Timeline for Homotopy theory of cospaces (or $\infty$-cogroupoids)
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 31, 2022 at 11:04 | history | edited | Emily | CC BY-SA 4.0 |
edited title
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| Oct 28, 2022 at 20:14 | comment | added | Emily | (Also thank you for the references and comments on the enrichment!) | |
| Oct 28, 2022 at 20:14 | comment | added | Emily | @მამუკაჯიბლაძე I think the homotopy theory of cosimplicial spaces goes in a different direction than what I have in mind: I'm looking for a notion of homotopy cogroups (or rather, "homotopy sets") of a "cospace"/$\infty$-cogroupoid, instead of the usual homotopy groups of (totalisations of) cosimplicial spaces. It seems to me that this might be a kind of "dual notion" of homotopy, obtained via subobjects rather than quotients, as happens at least for the $\pi_0$. | |
| Oct 28, 2022 at 9:36 | comment | added | მამუკა ჯიბლაძე | More concisely, $\operatorname{Hom}(F,G)(m)$, for $m\in\Delta$, is the set of natural transformations $F\to G^m$, where $G^m(n)=G(n)^{\hom_\Delta(n,m)}$. | |
| Oct 28, 2022 at 9:22 | comment | added | მამუკა ჯიბლაძე | This goes as follows. First, for $X:\Delta^{\mathrm{op}}\to\mathrm{Sets}$ and $G:\Delta\to\mathrm{Sets}$, define $G^X:\Delta\to\mathrm{Sets}$ via $(G^X)(n)=G(n)^{X(n)}$, with $n\in\Delta$. Next, define the above $\operatorname{Hom}(F,G)$ in such a way that for any $X:\Delta^{\mathrm{op}}\to\mathrm{Sets}$, natural transformations $X\to\operatorname{Hom}(F,G)$ are in one-to-one correspondence with natural transformations $F\to G^X$. | |
| Oct 28, 2022 at 9:19 | comment | added | მამუკა ჯიბლაძე | The latter is a generalization of a special case of the fact that for any small category $\Delta$, one can define, for $F,G:\Delta\to\mathrm{Sets}$, the functor $\operatorname{Hom}(F,G):\Delta^{\mathrm{op}}\to\mathrm{Sets}$. | |
| Oct 28, 2022 at 9:17 | comment | added | მამუკა ჯიბლაძე | Google reveals notes by Bert Guillou about homotopy theory of cosimplicial spaces. There was some stuff much earlier by Bousfield but I cannot remember any precise references, sorry. In any case there is an enrichment of cosimplicial objects of a nice enough category into simplicial sets. | |
| Oct 28, 2022 at 4:22 | history | asked | Emily | CC BY-SA 4.0 |