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
]
.
References
[Ste51]
Norman Steenrod.
The Topology of Fibre Bundles
. Princeton Mathematical Series, vol. 14. Princeton
University Press, Princeton, N. J., 1951.
[Wei05]
Michael Weiss. What does the classifying space of a
category classify?
Homology Homotopy Appl.
, 7(1):185–195 (electronic), 2005.
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