There are a number of model categories important in higher category theory, which provide a "presentation" of some $\infty$-category of $\infty$-categories. For example:
- The Joyal model structure on simplicial sets, which presents the $\infty $ (i.e. $(\infty, 1)$)-category of quasi-categories (the fibrant objects are the quasi-categories, the weak equivalences are the categorical equivalences, and the cofibrations are the injections).
- The covariant (or, dually, contravariant) model structure on simplicial sets over a base simplicial set $S$. This presents the $\infty$-category of quasicategories "cofibered in groupoids" over $S$, where "groupoids" really means Kan complexes and "cofibered" means "left fibration." The fibrant objects are the left fibrations over $S$.
- The coCartesian (or, dually, Cartesian) model structure on marked simplicial sets over a base simplicial set $S$; this presents the $\infty$-category of quasicategories "cofibered" over $S$ (in quasicategories, not necessarily in Kan complexes). The fibrant objects are precisely the coCartesian fibrations to $S$ with the coCartesian edges (and no others) marked.
All these model structures are efficiently constructed in HTT using a general machine, which starts with generating cofibrations and weak equivalences and builds a cofibrantly generated (even combinatorial) model structure, essentially by using a cardinality argument to produce the generating acyclic cofibrations (i.e., take all acyclic cofibrations between objects of bounded cardinality) and the small object argument to get the factorizations. Unfortunately, this machine doesn't provide much of a handle on what the generating acyclic cofibrations actually are, or equivalently what the fibrations look like.
I'm curious if an explicit description of the generating acyclic cofibrations in any of these cases is known. (According to the nLab, this is or at least was an open problem in the first case.) What I do know is that, in all three of the above situations, there is a very concrete and easily describable class of "anodyne" maps such that the fibrant objects are precisely those with the right lifting property with respect to these, and such that the fibrations between fibrant objects can be so described as well. (In case 1, the inner anodyne maps together with the inclusion $\ast \to N(J)$ for $J$ a contractible groupoid on two objects works; in case 2, the left anodyne maps in case 3, the marked anodyne maps -- actually, the opposite of the maps Lurie calls by that name.)
It'd be very nice if these were in fact generating acyclic cofibrations. At least in cases 2 and 3, that doesn't seem to be true, because the model structures are simplicial: given any object $X$ (over $S$), the map $\Delta^0 \times X \to \Delta^1 \times X$ is a trivial cofibration and if the inclusion $\Delta^0 \to \Delta^1$ is the inclusion of the second vertex, we don't have an anodyne map in this case. On the other hand, I don't see any obvious reason why taking the previous "anodyne" maps together with all $A \times X \to B \times X$ for $A \to B$ a trivial cofibration of simplicial sets wouldn't be sufficient. Are they in fact sufficient?
