In
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Quillen's Theorem A gives a sufficient condition for the
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free ...
s of two
categories
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
*Category (Vais ...
to be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting of classifying spaces of categories to be
homotopy Cartesian
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
. The two theorems play central roles in Quillen's
Q-construction In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact category ''C'', the construction creates a topological space B^+C so that \pi_0 (B^+C) is the Gr ...
in
algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sens ...
and are named after
Daniel Quillen
Daniel Gray Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978.
Fr ...
.
The precise statements of the theorems are as follows.
In general, the homotopy fiber of
is not naturally the classifying space of a category: there is no natural category
such that
. Theorem B constructs
in a case when
is especially nice.
References
*
*
*
*
Theorems in topology
{{topology-stub