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.