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Model structure on marked simplicial spaces compatible with Joyal and Cartesian structures

Construct a model structure on the category s^+ of marked simplicial spaces such that: (i) it is Quillen equivalent to the Joyal model structure on simplicial sets; (ii) the inclusion functor i^+_1: S^+ -> s^+ is left Quillen when S^+ carries the Cartesian model structure; and (iii) the inclusion functor i_1: S -> s^+ is left Quillen when S carries the Joyal model structure. This construction would enable a quasi-categorically enriched lift of the unstraightening construction for Cartesian fibrations.

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Background

The paper develops cosmological biequivalences implementing unstraightening for right and Cartesian fibrations across a range of ∞-cosmoi by working simplicially (Kan-enriched). To obtain an enrichment over quasi-categories (rather than just over Kan complexes) for the marked unstraightening, the authors explain that one would need a suitable model structure on marked simplicial spaces.

Specifically, they seek a model structure on s+ that is Quillen equivalent to the Joyal model structure and that makes the standard inclusions from marked simplicial sets (S+) and simplicial sets (S) into marked simplicial spaces (s+) left Quillen for the relevant model structures. The commonly used model structure on marked simplicial spaces nearly satisfies these conditions, but currently the inclusion i_1: S -> s+ is left Quillen from the Kan (not Joyal) model structure, leaving this construction out of reach.

References

Indeed, what we would need is a proof for the following conjecture. There exists a model structure on $s+$ with the following specifications: It is Quillen equivalent to the Joyal model structure. The inclusion $i+_1\colon S+ \to s+$ is left Quillen from the Cartesian model structure. The inclusion $i_1\colon S \to s+$ is left Quillen from the Joyal model structure.

— Cosmological Unstraightening (2505.16342 - Rasekh, 22 May 2025) in Conjecture, Section 4 (Challenges Towards A Quasi-categorically enriched cosmological Unstraightening)