Timeline for Can graphs of groups be thought of as "graph objects" in the category of groupoids?
Current License: CC BY-SA 4.0
9 events
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| Nov 29, 2022 at 2:17 | answer | added | Robbie Lyman | timeline score: 2 | |
| Oct 5, 2022 at 13:41 | comment | added | HJRW | By the way, the aforementioned paper by Scott and Wall is called "Topological methods in group theory". It's not that easy to find an official electronic copy, but googling seems to work... | |
| Oct 5, 2022 at 13:39 | comment | added | HJRW | Oh, I see, it's because you decided to model all graphs on a single category with 2 objects. An alternative point of view is to treat each graph as its own category. A graph of groups is then just a functor from a graph to the category of groups with injective maps. | |
| Oct 5, 2022 at 12:58 | comment | added | Antoine Labelle | @HJRW No, graph objects in the category of groups would have a group structure on their vertex set and edge set, that's not what graph of groups are. We need groupoids since they can be interpreted (up to equivalence) as sets with a group attached to each element. | |
| Oct 5, 2022 at 7:54 | comment | added | HJRW | Isn't a graph of groups exactly a graph object in the category of groups with injective maps? I'm not sure why you think you need groupoids... | |
| Sep 24, 2022 at 23:13 | comment | added | Antoine Labelle | I've never heard of homotopy colimits; that seems interesting. Could you elaborate on how the fundamental groupoid of a graph of groups can be interpreted as a homotopy colimit? | |
| Sep 24, 2022 at 17:57 | comment | added | Qiaochu Yuan | A graph of groups is a diagram of a certain shape in the 2-category of group(oids) and its fundamental group(oid) is given by taking the homotopy colimit. We can of course talk about homotopy colimits of diagrams in a lot of generality. | |
| Sep 24, 2022 at 17:39 | comment | added | Benjamin Steinberg | My feeling is that graphs of groups can be viewed as a kind of homotopy colimit in the category of groups. For instance, the HNN extension corresponds to a kind of mapping cyclinder and the amalgamated free product of a homotopy pushout. The graph of spaces approach to these by Scott and Wall is via homotopy colimits | |
| Sep 24, 2022 at 17:31 | history | asked | Antoine Labelle | CC BY-SA 4.0 |