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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)?

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Within the context of stacks in algebraic geometry, $BG$ (or $pt/G$) is absolutely the notation that would be used. I suppose you don't like this because it is also the notation for the classifying space of $G$? From my perspective, I don't particularly like the notation $Bun(G)$, only because $Bun_G(X)$ is typically used to denote the moduli of $G$-bundles on the space $X$. But maybe others disagree. I'm struggling to think of a good alternative though. –  Mike Skirvin Mar 6 '12 at 22:59
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I certainly have seen $\operatorname{Bun}(G)$, but I agree with @Mike. Since from a stacky perspective $\operatorname{Bun}_G(X)$ is the space of maps $X \to $ this space that Jim is after, it's not bad to call the target space itself by $\operatorname{Bun}_G$. –  Theo Johnson-Freyd Mar 7 '12 at 1:32
    
I have made a small change to the formatting. The issue is that Markdown considers the backwards single quote ` to be a special character. –  Theo Johnson-Freyd Mar 7 '12 at 1:33
    
May I suggest that my learned colleagues' comments constitute genuine answers to the question asked, and might therefore be submitted as answers? –  Tom Leinster Mar 7 '12 at 1:39
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2 Answers

up vote 6 down vote accepted

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.

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I can live with Spaces/BG since it is not previously used and will encourage the reader to understand what it denotes. –  Jim Stasheff Mar 7 '12 at 13:15
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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, $\operatorname{Bun}_G(X)(T) = BG(X \times T)$.

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Perhaps no confusion within alg-geom, but we are not all alg-geometers. So your BG is a category and alg-geom has no use for a classifying space for G-bundles? or it has some other symbol? –  Jim Stasheff Mar 7 '12 at 13:12
    
S. Carnahan's $BG$ is the classifying space in the algebro-geometric context. You might (I do) find it more intuitive to turn the category fibred in groupoids into a sheaf of groupoids/homotopy 1-types, using the canonical cleavage' construction. Then we are talking about $BG$ as a sheaf of homotopy types on affine schemes. For a scheme $X$, $\operatorname{Bun}_G(X)=\underline{\operatorname{Hom}}(X, BG)$ is another sheaf of homotopy types, and the global sections of $\pi_{0}\operatorname{Bun}_G(X)$ are the isomorphism classes of principal $G$-bundles over $X$, just like in topology. –  Chris Brav Mar 7 '12 at 14:58
    
That does clarify the relation, but notice how many words it required. –  Jim Stasheff Mar 8 '12 at 14:02
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