Timeline for Measure the failure of colimit to commute with taking free loops (or Hochschild homology)?
Current License: CC BY-SA 4.0
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| Jan 12, 2021 at 16:21 | answer | added | Phil Tosteson | timeline score: 3 | |
| Jan 12, 2021 at 15:30 | history | edited | Vivek Shende | CC BY-SA 4.0 |
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| Jan 12, 2021 at 15:06 | history | edited | Vivek Shende | CC BY-SA 4.0 |
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| Jan 12, 2021 at 14:41 | comment | added | Tyrone | You still need to be more careful: let $X$ be any space and consider $X_+=X\sqcup\{+\}$ (basepoint is the disjoint point). Again $\Omega(X_+)=\Omega(+)=\ast$. In general the statement is false even for connected spaces without some further assumptions. Let $X$ be the Warsaw circle. Then $X$ is not contractible. However $\Omega X$ is, and moreover the section $X\rightarrow\mathcal{L}X$ is an inverse homotopy equivalence to the evaluation map. (Of course $X$ is not of CW type). | |
| Jan 12, 2021 at 14:33 | comment | added | Maxime Ramzi | In that case I think Goodwillie calculus might be interesting, especially if your desired target ($\infty$-)category is stable (for instance chain complexes if you're working up to quasi-isomorphism) - then excisive functors are related to finite colimit-preserving functors. | |
| Jan 12, 2021 at 14:10 | history | edited | Vivek Shende | CC BY-SA 4.0 |
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| Jan 12, 2021 at 14:09 | comment | added | Vivek Shende | @Tyrone I meant weak equivalence. Also I meant “each connected component is contractible.” Now I edited it to clarify. | |
| Jan 12, 2021 at 14:07 | comment | added | Tyrone | Firstly, by isomorphism, you mean homeomorphism, or do you have something else in mind? Secondly, if $X=S^0$ is a two-point discrete space, then $\Omega S^0=\ast$ and $S^0\rightarrow\mathcal{L}S^0$ is a homeomorphism. | |
| Jan 12, 2021 at 9:35 | comment | added | Vivek Shende | @MaximeRamzi, that’s ok — for what I need, finite colimits are enough. | |
| Jan 12, 2021 at 9:02 | comment | added | Maxime Ramzi | I think (but I might be completely wrong) that usually the calculus of functors is related to finite colimits rather than arbitrary colimits (see the definition of excisive or $n$-excisive for instance) | |
| Jan 12, 2021 at 2:47 | history | edited | Vivek Shende | CC BY-SA 4.0 |
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| Jan 12, 2021 at 2:37 | history | edited | Vivek Shende | CC BY-SA 4.0 |
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| Jan 12, 2021 at 2:30 | history | edited | Vivek Shende | CC BY-SA 4.0 |
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| Jan 12, 2021 at 2:24 | history | edited | Vivek Shende | CC BY-SA 4.0 |
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| Jan 12, 2021 at 2:16 | history | asked | Vivek Shende | CC BY-SA 4.0 |