zbl 1520.18018
This paper aims to provide a description of the \(\infty\)-category of natural transformations \(\mathrm{Nat}\left( F,G\right) \)as an end for any functors \(F\)and \(G\)from an \(\infty\)-category to an \(\infty\)-bicategory.
The synopsis of the paper goes as follows.
The synopsis of the paper goes as follows.
- § 1
- is concerned with preliminaries.
- § 2
- gives the formal definition of Tw\(\left( \mathbb{C} \right) \), establishing that \[ \mathrm{Tw}\left( \mathbb{C}\right) \rightarrow\mathbb{C}\times\mathbb{C} \] is a Cartesian fibration.
- § 3
- proves that this Cartesian fibration classifies precisely the enhanced mapping functor \[ \mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathbb{C}^{\mathrm{op} }\times\mathbb{C}\rightarrow\mathfrak{C}at_{\infty} \] where \(\mathfrak{C}at_{\infty}\)is the \(\left( \infty,2\right) \)-category of \(\infty\)-categories. The proof is highly technical, freely uses results from [J. Lurie, “\((\infty,2)\)-categories and the Goodwillie calculus I”, Preprint, arXiv:0905.0462] and [A. Gagna et al., J. Lond. Math. Soc., II. Ser. 106, No. 3, 1920–1982 (2022; Zbl 1518.18020)].
- § 4
- turns to the true aim of the paper, establishing that, given two functors \[ F,G:\mathcal{C}\rightarrow\mathbb{D} \] from an \(\infty\)-category to an \(\left( \infty,2\right) \)-category, the \(\infty\)-category of natural transformations between them can be expressed as an end \[ \mathrm{Nat}\left( F,G\right) \simeq\mathrm{lim}_{\mathrm{Tw}\left( \mathcal{C}\right) ^{\mathrm{op}}}\mathrm{Map}_{\mathbb{D}}\left( F\left( -\right) ,G\left( -\right) \right) \] The proof is highly technical, making use of a wide variety of techniques native to the contexts of scaled simplicial sets and marked simplicial sets. This result allows of analyzing in greater detail the theory of weighted colimits of \(\mathfrak{C}at_{\infty}\)-valued functors exposed in [D. Gepner et al., Doc. Math. 22, 1225–1266 (2017; Zbl 1390.18021)], showing that this definition coincides with the definition provided by the first author [J. Homotopy Relat. Struct. 17, No. 1, 1–22 (2022; Zbl 1505.18030)].
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18N10 | 2-categories, bicategories, double categories |
| 18N60 | \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories |
| 18N65 | \((\infty, n)\)-categories and \((\infty,\infty)\)-categories |
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| [22] | Tim Van der Linden, Université catholique de Louvain: tim.vanderlinden@uclouvain.be |
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