Timeline for The most general context of Mather's Cube Theorems
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 16, 2012 at 23:33 | comment | added | some guy on the street | Oh my goodness wow! I'll have to keep studying these things. | |
| Mar 4, 2012 at 1:03 | comment | added | Jacob Lurie | Hi Tom; I meant statement 1 (the category of sets is an ordinary topos, rather than an $\infty$-topos). Statement 2 would be a special case of the blanket hypothesis "homotopy colimits in X commute with homotopy pullback". The more general property you describe also holds in any $\infty$-topos. | |
| Mar 3, 2012 at 21:35 | comment | added | Tom Goodwillie | Or, Jacob, I suppose by "the Mather cube theorem" you mean statement 2 in the question. As I mentioned in my edit to my answer, statement 1 fails for the category of sets. | |
| Mar 3, 2012 at 21:14 | comment | added | Tom Goodwillie | Jacob, I'm guessing that such things also have the following more general property: If $F$ and $G$ are functors $I\to \mathcal X$ such that for every $i$-morphism $i\to j$ you get a homotopy pullback square $(F(i)\to G(i))\to (F(j)\to G(j))$ then for every $i$-object $i$ you get a homotopy pullback square $(F(i)\to G(i))\to (hocolim F\to hocolim G)$. (The case of this when $I$ is the poset $a\leftarrow b\to c$ is the (first) Mather cube theorem.) | |
| Mar 3, 2012 at 16:09 | history | answered | Jacob Lurie | CC BY-SA 3.0 |