Timeline for What is an $\infty\text{-}E_{\infty}$ morphism?
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| Feb 14, 2023 at 5:07 | history | edited | ChesterX | CC BY-SA 4.0 |
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| Feb 12, 2023 at 14:07 | comment | added | ChesterX | @D.-C.Cisinski As for the rest of your comment, I think I get the idea, provided I am able to decipher the notion of $\infty$-morphisms in the first place! Just one question, by descent you mean some sort of gluing of the data right? That is, we can 'glue together' the $\infty$-morphisms on different charts (=Euclidean balls) to get a full morphism on the manifold. | |
| Feb 12, 2023 at 14:06 | comment | added | ChesterX | @D.-C.Cisinski As I am not very familiar with $\infty$-categories, I will repeat what I understood from the first line of your comment. Firstly, one can treat any $E_\infty$-operad $\mathcal{E}$ as an $\infty$-operad in the sense of, say, Moerdijk-Weiss. The category of $\mathcal{E}$-algebras gets an $\infty$-category structure. Then, an $\infty\text{-}E_\infty$-morphism is a 'weak morphism' in this $\infty$-category. If this is the correct interpretation, could you point me to some articles where this point of view is spelled out for $E_\infty$ or maybe even for $A_\infty$-operads? | |
| Feb 12, 2023 at 10:21 | comment | added | D.-C. Cisinski | The point of view of $\infty$-operads defines a notion of "weak morphism": this corresponds to morphisms in the $\infty$-category of $E_\infty$-algebras. They can be seen as the cochain level through rectification results, and they allow transfer of $E_\infty$-algebra structures. This is sufficient to promote the Stokes integration map to an $E_\infty$-algebra map: for $M$ a point, this is easy, which gives the case where $M$ is an euclidian ball by transfer, and we get the general case by descent. | |
| S Feb 12, 2023 at 8:47 | review | First questions | |||
| Feb 12, 2023 at 10:03 | |||||
| S Feb 12, 2023 at 8:47 | history | asked | ChesterX | CC BY-SA 4.0 |