The Bergner model structure on $sCat$ (simplicially enriched categories) has a "projective" flavor: fibrations are "levelwise" while cofibrations satisfy stringent relative "freeness" conditions.

This seems to be something inherent to modeling $\infty$-categories using strictly-associative composition. Similarly, Barwick and Kan's model structure on relative categories and Horel's model structure on simplicially-internal categories are induced projectively, and not very many objects are cofibrant.

Is this really a trade-off we're stuck with?

**Question 1:** Is there a model structure $M$ on $sCat$ other than the Bergner model structure such that the identity functor is a left Quillen equivalence $sCat_M \to sCat_{Bergner}$? More generally, is there a model $M$ of the homotopy theory of $\infty$-categories whose underlying category is $sCat$ which doesn't have fewer cofibrant objects than $sCat_{Bergner}$?

**Question 2:** Is there such an $M$ with a reasonable supply of cofibrant objects? Say, such that every ordinary category is cofibrant? Or perhaps at least such that every ordinary Reedy category is cofibrant?

**Question 3:** As above, but for relative categories or simplicially-internal categories?

Two data points:

If $C$ is a category and $M$ is a model category, then $M^C$ with levelwise weak equivalences has the correct homotopy theory. But from thinking about the Bergner model structure, you might think you'd have to consider the category of simplicial functors $C' \to M'$ where $C'$ is the standard resolution of $C$ and $M'$ is the simplicial localization of $M$. The fact that you don't have to do this hints at some alternate model category -type structure where $C$ is already cofibrant and $M$ is fibrant -- maybe this would be a model structure on relative categories.

A similar phenomenon happens with operads, if I recall correctly (which are also

*strict*monoids for the Kelly tensor product on symmetric sequences) -- the usual model structure (Berger-Moerdijk, I believe) is much more stringent in its cofibrancy requirements than what is needed in practice.As hinted at in the comments, it's a good idea to ask whether the existing "projective" model structures $M$ are left proper. For

Left properness of a model structure depends only on the weak equivalences (saying that the co-base change Quillen adjunctions between co-slice categories are Quillen equivalences).

A model structure with all objects cofibrant is left proper.

Well, it turns out that the Bergner, Lack, Barwick-Kan, and Horel model structures are all left proper! (I'm not sure about the Berger-Moerdijk model structure.) So this leaves open the possibility of model structures with the same weak equivalences and all objects cofibrant.

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