Timeline for Homotopy pullbacks and homotopy pushouts
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Dec 30, 2009 at 1:27 | comment | added | Joey Hirsh | Reid: Doesn't the mapping cone provide the kernel in homology? Shouldn't this make the mapping cone a homotopy kernel, and as such a homotopy pullback/limit (rather than pushout/colimit)? | |
| Dec 12, 2009 at 18:42 | vote | accept | Alicia Garcia-Raboso | ||
| Dec 12, 2009 at 18:42 | comment | added | Alicia Garcia-Raboso | I see a DG-scheme-type of thing going on... very interesting. Thanks for all your wonderful (and swift) explanations! | |
| Dec 12, 2009 at 18:22 | comment | added | Reid Barton | That is true with the usual formula for the homotopy pullback. The properly analogous statement would be for pullbacks of sets, but in that case the two notions of pullback agree. A more typical example is comparing the pullback of vector spaces A -> C <- B to the homotopy pullback in unbounded chain complexes. The homotopy pullback will be the ordinary pullback in degree 0, and in (homological) degree -1 it should be something like the cokernel of the map A+B -> C. Generally homotopy limits have "extra stuff in negative degrees", which we can't see in topological spaces. | |
| Dec 12, 2009 at 18:01 | comment | added | Alicia Garcia-Raboso | So I see why the homotopy pushout is a derived version of the ordinary pushout: the latter is π_0 of the former (at least in your example). I fail to see the corresponding statement for the homotopy pullback: it seems to me that the ordinary pullback is a subspace of the homotopy pullback. Is this correct? | |
| Dec 12, 2009 at 17:36 | comment | added | Reid Barton | Well, I more or less repeated it from the beginning of one of his talks :) As for homotopy pullbacks, of course they are formally dual, but maybe more helpful is this: The ordinary fiber product of X and Y over Z is the set of points of X x Y which have the same image in Z. In homotopy theory, we cannot talk about equality of points in a space--or rather, we have an entire space of "ways in which points may be considered equal", namely, the space of paths in Z between the two points. If you write out the common definition for homotopy pullbacks in spaces you'll see this is what it computes. | |
| Dec 12, 2009 at 17:25 | comment | added | Alicia Garcia-Raboso | Very nice! It reminds me of the introduction to Lurie's thesis, where he talks about Bezout's theorem. Do you have a similar picture for the pullback? | |
| Dec 12, 2009 at 16:49 | history | answered | Reid Barton | CC BY-SA 2.5 |