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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 11th 2013
• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 16th 2018

Added an explicit description of generating acyclic cofibrations for the Joyal model structure.

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeOct 17th 2018

Cleaned up statement of Stevenson’s theorem about the generating acyclic cofibrations and added the arXiv reference to his paper. (I talked with Danny about this yesterday and he agrees that the hypothesis can be stated as the equivalent ’countably-many simplices’, rather than ’countably-many non-degenerate simplices’)

Richard, if you are reading this, the LaTeX-like syntax that Dmitri used isn’t working. I left it in so this could be verified and fixed, if indeed there is abug.

• CommentRowNumber4.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 17th 2018

An element of B is a simplicial set, not a map of simplicial sets. What does it mean for a simplicial set to be an acyclic cofibration?

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeOct 17th 2018
• (edited Oct 17th 2018)

Gah, there’s a bijection $\mathcal{B}\to \{\{0\}\to B\mid B\in \mathcal{B}\}$, and I was implicitly casting along this.

1. Richard, if you are reading this, the LaTeX-like syntax that Dmitri used isn’t working. I left it in so this could be verified and fixed, if indeed there is a bug.

Thanks for the message. I asked Dmitri to leave this; it is not functionality that has been implemented yet, but I have been planning to add it very soon. But feel free to change it to the more conventional syntax in the meantime, I think I will remember that the reference was added here.

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeOct 17th 2018

Fixed the issue that Dmitri pointed out.

2. The Joyal model structure has, in fact, a simplicial enrichment compatible with the model structure (although of course not the canonical one). I’ll add the details later but I wanted to remove the wrong statement

Denis Nardin

3. I removed the incorrect (as I explained here: https://chat.stackexchange.com/transcript/message/49892813#49892813) generating set of trivial cofibrations.

I also corrected wording in a few places and added references for Joyal’s original construction of the model structure.

Alexander Campbell