After a relaxing and enjoyable break, we’re finally in a position to state what it means for structures to satisfy Galois descent.
Fix a field . The gadgets we want to study assign to each separable extension a category of “objects over ,” to each morphism of extensions an “extension of scalars” functor , and to each composable pair of morphisms of extensions a natural isomorphism
of functors (where again we’re taking compositions in diagrammatic order) satisfying the usual cocycle condition that the two natural isomorphisms we can write down from this data agree. We’ll also want unit isomorphisms satisfying the same compatibility as before. This is just spelling out the definition of a 2-functor from the category of separable extensions of to the 2-category , and in particular each naturally acquires an action of (where we mean automorphisms of extensions of , hence if is Galois this is the Galois group) in precisely the sense we described earlier.
We’ll call such an object a Galois prestack (of categories, over ) for short. The basic example is the Galois prestack of vector spaces , which sends an extension to the category of -vector spaces and sends a morphism to the extension of scalars functor
Every example we consider will in some sense be an elaboration on this example in that it will ultimately be built out of vector spaces with extra structure, e.g. the Galois prestacks of commutative algebras, associative algebras, Lie algebras, and even schemes. In these examples, fields are not really the natural level of generality, and to make contact with algebraic geometry we should replace them with commutative rings, but for now we’ll ignore this.
In order to state the definition, we need to know that if is an extension, then the functor naturally factors through the category of homotopy fixed points for the action of on . We’ll elaborate on why this is in a moment.
Definition: A Galois prestack satisfies Galois descent, or is a Galois stack, if for every Galois extension the natural functor (where ) is an equivalence of categories.
In words, this condition says that the category of objects over is equivalent to the category of objects over equipped with homotopy fixed point structure for the action of the Galois group (or Galois descent data).
(Edit, 11/18/15:) This definition is slightly incorrect in the case of infinite Galois extensions; see the next post and its comments for some discussion.
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