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.