Here is something introduced today:
- If G is an abstract group, then G-Sets denotes the category of sets endowed with left G-action
- If G is a topological group, then G-Sets denotes the category of sets X endowed with a continuous G-action where X is given the discrete topology.
- If G is an abstract group, then Mod_G denotes the abelian group objects in the category G-Sets.
- If G is a topological group, then Mod_G denotes the abelian group objects in the category G-Sets.
This works well in the sense that if G is the absolute Galois group of a field K, then G-Sets is equivalent to the category of sheaves of sets on the small etale site of Spec(K). Similarly, Mod_G is equivalent to the category of abelian sheaves on the small etale site of Spec(K).
I suppose that if G is a topological group, then one may also want to consider the category of topological spaces endowed with continuous G-action. For this category we could use Top_G, or G-Tops, or G-Spaces (although Spaces/S has been used for the category of algebraic spaces over S already… only a few times and it may be better to give that a really expensive name). Then there is the category of abelian group objects, usually called topological G-modules, in G-Tops/Top_G/G-Spaces, sigh! These already come up briefly in the chapter on etale cohomology when introducing group homology on the category of compact topological G-modules (Warning: This part is still very rough and not yet cleaned up).
I also changed the definition of a geometric point in the chapter on etale cohomology to require the field in question to be algebraically closed (from just requiring it to be separably closed). Sure, for discussing stalk functors of sheaves on the small etale site you only need separably algebraically closed, but it is just more convenient to have the same definition everywhere.