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Consider the category of simplicial presheaves on a category equipped with the global projective model structure. I am aware of the fact that objects of are represented in by cofibrant objects. Consider now a simplicial object in . I would like to know if the corresponding object in is cofibrant too.
For concreteness, take C to be the category of smooth manifolds and to be the nerve of an action Lie groupoid . Is cofibrant when regarded as an object of ?
A complete answer is available here: https://mathoverflow.net/questions/97690/necessary-conditions-for-cofibrancy-in-global-projective-model-structure-on-simp
In your case simplicial levels are representable presheaves, so one of the two conditions is satisfied. The other condition is violated, though: degenerate 1-simplices over S are smooth maps S→M×G whose second component is the constant map. A nondegenerate 1-simplex can restrict to a degenerate 1-simplex along some map S’→S, e.g., because the second component vanishes on S’ but not S.
Great! Somebody should now spell this out in some relevant nLab entry. Here is a start.
Thanks a lot for the clarification and for the reference!
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