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We have that $G-PBun(X)$, the category of topological principal bundles for a structure group $G$ is equivalent to $Top[X,BG]$ where $BG$ is the classifying space of $G$.

This almost looks like an adjunction - can it be turned into one?

Maybe in a higher-dimensional sense?

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I'm not sure I understand your question. You want adjunction between what what and what? If you're talking about categories $Bun_G(X)$ and $[X,BG]$ — sure, any equivalence of categories can be refined to an adjoint equivalence. –  Grigory M Jan 2 at 16:00
    
I am talking about those two categories. But I can't see how to turn $Bun_G(X)$ into a hom-space between $FX$ and $B$ in some category, for some functor $F$. –  Mozibur Ullah Jan 2 at 16:05

2 Answers 2

up vote 3 down vote accepted

This is not an adjunction, but rather the statement that the $2$-functor which sends a (nice) space $X$ to its groupoid of principal $G$-bundles is representable, namely by $BG$. Actually this is the functorial definition of $BG$.

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There is no left adjoint of $B$ as far as I know, which seems like what youre asking for, but there is in fact a right adjoint. Let $\Omega$ denote the loop space functor sending a space $S$ to the space of based loops in $S$. Then $\Omega$ is a functor to topological monoids where based loops are composed in the obvious way and we have a natural isomorphism

$$ \operatorname{Hom}(BG, X) \cong \operatorname{Hom}(G,\Omega X) $$

where the left $\operatorname{Hom}$ is as pointed topological spaces and the right $\operatorname{Hom}$ is as topological monoids.

This no longer classifies principal $G$-bundles since were looking at maps out of $BG$ but it is an adjunction of the $B$ functor.

For a reference, see this question.

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