Timeline for How to specify a finite group up to inner automorphism?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 3, 2013 at 23:24 | vote | accept | John Pardon | ||
| Mar 1, 2013 at 6:02 | answer | added | Will Sawin | timeline score: 2 | |
| Feb 15, 2013 at 9:25 | comment | added | Mark Grant | Dylan might be onto something...what about the classifying space functor from the category of groups defined in your previous comment to the category whose objects are topological spaces and whose morphisms are unbased homotopy classes of unbased maps? The construction of $BG$ depends on choices, as you say, but all such choices lead to isomorphic objects. | |
| Feb 15, 2013 at 7:50 | comment | added | John Pardon | @Dylan Wilson: well, it's not quite localization (since inner automorphisms are already isomorphisms in the category of groups). It's more like I set $\operatorname{Hom}(G_1,G_2)$ to be the set of group homomorphisms quotiented by the equivalence relation where two which differ by an inner automorphism of the target are considered the same. | |
| Feb 15, 2013 at 4:45 | comment | added | David Roberts♦ | Inner automorphisms correspond to natural transformations between functors between groups considered as one-object groupoids. So really there's something 2-categorical going on, and you're quotienting down to the 1-category of one-object (or possibly just connected) finite groupoids with isomorphism classes of functors as arrows. Probably you're trying to construct a category equivalent to this category. My suggestion is to keep track of the inner automorphisms and make your structure richer. | |
| Feb 15, 2013 at 1:41 | comment | added | Dylan Wilson | So what's the problem with the functor from groups to homotopy types given by taking the classifying space? (Essentially your (1)). This satisfies all of those criteria. I feel like you're asking a different question: Consider the category of groups and localize at inner automorphisms... now find a fully faithful functor from this new category into something nice. | |
| Feb 15, 2013 at 1:15 | answer | added | Ryan Budney | timeline score: -1 | |
| Feb 15, 2013 at 1:14 | comment | added | John Pardon | Since I say "is the identity", the data must be something combinatorial, not a category. | |
| Feb 15, 2013 at 1:14 | comment | added | John Pardon | I want a construction (read: functor) which takes in a finite group $G$ and outputs some data, so that (1) the map on this data associated to an inner automorphism of $G$ is the identity (2) given $G$ and the associated data, I should be able to construct a group $G'$ and an isomorphism $G'\to G$ which is well-defined up to inner automorphism. | |
| Feb 15, 2013 at 1:00 | comment | added | Dylan Wilson | What is a "group up to inner automorphism"? | |
| Feb 15, 2013 at 0:58 | history | asked | John Pardon | CC BY-SA 3.0 |