We use fibrations of complete Segal spaces as introduced in [Ras22, Ras23a] to construct four complete Segal spaces: Reedy fibrant simplicial spaces, Segal spaces, complete Segal spaces, and spaces. Moreover, we show each one comes with a universal fibration that classifies Reedy left fibrations, Segal coCartesian fibrations, coCartesian fibrations and left fibrations and prove these are representable fibrations in the sense of [Ras22]. Finally, we use equivalences between quasi-categories and complete Segal spaces constructed in [JT07, Ras21a] to present analogous constructions using fibrations of quasi-categories.
As part of establishing the results, we also develop a theory of minimal Reedy fibrations for elegant Reedy categories, which can be of independent interest.
Keywords: higher category theory, ∞-categories, complete Segal spaces, left fibrations, minimal fibrations
2020 MSC: 18N60, 18N50, 55U35, 18N45, 18N40
Theory and Applications of Categories, Vol. 41, 2024, No. 2, pp 21-70.
Published 2024-01-17.
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