Added:

André Joyal on the history of the Joyal model structure (also on MathOverflow):

]]>I became interested in quasi-categories (without the name) around 1980 after attending a talk by Jon Beck on the work of Boardman and Vogt. I wondered if category theory could be extended to quasi-categories. In my mind, a crucial test was to show that a quasi-category is a Kan complex if its homotopy category is a groupoid. All my attempts at showing this have failed for about 15 years, until I stopped trying hard! I found a proof after extending to quasi-categories a few basic notions of category theory. This was around 1995. The model structure for quasi-categories was discovered soon after. I did not publish it immediately because I wanted to show that it could be used for proving something new in homotopy theory. I am a bit of a perfectionist (and overly ambitious?). I was hoping to develop a synthesis between category theory and homotopy theory (hence the name quasi-categories). I met Lurie at a conference organised by Carlos Simpson in Nice (in 2001?). I gave a talk on the model structure and Lurie asked for a copy of my notes afterward.

Added:

There are analogues of the Joyal model structure for cubical sets (with or without connection):

]]>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.

]]>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 $sSet^+$)

]]>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).

]]>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 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

]]>The Joyal model structure has, in fact, a simplicial enrichment compatible with the model structure (although of course not the canonical one). I’ll add the details later but I wanted to remove the wrong statement

Denis Nardin

]]>Fixed the issue that Dmitri pointed out.

]]>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.

]]>Gah, there’s a bijection $\mathcal{B}\to \{\{0\}\to B\mid B\in \mathcal{B}\}$, and I was implicitly casting along this.

]]>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?

]]>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.

]]>Added an explicit description of generating acyclic cofibrations for the Joyal model structure.

]]>added some basics to *model structure for quasi-categories* at *general properties*