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Actually, there is an error in the Cole paper you're talking about, as recently discovered by Richard Williamson and very recently corrected by Tobi Barthel and Emily Riehl in "On the Construction of …

No, definitely not! They are much more complicated to characterize. You need to take horns on the set of morphisms you just wrote down. This is all detailed carefully in Hirschhorn's book, summarized …

This is Proposition 2.7 in Rezk's Every Homotopy Theory of Simplicial Algebras Admits a Proper Model

You asked if you can check it for less than the collection of all weak equivalences. In A.5 of Motivic Symmetric Spectra, Jardine proves a general right properness result from Corollary A.4, which is …

If I understand your situation correctly, the answer is yes, the cube lemma holds, but you cannot use that to get a full model structure. Your situation often arises when trying to transfer a model st …

I am glad that by putting our heads together in Munster this week we finally have a proof and can answer this question (a full 2 years after it was asked!). The answer is yes, and I want to sketch her …

There are a ton of papers about what you are asking. Another is the thesis of Richard Williamson (arXiv:1304.0867v1). Also, Valery Isaev has a paper that produces a model structure with all objects fi …

The model structure you are looking for is developed in a paper by Christensen and Hovey called Quillen model structures for relative homological algebra. They go into the factorization systems in gre …

Yes. According to the nLab page for the Strom model structure, this is proven in Section 6.4 of May's Concise Course in Algebraic Topology. The point is that a closed inclusion is a cofibration if and …

They have certainly not all been resolved one way or the other. As the other thread points out, Tyler Lawson and others were working on this a few years ago. During my last years of grad school, I sta …

Yes, and this is one of the main results in a paper I hope to put on arxiv very soon. I wrote about this result in a previous mathoverflow answer here. You are right that the way to do it is to focus …

There are many references where model categories, and their connection to homology, are described more. See this MO question for a list. For the example of $Ch(R)$, there are several model structures. …

I think the answer is yes (to both questions). In Emily Riehl's book "Categorical homotopy theory", chapter 13 is all about the enriched small object argument. Theorem 13.2.1 on page 177 (I hope I'm l …

Very recently a paper appeared on the arxiv by Barthel-May-Riehl which addresses this question in a very complete way. It discusses the three model structures on DG-algebras (answering the OPs questio …

EDIT: Now that the OP has edited his question to make clearer what he wants as an answer, I'm removing speculation about what he wanted. The answer is: yes, you can characterize homotopy equivalences …