What is a good notation for the 'set' (or stack if you insist)
of all principal G bundles over 'all' spaces for given G?
BG is way over used. How about Bun(G)?



The `set' of all principal G bundles over `all' spaces
In my comment to the question, I may have misinterpreted Jim's intent. Upon a second reading, I think Jim would like to consider the collection of all pairs consisting of a space $X$ and a $G$-bundle over $X$. If this is the correct interpretation, then I think the correct notation for this category is $\text{Spaces}\_{/\mathrm{B}G}$. The meaning of this notation is: $\mathrm{B}G = \operatorname{Bun}\_G$ is the stack of $G$-bundles, defined so that a map $X \to \mathrm{B}G$ *is* a $G$-bundle over $X$. The category $\text{Spaces}$ is whatever category of spaces you want to consider (topological spaces, manifolds, schemes, ...). The category of pairs $(X,$ $G$-bundle over $X)$ is the category of pairs $(X,f: X \to \mathrm{B}G)$, which is precisely the "comma" category of "spaces over $\mathrm{B}G$", and that's what's notated by the subscript.


