Timeline for mapping spaces of diagrams
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 15, 2012 at 17:27 | comment | added | Justin Noel | @Cary: You're right, your category is not Reedy, but it is generalized Reedy: arxiv.org/pdf/0809.3341v1.pdf. Also Quillen equivalences will induce homotopy equivalences between Dwyer-Kan function complexes. The DK complex from $X$ to $Y$ will have the same homotopy type as the mapping space from a cofibrant replacement of $X$ to a fibrant replacement for $Y$ in any of the model structures if the model structure makes the category a simplicial/topological model category. | |
| Feb 14, 2012 at 22:46 | comment | added | Cary | If we make the degree of $\{1,\ldots,n\}_+$ equal to $n$, then $R_+$ can be all injective order-preserving maps, and $R_-$ can be all surjective maps. Then every map factors uniquely into a map in $R_-$ followed by a map in $R_+$. Unfortunately not every map in $R_-$ lowers degree, and not every map in $R_+$ raises degree. | |
| Feb 14, 2012 at 21:46 | comment | added | Cary | I need a weak homotopy equivalence between my two constructions of mapping spaces, and I think that's stronger than saying that the homotopy categories are equivalent. Also, my A is the category of based sets $\emptyset_+$, $\{1\}_+$, $\ldots$, $\{1,\ldots,n\}_+$ and based maps between them. I believe this is not a Reedy category but I could be mistaken. | |
| Feb 14, 2012 at 21:44 | comment | added | David White | In all seriousness, I am not an expert on Reedy things. I don't know how far the average small category $A$ is from being Reedy or if you can somehow move from your situation into this situation. I should also mention that the functor which gives the Quillen equivalence is the identity, so Map($X,Y)_{Proj} \cong$ Map$(X,Y)_{Reedy} \cong$ Map$(X,Y)_{Inj}$ | |
| Feb 14, 2012 at 21:22 | comment | added | David White | Link to Lurie's HTT, if you dare: arxiv.org/abs/math.CT/0608040 | |
| Feb 14, 2012 at 21:22 | history | answered | David White | CC BY-SA 3.0 |