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1. • CommentRowNumber1.
   • CommentAuthormb
   • CommentTimeMay 2nd 2018
   • PermaLink
   Author: mb
   Format: MarkdownItexConsider the category $PSh(C,sSet)$ of simplicial presheaves on a category $C$ equipped with the global projective model structure. I am aware of the fact that objects of $C$ are represented in $PSh(C,sSet)$ by cofibrant objects. Consider now a simplicial object $c_\bullet \in C^{\Delta^{op}}$ in $C$. I would like to know if the corresponding object in $PSh(C,sSet)$ is cofibrant too. For concreteness, take C to be the category of smooth manifolds and $c_\bullet$ to be the nerve of an action Lie groupoid $M \rtimes G$. Is $M \rtimes G$ cofibrant when regarded as an object of $PSh(C,sSet)$?

   Consider the category $PSh(C,sSet)$ of simplicial presheaves on a category $C$ equipped with the global projective model structure. I am aware of the fact that objects of $C$ are represented in $PSh(C,sSet)$ by cofibrant objects. Consider now a simplicial object $c_\bullet \in C^{\Delta^{op}}$ in $C$. I would like to know if the corresponding object in $PSh(C,sSet)$ is cofibrant too.

   For concreteness, take C to be the category of smooth manifolds and $c_\bullet$ to be the nerve of an action Lie groupoid $M \rtimes G$. Is $M \rtimes G$ cofibrant when regarded as an object of $PSh(C,sSet)$?

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 2nd 2018
   • (edited May 2nd 2018)
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   Author: Dmitri Pavlov
   Format: MarkdownItexA 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 2nd 2018
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   Author: Urs
   Format: MarkdownItexGreat! Somebody should now spell this out in some relevant nLab entry. [Here](https://nforum.ncatlab.org/discussion/1352/model-structure-on-simplicial-presheaves/?Focus=68786#Comment_68786) is a start.

   Great! Somebody should now spell this out in some relevant nLab entry. Here is a start.

4. • CommentRowNumber4.
   • CommentAuthormb
   • CommentTimeMay 2nd 2018
   • PermaLink
   Author: mb
   Format: MarkdownItexThanks a lot for the clarification and for the reference!

   Thanks a lot for the clarification and for the reference!