Timeline for Is the infinity-groupoid of a finite CW complex finitely-presented?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 9, 2015 at 18:10 | vote | accept | Jamie Vicary | ||
| Apr 9, 2015 at 6:20 | comment | added | Zhen Lin | It appears to me that the OP is thinking in terms of homotopy type theory, so here "finitely presented ∞-groupoid" should be "higher inductive type with finitely many constructors". | |
| Apr 9, 2015 at 1:14 | comment | added | Qiaochu Yuan | Another way to say it: a presentation of a group is a description of that group as the cokernel of a map between free groups. Keeping in mind the analogy to free resolutions of modules, we might say more generally that "presentation" means "description of an object as a colimit of free objects." And a CW decomposition is precisely a description of a space as an iterated homotopy pushout of spheres (which themselves are iterated homotopy pushouts of points). | |
| Apr 9, 2015 at 1:08 | comment | added | Qiaochu Yuan | ...new "generators" in the sense that they may (before we kill them again) give rise to higher homotopy and new "relations" in the sense that they themselves kill homotopy. | |
| Apr 9, 2015 at 1:07 | comment | added | Qiaochu Yuan | This is a model-independent statement. Of course it depends on what the OP means by "free $\infty$-groupoid on a finite family of generators," but I think this is a reasonable interpretation. Think first about how one presents a group $G$ in terms of generators and relations. Topologically generators correspond to $1$-cells and relations correspond to $2$-cells. So in fact a presentation describes a CW complex of dimension $2$ (the presentation complex). This is the beginning of a presentation / CW description of $BG$, and one now needs to add $3$-cells, etc. These are simultaneously... | |
| Apr 9, 2015 at 0:50 | comment | added | David Roberts♦ | What model of an oo-groupoid are you using? Clearly n-cells are generators in dimension n, but how does one interpret the attaching maps? Split in half and let the halves be source and target in a globular oo-groupoid? | |
| Apr 9, 2015 at 0:22 | comment | added | Jamie Vicary | thanks. This makes sense to me, although I do not quite see why it is a trivial statement... I will think further. | |
| Apr 8, 2015 at 23:43 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |