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added some basics to model structure for quasi-categories at general properties
Cleaned up statement of Stevenson’s theorem about the generating acyclic cofibrations and added the arXiv reference to his paper. (I talked with Danny about this yesterday and he agrees that the hypothesis can be stated as the equivalent ’countably-many simplices’, rather than ’countably-many non-degenerate simplices’)
Richard, if you are reading this, the LaTeX-like syntax that Dmitri used isn’t working. I left it in so this could be verified and fixed, if indeed there is abug.
An element of B is a simplicial set, not a map of simplicial sets. What does it mean for a simplicial set to be an acyclic cofibration?
Gah, there’s a bijection $\mathcal{B}\to \{\{0\}\to B\mid B\in \mathcal{B}\}$, and I was implicitly casting along this.
Richard, if you are reading this, the LaTeX-like syntax that Dmitri used isn’t working. I left it in so this could be verified and fixed, if indeed there is a bug.
Thanks for the message. I asked Dmitri to leave this; it is not functionality that has been implemented yet, but I have been planning to add it very soon. But feel free to change it to the more conventional syntax in the meantime, I think I will remember that the reference was added here.
I removed the incorrect (as I explained here: https://chat.stackexchange.com/transcript/message/49892813#49892813) generating set of trivial cofibrations.
I also corrected wording in a few places and added references for Joyal’s original construction of the model structure.
Alexander Campbell
Is comment #8 correct, in that the Joyal model structure is simplicial with regard to another enrichment? Is the idea using $Map(X,Y)_n = Hom(\Delta'[n] \times X, Y)$, or do you need something more complicated?
I’m not sure what Denis had in mind, but intuitively one should be able to enrich over the Quillen model structure by replacing the internal homs of the Joyal model structure by their ’cores’ (i.e. throw away non-invertible 1-arrows).
Enriching $qCat$ like that (which I will call $Map$) does give a useful simplicially enriched category (which Lurie uses in his definition of $Cat_\infty$) but that enrichment doesn’t correspond in an immediately obvious way to a Quillen-simpicially enriched category structure on $sSet$.
(Lurie’s exposition implies that the reason for considering marked simplicial sets is precisely to realize that enrichment of $qCat$ as the fibrant-cofibrant subcategory of a simplicial model structure on $sSet^+$)
Ah, I see the kind of issue that can arise now. It is only reasonable for $Core(Y^X)$ itself to be appropriate when both $X$ and $Y$ are quasi-categories. Perhaps one can tweak things in the case that $X$ is not a quasi-category? Maybe $X$ should be replaced by a Kan complex, for example using the $Ex^\infty$ functor, which has good preservation properties? Of course one has to check that this even makes sense, i.e. gives an enrichment of the original Hom set.
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