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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 …

I don't believe it is possible to recover the "correct" notion of "functor $\mathcal{C}\to \rm Top$", as described at the other question you linked to, by viewing $\rm Top$ only as a topologically enr …

Section 8 of my paper All (∞,1)-toposes have strict univalent universes shows that under fairly general conditions, injective fibrant replacements can be given by cobar constructions (e.g. the dual of …

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 …

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 …

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. …

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 …

I think the paper I was thinking of when I asked this question was probably Markus Spitzweck, Homotopy limits of model categories over inverse index categories, JPAA 214:6 (2010) although he only …

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 …

This is probably not the sort of answer that the original question was going for, since you said you were happy to assume cofibrant generation but not combinatoriality (i.e. not local presentability), …

The general case is noted in A.3.3.4 in Higher topos theory. Actually it's quite easy to prove, using the characterization of simplicial model categories in terms of powers (cotensors), since the pow …

I think it shouldn't be too hard to show using Dugger's technology that $N$ is full, i.e. essentially surjective on 1-morphisms. Suppose $C,D$ are combinatorial model categories and $f : N C \to N D$ …

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 …

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 …