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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 …
David White's user avatar
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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 …
David White's user avatar
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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. …
David White's user avatar
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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 …
David White's user avatar
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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 …
David White's user avatar
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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 …
David White's user avatar
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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 …
David White's user avatar
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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 …
David White's user avatar
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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 …
David White's user avatar
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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 …
David White's user avatar
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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 …
David White's user avatar
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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 …
David White's user avatar
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