TITLE: The `set' of all principal G bundles over `all' spaces
QUESTION [5 upvotes]: 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)?

REPLY [2 votes]: In algebraic geometry, $BG$ only ever refers to a stack, viewed as a fibered category over the category of schemes, or affine schemes, depending on your conventions.  The objects are principal $G$-bundles $P \to X$ of schemes (or if $G$ is a group sheaf, it is a map of sheaves).  The morphisms are $G$-equivariant fiber squares.
More to the point, there is no alternative object in the algebro-geometric universe that takes the name $BG$, so there is no source of confusion there.
The name $\operatorname{Bun}_G$ is usually reserved for the enriched Hom-stack construction $\underline{\operatorname{Hom}}(-, BG)$.  In other words, $\operatorname{Bun}_G(X)(T) = BG(X \times T)$.