Timeline for How to compute Homotopy Pullback
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 11, 2021 at 7:49 | vote | accept | Bugs Bunny | ||
| Mar 11, 2021 at 5:04 | history | became hot network question | |||
| Mar 10, 2021 at 23:35 | answer | added | Dmitri Pavlov | timeline score: 10 | |
| Mar 10, 2021 at 22:41 | comment | added | David Roberts♦ | @TylerLawson I think that is worth putting as an answer, in the absence of more specifics. | |
| Mar 10, 2021 at 21:57 | comment | added | Denis Nardin | Another approach (more or less equivalent to Tyler Lawson's suggestion) is to use the Bousfield-Kan formula if you are comfortable with pullbacks and totalizations (the latter is admittedly quite subtle in general). | |
| Mar 10, 2021 at 21:50 | comment | added | Tyler Lawson | If your category is cotensored over spaces and has pullbacks, you can often construct a homotopy pullback as the fiber product $(A \times C) \times_{B \times B} B^{[0,1]}$ (i.e. apply the "classical" homotopy pullback formula from spaces). Anything isomorphic to that in the homotopy category of $\mathcal{V}$ will also be a homotopy pullback. | |
| Mar 10, 2021 at 21:32 | comment | added | Maxime Ramzi | But ultimately it will depend on your specific situation | |
| Mar 10, 2021 at 21:31 | comment | added | Maxime Ramzi | Well very often you can replace your diagram, up to equivalence, by a "good" diagram where the pullback is also a homotopy pullback. This is one of the raisons d'être of model structures, although you need much less than that. For instance if you can find a diagram $A' \to B' \leftarrow C'$ which is related to your original diagram by a zigzag of maps of diagrams, each of which inducing equivalences on mapping spaces, and such that $V(X,C')\to V(X,B')$ is a fibration for all $X$, then you can just take the pullback $A'\times_{B'} C'$. | |
| Mar 10, 2021 at 21:24 | history | edited | Bugs Bunny | CC BY-SA 4.0 |
added 18 characters in body; edited title
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| Mar 10, 2021 at 21:22 | comment | added | Bugs Bunny | Yes, sorry. I will correct. But the problem is valid for both... | |
| Mar 10, 2021 at 21:18 | comment | added | Maxime Ramzi | Do you mean pullback ? | |
| Mar 10, 2021 at 21:15 | history | edited | Bugs Bunny | CC BY-SA 4.0 |
edited title
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| Mar 10, 2021 at 21:09 | history | edited | Bugs Bunny | CC BY-SA 4.0 |
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| Mar 10, 2021 at 21:03 | history | asked | Bugs Bunny | CC BY-SA 4.0 |