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Initial version:
Thomason-type model categories provide simple 1-categorical models for (∞,1)-categorical objects.
The provide a particularly convenient setting for results like Quillen’s Theorem A and Theorem B.
|(∞,1)-categorical structure|1-categorical structure|model structure| |∞-groupoid|category|Thomason model structure| |∞-groupoid|poset|model structure on posets| |(∞,1)-category|relative category|Barwick–Kan model structure| |connective spectra|symmetric monoidal groupoid|Fuentes-Keuthan model structure|
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For the example of the model structure on RelCat… I think the “Barwick-Kan” model structure better applies for the model structure that presents simplicial spaces, since that’s the one they lay out in “RELATIVE CATEGORIES: ANOTHER MODEL FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES”
As for the model structure presenting (∞,1)-categories, Barwick-Kan, in “IN THE CATEGORY OF RELATIVE CATEGORIES THE REZK EQUIVALENCES ARE EXACTLY THE DK-EQUIVALENCES”, call them Rezk equivalences since they are transferred from Rezk’s model structure for complete Segal spaces. (they also define Dwyer-Kan equivalences and prove they are the same thing)
So, I’ve been calling that the “Rezk” model structure on relative categories. Maybe “Barwick-Kan-Rezk” would be better.
@Hurkyl: No, the correct notion of a weak equivalence between relative categories was identified by Dwyer and Kan way back in the 1980s: it is a relative functor that induces a weak equivalence of simplicial categories on its hammock localization. All the other notions, e.g., the one induced from Rezk’s model structure, are equivalent to this one, and were introduced much later.
But it was Barwick and Kan that promoted this notion of a weak equivalence to an actual model structure. So it can be rightfully called the Barwick–Kan model structure.
Fine, Barwick-Dwyer-Kan. Or Barwick-Dwyar-Kan-Rezk. Or whatever. The point is to distinguish the model structure Barwick-Kan showed existed and fleshed out its properties (and which presents a useful infinity category) from its localization which presents infinity categories.
Having the specific label Barwick-Kan apply to that model structure rather than its localization seems by far the most natural meaning. shrug
In their paper “Relative categories: Another model for the homotopy theory of homotopy theories”, Barwick and Kan not only showed that the Reedy model structure on simplicial spaces transfers to RelCat, they also proved that any left Bousfield localization of the Reedy model structure transfers to RelCat.
However, I have never seen any applications of the transferred Reedy model structure on RelCat, whereas the localized model structure has tons of applications.
I am not sure why we need a designated name for a model structure that is never used outside of Barwick and Kan’s paper.
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