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Jan, 2021
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Classifying Spaces

We say a space B is a classifying space if it classifies a certain geometric objects (structures) on a space X as the homotopy set [ X,B ].

One of the origins is the classification theorem of principal G -bundles by Steenrod [ Ste51 ] . And then the theorem and the concept of classifying spaces have been extended in various directions.

In general, the classifying space of a group object in a category of geometric objects cannot be constructed in the category. Nevertheless there are attempts to study geometric structures on classifying spaces. When G is a discrete group, the classifying space of G is K ( G, 1).

One of the most general forms is Brown’s representability theorem.

On the other hand, classifying spaces of small categories are defined by mimicking the construction of the classifying space of a topological group. There seemed to be no motivation for classifying geometric objects.

One of interpretations is given by M. Weiss [ Wei05 ] .