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Mark's answer explains why (1) cofibrations and fibrations do determine the model structure, and Charles' and Tom's examples show that (4) cofibrant objects and fibrant objects do not. For (2), any w …

Just to be a little bit contrary, let me point out one concrete reason that locally presentable categories are not aesthetic: it is unclear whether they have any analogue in constructive mathematics. …

Yes. The theory of categories is defined by a finite limit sketch, i.e. an essentially algebraic theory, and the category of models of any finite limit sketch is locally finitely presentable. See, f …

I don't think the existence of the dual "injective" model structure merits an "of course," since its generators are much less obvious to construct. However, it turns out that injective model structur …

As Denis-Charles said in a comment, if you require the unique lifting property on a weak factorization system, it becomes a "unique" or "orthogonal" factorization system. In the paper Bousfield local …

I don't know whether set-theoretic hypotheses are necessary to answer this question. But if we assume the negation of Vopěnka's principle, here is an example: by Example 6.12 of Adamek-Rosicky Locall …

There is, of course, at least one model structure on any category in which every object is both fibrant and cofibrant, but it's not that interesting: the weak equivalences are just the isomorphisms an …

You might be interested in looking into enriched model categories. If $\mathcal{V}$ is a monoidal model category (with cofibrant unit, for simplicity) and $\mathcal{C}$ is a $\mathcal{V}$-model categ …

Addressing this sort of question was one of the main goals of math/0610194. The best answer I was able to give is that if $B$ has a suitable model structure with respect to which $F$ is objectwise fi …

Indeed, COSHEP (more traditionally called the "presentation axiom" by constructivists) does seem to be what you need in order to get a model structure on Set, or Cat. That's true in a lot of similar …

No, it is not. If what you mean by $\rm colim_n$ is the actual colimit of $F$ as a diagram of shape $\Delta$, then this colimit is isomorphic to the coequalizer of the two maps $F([1]) \rightrightarr …

The weak equivalences in the Reedy model structure are levelwise, and in a simplicial set if there is a simplex of any dimension then there are simplices of all dimensions. Thus, the weak equivalence …

In the canonical model structure on $\omega\mathrm{Cat}$, the weak equivalences are not defined as preimages of isomorphisms under any functor, and are not even even closely related to any such preima …

It seems very unlikely to me that you will be able to get any useful handle on the generating acyclic cofibrations in the injective model structure, even in simple cases like when $I$ is the delooping …

The canonical model structure on 2-Cat is left proper. This is proven in Steve Lack's original paper A Quillen model structure for 2-categories that constructs this model category. The proof involve …