2-Cartesian fibrations I: A model for $\infty$-bicategories fibred in $\infty$-bicategories
| Autor: | García, Fernando Abellán, Stern, Walker H. |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | In this paper, we provide a notion of $\infty$-bicategories fibred in $\infty$-bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled simplicial sets equipped with an additional collection of triangles containing the scaled 2-simplices, which we call lean triangles, in addition to a collection of edges containing all degenerate 1-simplices. We prove the existence of a left proper combinatorial simplicial model category whose fibrant objects are precisely the 2-Cartesian fibrations over a chosen scaled simplicial set $S$. Over the terminal scaled simplicial set, this provides a new model structure modeling $\infty$-bicategories, which we show is Quillen equivalent to Lurie's scaled simplicial set model. We conclude by providing a characterization of 2-Cartesian fibrations over an $\infty$-bicategory. This characterization then allows us to identify those 2-Cartesian fibrations arising as the coherent nerve of a fibration of $\operatorname{Set}^+_{\Delta}$-enriched categories, thus showing that our definition recovers the preexisting notions of fibred 2-categories. Comment: Comments welcome! |
| Databáze: | arXiv |
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