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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 …

Consider sSet×sSet with the induced structure from sSet. The monoidal unit is (1,1) (which is trivially fibrant, as is the unit in any cartesian monoidal model category), and so an object of the form …

Charles and Jonathan have given good answers to (1), but here's another way to recover the weak equivalences from the cofibrations and fibrations, which I think is sometimes convenient. First of all, …

The $(\infty,1)$-category of presheaves on any small $(\infty,1)$-category $C$ is presented by the model structure of simplicial presheaves on any simplicial category which incarnates $C$. So if $M$ …

Whether or not there are nontrivial examples, I would expect such a notion to be less useful than the classical one, because one of the points of the classical definition is to be able to use strict 1 …

If I'm not mistaken, here is an "even worse" counterexample than Charles'. Let $R$ be the walking isomorphism $(0\cong 1)$, let $R_+$ be $(0\to 1)$, and let $R_-$ be $(1\to 0)$. The conditions on de …

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 …

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 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 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 …

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 …

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 …

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 …

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 …

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 …