Theorem A for marked 2-categories. (English) Zbl 1495.18026
Summary: In this work, we prove a generalization of Quillen’s Theorem A to 2-categories equipped with a special set of morphisms which we think of as weak equivalences, providing sufficient conditions for a 2-functor to induce an equivalence on \((\infty, 1)\)-localizations. When restricted to 1-categories with all morphisms marked, our theorem retrieves the classical Theorem A of Quillen. We additionally state and provide evidence for a new conjecture: the cofinality conjecture, which describes the relation between a conjectural theory of marked \((\infty, 2)\)-colimits and our generalization of Theorem A.
MSC:
| 18N10 | 2-categories, bicategories, double categories |
| 18N55 | Localizations (e.g., simplicial localization, Bousfield localization) |
| 18N60 | \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories |
| 18N65 | \((\infty, n)\)-categories and \((\infty,\infty)\)-categories |
| 18N50 | Simplicial sets, simplicial objects |
| 18A30 | Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) |
Keywords:
Quillen’s theorem A; infinity category; 2-category; localization; cofinality; marked colimitReferences:
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