Posted in math.CT, math.GR, tagged group actions on April 1, 2013 |
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A common theme in mathematics is to replace the study of an object with the study of some category that can be built from that object. For example, we can

- replace the study of a group with the study of its category of linear representations,
- replace the study of a ring with the study of its category of -modules,
- replace the study of a topological space with the study of its category of sheaves,

and so forth. A general question to ask about this setup is whether or to what extent we can recover the original object from the category. For example, if is a finite group, then as a category, the only data that can be recovered from is the number of conjugacy classes of , which is not much information about . We get considerably more data if we also have the monoidal structure on , which gives us the character table of (but contains a little more data than that, e.g. in the associators), but this is still not a complete invariant of . It turns out that to recover we need the *symmetric* monoidal structure on ; this is a simple form of Tannaka reconstruction.

Today we will prove an even simpler reconstruction theorem.

**Theorem:** A group can be recovered from its category of -sets.

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