2023/06/19 by Moser, Lyne, Sarazola, Maru
#18A25 #18D30 #18N40 #18N60 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2306.11076
We construct two model structures, whose fibrant objects capture the notions of discrete fibrations and of Grothendieck fibrations over a category C. For the discrete case, we build a model structure on the slice Cat/C, Quillen equivalent to the projective model structure on [Cop,Set] via the classical category of elements construction. The cartesian case requires the use of markings, and we define a model structure on the slice Cat+/C, Quillen equivalent to the projective model structure on [Cop,Cat] via a marked version of the Grothendieck construction. We further show that both of these model structures have the expected interactions with their ∞-counterparts; namely, with the contravariant model structure on sSet/ NC and with Lurie's cartesian model structure on sSet+/ NC.