Timeline for Dold-Kan correspondence in the category of symmetric spectra
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 9, 2019 at 7:51 | comment | added | Surojit Ghosh | One thing I am thinking that we know we have a map $Sp \to Ho(Sp)$. Since $Ho(Sp)$ is an additive category so we can add maps. Therefore, we can take the alternative sum of face maps in $Ho(Sp)$ then take a representative in $Sp$ and define that as the differential. Now if we have a cosimplicial object $W^\bullet$ in $Sp$. Here each $W^n$ is an $E$-module spectra. Then we have a map $W^n \to M^n W^\bullet$, the Matching object. Then can we claim that the kernel of the map $W^n \to M^n W^\bullet$ is a summand of $W^n?$ I think the kernel should be $\bigcap_{i=0}^{n-1} ker(s^i).$ | |
| Jun 30, 2019 at 9:57 | comment | added | Dylan Wilson | As I explained above, I don’t see what the “Moore complex functor” is because the category of symmetric spectra is not additive. In response to your second comment: That direction is easy. Take the skeletal filtration of the simplicial object, take the (cofiber) associated graded, and then use the boundary maps coming from the cofiber sequences. | |
| Jun 30, 2019 at 8:41 | comment | added | Surojit Ghosh | Btw, I don't need a formal Dold-Kan equivalence: all I need is a procedure for producing a chain complex of spectra from a simplicial object in symmetric spectra. | |
| Jun 30, 2019 at 8:16 | comment | added | Surojit Ghosh | For any pointed simplicial category we can define the chain compexes are those such that composition of consecutive differentials is nullhomotopic. We can do this for the category of symmetric spectra. With these notions in our hand can we say something about the normalized complex functor or the Moore complex functor from simplicial objects in symmetric spectra to the chain complexes in symmetric spectra? | |
| Jun 24, 2019 at 15:39 | history | answered | Dylan Wilson | CC BY-SA 4.0 |