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A model category is a category equipped with notions of weak equivalences, fibrations and cofibrations allowing to run arguments similar to those of classical homotopy theory.
2
votes
Accepted
Fibrations in a model structure for homotopy $n$-types of simplicial sets
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 …
2
votes
Accepted
Simplicial enrichment on unbounded algebras over an operad
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 …
6
votes
Bar construction in commutative algebras is calculated by pushout
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. …
5
votes
Accepted
Is hammock localization a localization in the sense of Lurie?
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, …
2
votes
Reference request for equivalences between different models of lax limits
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 …
2
votes
Accepted
Minimal cell structures in combinatorial model categories
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 …
12
votes
Accepted
Model categories as a tool to resolve size issues for localizing categories
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 …
7
votes
Accepted
Is the mapping cylinder a replacement for morphism by cofibration in model categories?
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 …
1
vote
Accepted
Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations
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 …
5
votes
Non-examples of model structures, that fail for subtle/surprising reasons?
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 …
2
votes
A category with weak equivalences that is not a model category
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 …
12
votes
Categories on which one can determine all model structures?
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 …
2
votes
Strøm model structures on the category of simplicial sets
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 …
2
votes
$n$-truncation of a Simplicial Model Category
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 …
0
votes
Cofibrancy of a right module over an operad
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 …