Timeline for Fibrations of the injective model structure on G-simplicial sets
Current License: CC BY-SA 3.0
13 events
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| Nov 15, 2014 at 0:38 | comment | added | Simon Henry | Oh indeed you're right: I forgot half of the argument. you also need to use the assumption that the stabilizer of each points not in $A$ is trivial, then you chose a lifting of one point in each orbit and you extend to the rest of the orbit in the unique way that respect the $G$-action. | |
| Nov 14, 2014 at 22:25 | comment | added | user2664 | @SimonHenry: thanks for your answer, your construction of the diagonal filler is "pointwise" as I understand, so it's not clear for me that at the end you get a natural transformation compatible with the G-action. | |
| Nov 6, 2014 at 8:11 | comment | added | Simon Henry | @Asymptotik: No. I computed that my self. Roughly, Its easy to check that the generating cofib satisfies these properties and that they are stable under retract,pushout and transfinite composition, and then you have to check that maps with these property satisfy the LLP with respect to trivial kan fibration and this can be donne by construction the diagonal filing on $n$-simplicies by induction on $n$. With those two thing in hand, the small object argument conclude the proof. | |
| Nov 6, 2014 at 1:27 | comment | added | user2664 | @SimonHenry: do you have a reference for the characterization of the cofibrations you mentioned in the projective model structure ? Thanks. | |
| Oct 19, 2014 at 11:42 | comment | added | Dmitri Pavlov | The mixed model structure, whose fibrations and acyclic fibrations are as described in the main post, is explored by Shipley, see Proposition 1.3 in “A convenient model category for commutative ring spectra”. | |
| Oct 18, 2014 at 16:29 | history | edited | David White | CC BY-SA 3.0 |
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| Oct 18, 2014 at 15:44 | comment | added | Charles Rezk | @SimonHenry The model structure is the "usual" equivariant model structure. There are surely many references: maybe Dwyer & Kan, "Singular functors and realization functors". www3.nd.edu/~wgd/Dvi/SingularAndRealization.pdf | |
| Oct 18, 2014 at 14:41 | history | edited | Simon Henry | CC BY-SA 3.0 |
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| Oct 18, 2014 at 8:05 | comment | added | Simon Henry | Matthias Wendt: No for the projective model structure the fibration and trivial fibration are defined by forgeting the G action, and the cofibration are inclusions $A \subset B$ where the elements of $B$ which are not in $A$ have a stabilizer reduced to $\{0\}$. Charles Rezk: thanks for that remark, this is really helpful ! (If someone can point a reference for this it might also be helpful) | |
| Oct 17, 2014 at 14:52 | comment | added | Charles Rezk | There's a model structure on $G$-simplicial sets, where the cofibrations and fibrations are as you suggest, but the weak equivalences are equivariant ones (i.e., weak equivalence on $H$-invariants for all $H$). The fibrations in your injective structure will be characterized by more restrictive property. | |
| Oct 17, 2014 at 13:28 | history | edited | Neil Strickland | CC BY-SA 3.0 |
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| Oct 17, 2014 at 12:32 | comment | added | Matthias Wendt | Wouldn't your description imply that the projective and injective model structures coincide? | |
| Oct 17, 2014 at 11:44 | history | asked | Simon Henry | CC BY-SA 3.0 |