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I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few answers which satisfy the first requirement, but not the second.

  1. Give a topological space $X$. The fundamental group of $X$ is a "group up to inner automorphism": you can get a group by picking a basepoint $p$, but given two different basepoints $p$ and $q$, there is not a canonical isomorphism $\pi_1(X,p)\to\pi_1(X,q)$, but any two automorphisms coming from a path from $p$ to $q$ differ by an inner automorphism. This answer is unsatisfying because there are many topological spaces one can pick, even if we restrict them to be $K(\pi,1)$'s.

  2. Give a groupoid $X$ (this generalizes the first example by taking the fundamental groupoid). This answer is unsatisfying because again there are many (say, finite) groupoids corresponding to a given group up to inner automorphism.

  3. Give a tensor category which is isomorphic to the category of finite dimensional representations of a finite group over an algebraically closed field of characteristic zero. This is almost a good answer. It looks at first that specifying such a category is a finite amount of combinatorial data: we specify the isomorphism classes of objects and specify how the tensor product of any two of them decomposes. However we also have to specify isomorphisms $(A\otimes B)\otimes C\to A\otimes(B\otimes C)$, and this means specifying some matrix which depends on the specific bases of the isotypic components of the tensor products that we chose when specifying how they decompose. So it is unsatisfying as well, although it is the best I have come up with. It is almost good enough since it is easy to say what an equivalence between two such finite categories is in terms of matrices.

  4. Give an orbifold whose coarse space is a single point. This answer is unsatisfying because I am hoping the answer to this question will be useful for giving a nice definition of an orbifold. Also, by unraveling the definition of an orbifold, this answer really just reduces to (1) or (2) and is thus also unsatisfying.

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What is a "group up to inner automorphism"? –  Dylan Wilson Feb 15 '13 at 1:00
    
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. –  John Pardon Feb 15 '13 at 1:14
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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. –  Dylan Wilson Feb 15 '13 at 1:41
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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. –  David Roberts Feb 15 '13 at 4:45
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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. –  Mark Grant Feb 15 '13 at 9:25

2 Answers 2

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You can give a monoidal category with a tensor functor to the category of finite sets where tensor is Cartesian product that is equivalent to the category of permutation representations of the group. By Grothendieck's Galois theory, one can recover the group from this. This improves on linear representations because your association data is finite, and should be a bit more canonical.

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Wouldn't the 2-skeleton of a finite CW-complex satisfy both your first and second conditions? You could choose a finite 2-dimensional simplicial complex if you wanted everything to be finitary, or you could add some restrictions to the attaching maps of the CW-complex, that they're PL.

From a finite simplicial complex you have the fundamental groupoid. Similarly, given a finitely-presentable groupoid, its presentation complex can be taken to be a finite 2-dimensional CW-complex.

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I admit my question is sort of ill-defined, but for me there are still too many choices needed for such a construction (like, picking a presentation). Also, your presentation complex probably has a single 0-cell, so I can pick it as a well-defined basepoint and recover the group, whereas I want to only be able to recover it up to inner automorphism. –  John Pardon Feb 15 '13 at 1:34
    
The presentation complex would not have a single vertex if your groupoid's arrows were not all composable. I think I understand what you're shooting for and see how my response isn't quite on the mark, though. –  Ryan Budney Feb 15 '13 at 3:14
    
You don't need to choose a presentation, you can just take all the elements to be generators! But you still don't get the inner automorphism acting trivially. They act trivially in the basepoint-free homotopy category, though, I think. –  Will Sawin Mar 1 '13 at 14:14

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