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No, this is not true. A counterexample is provided in Hirschhorn's book, Example 2.1.6 on page 36. See also the text on page 71: "Unfortunately, Example 2.1.6 shows that not all S-local trivial cofibr …

There is no obstruction. If $M$ is a simplicial monoidal model category, and $O$ is an operad in $M$, then the category of $O$-algebras is simplicially enriched, tensored, and cotensored. If it's a mo …

Welcome to MathOverflow! First, let me point out that what you're asking is already true at the 1-categorical level. The pushout in the category of commutative rings is computed by the tensor product. …

It's generally best not to leave questions without an answer, even if they are answered in the comments. MO best practice is to post a CW answer summarizing the answer from the comments. In this case, …

This is a great question. Let me start with limits and discuss lax limits later. Given a $D$-shaped diagram $X$ of model categories (where $D$ is a small category), one can ask whether the two ways (B …

If you want to generalize the intuition from "minimal cell structures" in topological spaces to model categories, you actually probably want to be thinking about cellular model categories rather than …

I guess I'm the canonical person to answer this question. I wrote those notes as a PhD student, a long time ago, to go along with a talk I was giving at a grad student conference. They were basically …

The short answer is "yes," it is true that the induced map $X\to M_f$ is a cofibration. I refer you to Section IX of Williamson's thesis Cylindrical model structures, page 114 of the pdf. He says that …

This question was already answered in the comments, but I don't want it to linger forever on the unanswered queue, so I'm making a CW answer summarizing the comments and adding my own example. Tyrone …

This week, we learned that another example is the Strøm (aka Hurewicz) model structure on the category of simplicial sets. Specifically, there is no model structure on $sSet$ whose class of weak equiv …

This week, we learned that another example is the category of simplicial sets, and the class of weak equivalences the simplicial homotopy equivalences. All credit to Tom Goodwillie, Tim Campion, and T …

Yes, this has been done in other settings. For example, Scott Balchin, Kyle Ormsby, Angélica M. Osorno, and Constanze Roitzheim wrote a paper, Model structures on finite total orders, that enumerates …

EDIT: The answer below suggests that there is a model structure on $sSet$ whose weak equivalences are the simplicial homotopy equivalences, but seems to have a problem. I'm going to leave what I wrote …

The OP wrote "I was hoping to find a reference that deals with truncation in simplicial model categories." In 2022, Michael Batanin and I published a paper, Homotopy theory of algebras of substitudes …

I think the OP does a good job answering his own question, in the context he had in mind. But, the original question asked "are there general methods to determine if $M$ is cofibrant with respect to t …