A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set X preserves no structure: or, in other words, it preserves only the structure of being a set. When X is finite, what structure can the alternating group be said to preserve?As a way of making the question precise, is there a natural definition of a category C equipped with a faithful functor to FinSet such that the skeleton of the underlying groupoid of C is the groupoid with objects X n such that Aut ( X n ) ≃ A n ?

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A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set   X preserves no structure: or, in other words, it preserves only the structure of being a set. When   X is finite, what structure can the alternating group be said to preserve?As a way of making the question precise, is there a natural definition of a category   C equipped with a faithful functor to   FinSet such that the skeleton of the underlying groupoid of   C is the groupoid with objects       X    n   such that   Aut  (      X    n    )  ≃      A    n  ?

atoandro8f04v · Accepted Answer

The alternating group preserves orientation, more or less by definition. I guess you can take   C to be the category of simplices together with an orientation. I.e., the objects of   C are affinely independent sets of points in some         R    n   together with an orientation and the morphisms are affine transformations taking the vertices of one simplex to the vertices of another. Of course this is cheating since if you actually try to define orientation you&#039;ll probably wind up with something like &quot;coset of the alternating group&quot; as the definition. On the other hand, some people find orientations of simplices to be a geometric concept, so this might conceivably be reasonable to you.

Vicente Duran · Accepted Answer

Here is one idea, although I do not find it very satisfying. An object of   C is a finite set   X equipped with                 |      X          |      !        2   (or   1 if   |  X  |  =  1) total orders, all of which are even with respect to each other (in other words, basically a coset of       A    n   in       S    n  ). A morphism between two objects in   X is a map of sets preserving these orders (in other words, take one of the orderings on   X and apply a function   f  :  X  →  Y to its elements. The result, after throwing out repeats, must be compatible with an ordering on   Y.)

A common way to define a group is as the group of structure-preserving transformations on some struc

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