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I recently read the following statement:

"For any type of mathematical object, an object of that type with $G$ symmetry “is” a map from [its classifying space] $BG$ to the space of all objects of that type".

The quote is taken from https://arxiv.org/abs/1712.07950 (page 10).

To give a specific example, if $P$ denotes a group of gapped phases of matter, then maps from $BG$ to $P$ correspond to gapped phases of matter with $G$ symmetry.

Could you please explain why this is true? I am unfamiliar with the notion of classifying spaces.

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