Timeline for Does a homotopy sheaf functor commute with group completion
Current License: CC BY-SA 4.0
11 events
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| Sep 27, 2019 at 13:49 | comment | added | Simon Henry | @Nicky : One first check that $\pi_0: \widehat{\Delta} \rightarrow Set $ is left adjoint to the constant simplicial set functor. Then one deduce that applying level wise $\pi_0$ : $sPre(C) \rightarrow Pre(C)$ is left adjoint to the inclusion $ Pre(C) \rightarrow sPre(C)$. Then one uses that sheaficication is left adjoint to the forgetful inclusion of sehaves into presheaves, and that adjoint compose. | |
| Sep 6, 2019 at 17:10 | comment | added | Nicky | Thanks! Can you explain a bit why this is true in 1-category level please? | |
| Sep 6, 2019 at 14:01 | vote | accept | Nicky | ||
| Sep 6, 2019 at 11:31 | comment | added | Simon Henry | Yes it is. Both at 1-category level and homotopicaly speaking. | |
| Sep 5, 2019 at 20:36 | comment | added | Nicky | For ordinary category $C$, $\pi_0^{\tau}$ is the left adjoint of the inclusion $Sh(C)\to sPre(C)$, where a sheaf is regarded as discrete simplicial sheaf? | |
| Sep 5, 2019 at 19:09 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Sep 5, 2019 at 18:57 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Sep 5, 2019 at 18:53 | comment | added | Simon Henry | Absolutely, I'm just not sure what language you would like, but I've added a rephrasing of the argument at the end that will be easier to translate into model theoretic or traditional homotopy theoretic language. I hope this helps. Anyway the point I'm making here is that what make thing work is not so much the description of the group completion as $\Omega B$ but rather its universal property with respect to 'group' object (in the sense monoid whose $\pi_0$ is a group) which somehow make this commutation trivial. | |
| Sep 5, 2019 at 18:49 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Sep 5, 2019 at 18:42 | comment | added | Nicky | Is there a way to see this without using $\infty$-category? For an ordinary category $C$, how can I see that the homotopy sheaf defined in my question is a left adjoint? | |
| Sep 5, 2019 at 15:18 | history | answered | Simon Henry | CC BY-SA 4.0 |