

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
]
.


