Abstract
We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]),On bisimplicial spaces (due to deBrito [12]),On bisimplicial sets,On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal–Tierney and the straightening construction due to Lurie.
Highlights
0.1 Functors via fibrationsFor a given category C, we are often interested in studying pseudo-functors of the form P : Cop → CatCommunicated by Emily Riehl
[36] In particular, the first paper includes a detailed analysis of the Cartesian model structure on bisimplicial spaces, which we only review here (Sect. 1.6)
There exists a new left proper combinatorial simplicial model structure CM on the category C with the following properties: – A morphism f in CM is a cofibration if and only if it is a cofibration in CM . – A morphism f in CM is a weak equivalence if and only if F f is a weak equivalence in DN . – A morphism in f in CM is a fibration if and only if it has the right lifting property with respect to trivial cofibrations
Summary
For a given category C, we are often interested in studying pseudo-functors of the form. Unlike Cartesian fibrations we can endow the category of simplicial sets over a fixed simplicial set with a model structure, the contravariant model structure, such that the fibrant objects are precisely the right fibrations It was first proven by Lurie [31] (and later many other authors [23,24,44]), that this model structure is Quillen equivalent to presheaves valued in spaces. In the complete Segal object approach to Cartesian fibrations every proof of the equivalence between right fibrations and space valued presheaves immediately generalizes to a new proof of the Grothendieck fibration for Cartesian fibrations. Quasi-categories vs segal spaces: Cartesian edition construct a direct equivalence between marked simplicial objects and bisimplicial objects that induces an equivalence between the associated Cartesian model structures. In order to help the reader, here is a list of all relevant model structures (except the four Cartesian model structures already mentioned), the underlying category, along with the abbreviation and a reference to their definition: Model structure
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