Previously we described what it means for a group to act on a category (although we needed to slightly correct our initial definition). Today, as the next step in our attempt to understand Galois descent, we’ll describe what the fixed points of such a group action are.
John Baez likes to describe (vertical) categorification as replacing equalities with isomorphisms, which we saw on full display in the previous post: we replaced the equality with isomorphisms , and as a result we found 2-cocycles lurking in this story.
I prefer to describe categorification as replacing properties with structures, in the nLab sense. That is, the real import of what we just did is to replace the property (of a function between groups, say) that with the structure of a family of isomorphisms between and . The use of the term “structure” emphasizes, as we also saw in the previous post, that unlike properties, structures need not be unique.
Accordingly, it’s not surprising that being a fixed point of a group action on a category is also a structure and not a property. Suppose is a group action as in the previous post, and is an object. The structure of a fixed point, or more precisely a homotopy fixed point, is the data of a family of isomorphisms
which satisfy the compatibility condition that the two composites
are equal, as well as the unit condition that
where is the unit isomorphism . This is, in a sense we’ll make precise below, a 1-cocycle condition, but this time with nontrivial (local) coefficients.
Curiously, when the action is trivial (meaning both that and that ), this reduces to the definition of a group action of on in the usual sense. In general, we can think of homotopy fixed point structure as a “twisted” version of a group action on where the twist is provided by the group action on .
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