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Quotient of a weakly contractible space by a free action
notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete
Classifying_space
indeed the classifying space of O {\displaystyle \operatorname {O} } . Classifying space for U(n) Classifying space for SO(n) Classifying space for SU(n) Milnor
Classifying_space_for_O(n)
Simplicial set constructed from the objects and morphisms of a small category
geometric realization of this simplicial set is a topological space, called the classifying space of the category C. These closely related objects can provide
Nerve_(category_theory)
Exact homotopy case
mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact
Classifying_space_for_U(n)
indeed the classifying space of SO {\displaystyle \operatorname {SO} } . Classifying space for O(n) Classifying space for U(n) Classifying space for SU(n)
Classifying_space_for_SO(n)
Vector bundle of rank 1
universal bundle for complex line bundles. According to general theory about classifying spaces, the heuristic is to look for contractible spaces on which there
Line_bundle
indeed the classifying space of SU {\displaystyle \operatorname {SU} } . Classifying space for O(n) Classifying space for SO(n) Classifying space for U(n) Hatcher
Classifying_space_for_SU(n)
Special tangential structure
is described by a classifying map M → BSO ( n ) {\displaystyle M\rightarrow \operatorname {BSO} (n)} into the classifying space BSO ( n ) {\displaystyle
Spinc_structure
Describes a periodicity in the homotopy groups of classical groups
\ldots \end{aligned}}} For the theory associated to the infinite unitary group, U, the space BU is the classifying space for stable complex vector bundles
Bott_periodicity_theorem
Indication of topological symmetry groups to topological condensed matter
considering the symmetric space in which such Hamiltonians live. These classifying spaces are shown for each symmetry class: For example, a (real symmetric)
Periodic table of topological insulators and topological superconductors
Periodic_table_of_topological_insulators_and_topological_superconductors
Theorem in homotopy theory
Burnside ring of a finite group G to the stable cohomotopy of the classifying space BG. The conjecture was made in the mid 1970s by Graeme Segal and proved
Segal's_conjecture
Finite topological space with two points, only one of which is closed
Sierpiński. The Sierpiński space has important relations to the theory of computation and semantics, because it is the classifying space for open sets in the Scott
Sierpiński_space
Branch of mathematics
of classifying spaces. The idea that a classifying space classifies principal bundles can be pushed further. For example, one might try to classify cohomology
Homotopy_theory
Mathematical concept
projective space plays an important role as a classifying space for complex line bundles: families of complex lines parametrized by another space. In this
Complex_projective_space
Springer-Verlag, ISBN 0-387-97710-4, MR 1300636 Moerdijk, I. (1995), Classifying spaces and classifying topoi, Lecture Notes in Mathematics, vol. 1616, Berlin: Springer-Verlag
Classifying_topos
Group of unitary matrices
identically zero. The classifying space for U ( n ) {\displaystyle \operatorname {U} (n)} is described in the article classifying space for U(n). Orthogonal
Unitary_group
Spherical fibrations over a space X are classified by the homotopy classes of maps X → B G {\displaystyle X\to BG} to a classifying space B G {\displaystyle BG}
Stable_normal_bundle
Topics referred to by the same term
\operatorname {BO} (n)} , Classifying space for orthogonal group BO {\displaystyle \operatorname {BO} } , Classifying space for infinite orthogonal group
Bo
Theorem about cohomology rings
due to Armand Borel (1953), says the cohomology ring of a classifying space or a classifying stack is a polynomial ring. Atiyah–Bott formula Behrend 2003
Borel's_theorem
Quotient of special unitary group by its center
a contractible space with a U(1) action, which identifies it as EU(1) and the space of U(1) orbits as BU(1), the classifying space for U(1). P U ( H )
Projective_unitary_group
Special tangential structure
is described by a classifying map M → BSO ( n ) {\displaystyle M\rightarrow \operatorname {BSO} (n)} into the classifying space BSO ( n ) {\displaystyle
Spinh_structure
Concept in mathematics
classifying space for the Artin braid group, and Conf n ( R 2 ) {\displaystyle \operatorname {Conf} _{n}(\mathbf {R} ^{2})} is a classifying space for
Configuration space (mathematics)
Configuration_space_(mathematics)
Construction for vector bundles
vector bundle over paracompact spaces a line bundle. Its name comes from using the determinant on their classifying spaces. Determinant line bundles naturally
Determinant_line_bundle
Mathematical set with some added structure
points can vary widely: for example, the points can represent numbers, functions on another space, or subspaces of another space. It is the relationships
Space_(mathematics)
structure group a given topological group G, is a specific bundle over a classifying space BG, such that every bundle with the given structure group G over M
Universal_bundle
Tools for studying groups based on techniques from algebraic topology
(usually called an Eilenberg–MacLane space K ( G , 1 ) {\displaystyle K(G,1)} ). For example, classifying spaces for Z , Z / 2 {\displaystyle \mathbb {Z}
Group_cohomology
Symmetry group of a configuration in space
finite group acting faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by
Space_group
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
unitary group U in Bott's sense has a classifying space BU for complex vector bundles (see Classifying space for U(n)). A deeper application coming from
Kuiper's_theorem
Algebraic topology theory
{\displaystyle X} is contractible, it reduces to the cohomology ring of the classifying space B G {\displaystyle BG} (that is, the group cohomology of G {\displaystyle
Equivariant_cohomology
Topological space with only one nontrivial homotopy group
of K ( G , 1 ) {\displaystyle K(G,1)} is identical to that of the classifying space of the group G {\displaystyle G} . Note that if G has a torsion element
Eilenberg–MacLane_space
Subject area in mathematics
maps from the classifying spaces BGL(Fq) to the homotopy fiber of ψq − 1, where ψq is the qth Adams operation acting on the classifying space BU. This map
Algebraic_K-theory
Characteristic classes of vector bundles
mapping associated with a vector bundle to a classifying space (an infinite Grassmannian in this case). For any complex vector bundle V over a manifold
Chern_class
Structure group sub-bundle on a tangent frame bundle
{\displaystyle \pi \colon X\to BG} , where B G {\displaystyle BG} is the classifying space for G {\displaystyle G} -bundles, a reduction of the structure group
G-structure_on_a_manifold
Group whose operation is a composition of braids
{\displaystyle G} up to homotopy. A classifying space for the braid group B n {\displaystyle B_{n}} is the nth unordered configuration space of R 2 {\displaystyle \mathbb
Braid_group
{\displaystyle [S/G]} is called the classifying stack of G {\displaystyle G} (in analogy with the classifying space of G {\displaystyle G} ) and is usually
Quotient_stack
important for higher-dimensional versions of groups. Any crossed module M = ( d : H ⟶ G ) {\displaystyle M=(d\colon H\longrightarrow G)\!} has a classifying space
Crossed_module
Conjecture linking two mathematical areas
K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of proper actions of that group. The conjecture sets up a correspondence
Baum–Connes_conjecture
group of X. Also directly from the definition, an aspherical space is a classifying space for its fundamental group (considered to be a topological group
Aspherical_space
Fiber bundle whose fibers are group torsors
group G admits a classifying space BG: the quotient by the action of G of some weakly contractible space, e.g., a topological space with vanishing homotopy
Principal_bundle
Two theorems needed for Quillen's Q-construction in algebraic K-theory
sufficient condition for the classifying spaces of two categories to be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting
Quillen's_theorems_A_and_B
Association of cohomology classes to principal bundles
this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear
Characteristic_class
Topological spaces whose union is a boundary
spectrum is composed from the Thom spaces MGn of the standard vector bundles over the classifying spaces BGn. Note that even for similar groups, Thom spectra
Cobordism
Algebraic construct classifying topological spaces
mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group
Homotopy_group
Tranquil, hypnotic subgenre of electronic music
Rhapsody all classify space music as a subgenre of new-age music. Rhapsody's editorial staff writes in their music genre description for space music (listed
Space_music
American mathematician
Project Miller, Haynes (1984). "The Sullivan Conjecture on Maps from Classifying Spaces". Annals of Mathematics. 120 (1): 39–87. doi:10.2307/2007071. JSTOR 2007071
Haynes_Miller
Topics referred to by the same term
common shorthand for butyl, a functional group in organic chemistry BU ( n ) {\displaystyle \operatorname {BU} (n)} , Classifying space for unitary group
BU
Mathematical construction used in homotopy theory
descriptions of classifying spaces of groups. This idea was vastly extended by Grothendieck's idea of considering classifying spaces of categories, and
Simplicial_set
space machine produces a group completion of X together with infinite loop space structure. For example, one can take X to be the classifying space of
Infinite_loop_space_machine
Embedding of data within a manifold based on a similarity function
as feature spaces in machine learning models, including classifiers and other supervised predictors. The interpretation of latent spaces in machine learning
Latent_space
related to Quillen's plus construction on the classifying space BG. An acyclic group is a group G whose classifying space BG is acyclic; in other words, all
Acyclic_space
Japanese mathematician
Segal conjecture, transfer homomorphisms, and stable splittings of classifying spaces of groups. The ideas in this series of papers have by now grown into
Goro_Nishida
Modular space station in low Earth orbit
The International Space Station (ISS) is a space station in low Earth orbit (LEO). It is the product of the International Space Station program and is
International_Space_Station
Connects the homology of the symmetric groups with mapping spaces of spheres
the sphere spectrum and the classifying spaces of the symmetric groups via Quillen's plus construction. The mapping space Map 0 ( S n , S n ) {\displaystyle
Barratt–Priddy_theorem
Mathematical conjecture
mathematics, Sullivan conjecture or Sullivan's conjecture on maps from classifying spaces can refer to any of several results and conjectures prompted
Sullivan_conjecture
Mathematical object in category theory
the classifying morphism for the subobject represented by j. The category of sheaves of sets on a topological space X has a subobject classifier Ω which
Subobject_classifier
Vector bundle existing over a Grassmannian
bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles. Because
Tautological_bundle
NASA/ESA space telescope launched in 1990
the first space telescope, but it is one of the largest and most versatile, renowned as a vital research tool and as a public relations boon for astronomy
Hubble_Space_Telescope
Set on which a group acts freely and transitively
the classifying space B G {\displaystyle BG} . Homogeneous space Heap (mathematics) Serge Lang and John Tate (1958). "Principal Homogeneous Space Over
Principal_homogeneous_space
Statistics and machine learning technique
output of each individual classifier or regressor for the entire dataset can be viewed as a point in a multi-dimensional space. Additionally, the target
Ensemble_learning
Type of topological space
_{n}\mathbf {RP} ^{n}.} This space is classifying space of O(1), the first orthogonal group. The double cover of this space is the infinite sphere S ∞ {\displaystyle
Real_projective_space
Topological construct
Homeo ( F ) ) {\displaystyle B(\operatorname {Homeo} (F))} is the classifying space of Homeo ( F ) {\displaystyle \operatorname {Homeo} (F)} . Here
Clutching_construction
Definition of continuity for functions between posets
Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets
Scott_continuity
Topics referred to by the same term
\operatorname {BSU} (n)} , Classifying space for special unitary group BSU {\displaystyle \operatorname {BSU} } , Classifying space for infinite special unitary group
BSU
Probabilistic classification algorithm
especially popular for classifying short texts. It has the benefit of explicitly modelling the absence of terms. Note that a naive Bayes classifier with a Bernoulli
Naive_Bayes_classifier
American mathematician (born 1941)
conjecture, proved in its original form by Haynes Miller, states that the classifying space BG of a finite group G is sufficiently different from any finite CW
Dennis_Sullivan
Links the homology groups of a product space with those of the individual spaces
result on the (simplicial) classifying space of a crossed complex stated but not proved in the paper by Ronald Brown and Philip J. Higgins on classifying spaces
Eilenberg–Zilber_theorem
Mathematical property
} (a classifying space for Γ {\displaystyle \Gamma } ) and whose n-skeleton is finite. A group is said to be of type F∞ if it is of type Fn for every
Finiteness properties of groups
Finiteness_properties_of_groups
Mathematician
theory of the classifying space of a finite group. It is the analogue for cohomotopy of the work of Michael Atiyah and Graeme Segal on the K-theory of
Gunnar_Carlsson
Set of topological invariants
the notion of classifying space. For any vector space V, let G r n ( V ) {\displaystyle Gr_{n}(V)} denote the Grassmannian, the space of n-dimensional
Stiefel–Whitney_class
G-bundle over S is the same as to give a map (called a classifying map) from S to the classifying space B G {\displaystyle BG} . A similar phenomenon in algebraic
Functor represented by a scheme
Functor_represented_by_a_scheme
Algebraic topology uses abstract algebra to study topological spaces
topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to
List of algebraic topology topics
List_of_algebraic_topology_topics
Lie group of complex numbers of unit modulus; topologically a circle
ISBN 978-0-387-90613-3. Mitchell, Stephen A. (2011). "Notes on Principal Bundles and Classifying Spaces" (PDF). University of Washington. Nakahara, Mikio (2003)
Circle_group
Combination of higher category theory with Chern–Weil theory
{Z} )} BU ( n ) {\displaystyle \operatorname {BU} (n)} is the classifying space for the unitary group U ( n ) {\displaystyle \operatorname {U} (n)}
∞-Chern–Weil_theory
Type of word or affix that is used to accompany nouns
A classifier (abbreviated clf or cl) is a word or affix that accompanies nouns and can be considered to "classify" a noun depending on some characteristics
Classifier_(linguistics)
Statistical classification in machine learning
In machine learning, a linear classifier makes a classification decision for each object based on a linear combination of its features. A simpler definition
Linear_classifier
Space of all possible states that a system can take
state corresponds uniquely to a point in the phase space. For mechanical systems, the phase space usually consists of all possible values of the position
Phase_space
notation "+" is meant to suggest the construction adds more to the classifying space BC.) One puts K i ( C ) = π i ( B + C ) {\displaystyle K_{i}(C)=\pi
Q-construction
of the assembly maps are equivariant homology theory evaluated on the classifying space of G with respect to the family of virtually cyclic subgroups of
Farrell–Jones_conjecture
Hong Kong mathematician, spouse of Carrie Lam
Archived from the original on 2 April 2017. Retrieved 28 March 2017. "短期课程班--李群分类空间的同调群 Homology of classifying spaces of Lie groups". School of Mathematical
Lam_Siu-por
American mathematician
space Xn is contractible. Thus the quotient space Xn /Out(Fn) is "almost" a classifying space for Out(Fn) and it can be thought of as a classifying space
Karen_Vogtmann
instance space decomposition, which splits a complete multi-class problem into a set of smaller classification problems. Deductive classifier Cascading
Hierarchical_classification
Mathematical theory
either compact or semi-simple, then the cohomology ring of the classifying space for G-bundles, B G {\displaystyle BG} , is isomorphic to the algebra
Chern–Weil_homomorphism
Mathematical result about equivariant K-theory in homotopy theory
K^{*}(BG)\cong R(G)_{\widehat {I\,}}} between the K-theory of the classifying space of G and the completion of the representation ring. The theorem can
Atiyah–Segal completion theorem
Atiyah–Segal_completion_theorem
Type of mathematical space
→ G/H is a principal H-bundle, there exists a classifying map G/H → BH with target the classifying space BH. If we replace G/H with the homotopy quotient
Generalized_flag_variety
arXiv:math.QA/0307200 Baez, John C.; Stevenson, Danny (2009), "The classifying space of a topological 2-group", in Baas, Nils; Friedlander, Eric; Jahren
2-group
Set of methods for supervised statistical learning
grid search. The final model, which is used for testing and for classifying new data, is then trained on the whole training set using the selected parameters
Support_vector_machine
Non-parametric classification method
neighbour type classifier is the one nearest neighbour classifier that assigns a point x to the class of its closest neighbour in the feature space, that is
K-nearest_neighbors_algorithm
Branch of mathematics that studies the properties of groups
prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces of groups. Finally, the name of the torsion subgroup of an infinite
Group_theory
Branch of algebraic topology
] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups: B U ( n ) ≅ Gr ( n , C ∞ ) . {\displaystyle
Topological_K-theory
(pseudo-)Riemannian manifold whose geodesics are reversible
Riemannian symmetric space is a Riemannian product of irreducible ones. Therefore, we may further restrict ourselves to classifying the irreducible, simply
Symmetric_space
Type of topology
General topology. Springer-Verlag. p. 230. McCord, M. C. (1969). "Classifying Spaces and Infinite Symmetric Products". Transactions of the American Mathematical
Compact-open_topology
Mathematics glossary
classifying space Loosely speaking, a classifying space is a space representing some contravariant functor defined on the category of spaces; for example
Glossary of algebraic topology
Glossary_of_algebraic_topology
{K}} , that induces identity on the homotopy groups up to the dimension of K {\displaystyle {K}} . Then the classifying space B a u t I K {\displaystyle
Nilpotent_space
Tongue-in-cheek description of category theory and abstract mathematics
domains, unified by category theory. Typical methods include the use of classifying spaces and universal properties, use of the Yoneda lemma, natural transformations
Abstract_nonsense
Infinite series summing alternating 1 and -1 terms
Euler characteristic of any category which bypasses the classifying space and reduces to 1/|G| for any group when viewed as a one-object category. In this
Grandi's_series
Unsolved problem in topology
a discrete group and B G {\displaystyle BG} its classifying space, which is an Eilenberg–MacLane space of type K ( G , 1 ) {\displaystyle K(G,1)} , and
Novikov_conjecture
Mathematical space
classifying spaces in K-theory, notably the classifying space for U(n). In the homotopy theory of schemes, the Grassmannian plays a similar role for algebraic
Grassmannian
Morphological system
single external argument) There have been many attempts at classifying the types of classifiers. The number of proposed types have ranged from two to seven
Classifier constructions in sign languages
Classifier_constructions_in_sign_languages
weak Hausdorff spaces", Archiv der Mathematik, 32 (5): 487–504, doi:10.1007/BF01238530, MR 0547371. McCord, M. C. (1969), "Classifying spaces and infinite
Weak_Hausdorff_space
Concept in mathematics
P n {\displaystyle \mathbb {HP} ^{n}} 's under inclusion, is the classifying space BS3. The homotopy groups of H P ∞ {\displaystyle \mathbb {HP} ^{\infty
Quaternionic_projective_space
CLASSIFYING SPACE-FOR-ON
CLASSIFYING SPACE-FOR-ON
Boy/Male
Hindu
Space
Boy/Male
Tamil
Antrix | அஂதà¯à®°à¯€à®•à¯à®·
Space
Antrix | அஂதà¯à®°à¯€à®•à¯à®·
Surname or Lastname
English and Irish
English and Irish : variant of Stacey.
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Surname or Lastname
English
English : metonymic occupational name for a spicer (see Spicer).
Male
English
English surname transferred to forename use, derived from the French personal name Pascal, PACE means "Passover; Easter."
Boy/Male
Tamil
Antareeksh | அஂதரீகà¯à®·
Space
Antareeksh | அஂதரீகà¯à®·
Surname or Lastname
English
English : from the Old Norse personal name Spakr.Respelling of Jewish, Ukrainian, and Belorussian Shpak, a nickname from Ukrainian and Belorussian shpak ‘starling’. In the case of Jewish bearers, it is generally an ornamental name.
Boy/Male
Hindu, Indian
Space; Outer Space; Sky
Boy/Male
Tamil
Antariksh | அஂதரிகà¯à®·
Space
Antariksh | அஂதரிகà¯à®·
Boy/Male
Hindu
Space
Girl/Female
Indian, Telugu
Space
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu
Space; Sky
Girl/Female
Indian, Japanese, Tamil
Space; Star
Boy/Male
Hindu
Space
Surname or Lastname
English
English : variant of Speake.
Surname or Lastname
English
English : from a vernacular short form of the Latin personal name Paschalis (see Pascal, Italian Pasquale).nickname for a mild-mannered and peaceable person, from Middle English pace, pece ‘peace’, ‘concord’, ‘amity’ (via Anglo-Norman French from Latin pax, genitive pacis).Italian : from the medieval personal name Pace, used for both men and women, from the word pace ‘peace’ (see 1).
Surname or Lastname
English or Scottish
English or Scottish : unexplained.
Boy/Male
British, Christian, English, Italian
Form of Pascal; Passover
Girl/Female
Tamil
Antariksha | அஂதரிகà¯à®·
Space, Sky
CLASSIFYING SPACE-FOR-ON
CLASSIFYING SPACE-FOR-ON
Male
Greek
(Λευίς) Greek name LEUIS means "joined." In the bible, this is the name of the son of Halphaios (Latin Alphaeus), a collector of customs.
Boy/Male
Afghan, Arabic, Australian, Muslim
Accounter; Omnipotent; Another Name of Allah
Surname or Lastname
English (County Durham)
English (County Durham) : most probably a habitational name from a lost or unidentified place in northern England.
Boy/Male
Gujarati, Hindu, Indian, Kannada
Famous
Male
Scottish
Scottish Gaelic form of Latin Alexandrus, ALASTAIR means "defender of mankind."
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Moon
Girl/Female
Indian
Night
Female
English
French form of Latin Susanna, SUZANNE means "lily."
Boy/Male
Indian
Part of Mud
Girl/Female
Muslim
Prosperous
CLASSIFYING SPACE-FOR-ON
CLASSIFYING SPACE-FOR-ON
CLASSIFYING SPACE-FOR-ON
CLASSIFYING SPACE-FOR-ON
CLASSIFYING SPACE-FOR-ON
v. t.
To dig with a spade; to pare off the sward of, as land, with a spade.
v. t.
To season with spice, or as with spice; to mix aromatic or pungent substances with; to flavor; to season; as, to spice wine; to spice one's words with wit.
p. pr. & vb. n.
of Classify
n.
Figuratively, that which enriches or alters the quality of a thing in a small degree, as spice alters the taste of food; that which gives zest or pungency; a slight flavoring; a relish; hence, a small quantity or admixture; a sprinkling; as, a spice of mischief.
v. t.
Being over and above what is necessary, or what must be used or reserved; not wanted, or not used; superfluous; as, I have no spare time.
prep.
Indicating the space or time through which an action or state extends; hence, during; in or through the space or time of.
imp. & p. p.
of Space
n.
Space.
n.
One of that suit of cards each of which bears one or more figures resembling a spade.
adv.
With a quick pace; quick; fast; speedily.
v. t.
Scanty; not abundant or plentiful; as, a spare diet.
v. t.
To measure by steps or paces; as, to pace a piece of ground.
n.
The right of bowling again at a full set of pins, after having knocked all the pins down in less than three bowls. If all the pins are knocked down in one bowl it is a double spare; in two bowls, a single spare.
n.
A quantity or portion of extension; distance from one thing to another; an interval between any two or more objects; as, the space between two stars or two hills; the sound was heard for the space of a mile.
adv.
To a great extent or distance of space; widely; as, we are separated far from each other.
v. t.
To develop, guide, or control the pace or paces of; to teach the pace; to break in.
n.
Manner of stepping or moving; gait; walk; as, the walk, trot, canter, gallop, and amble are paces of the horse; a swaggering pace; a quick pace.
v. t.
Held in reserve, to be used in an emergency; as, a spare anchor; a spare bed or room.
a.
Distant in any direction; not near; remote; mutually separated by a wide space or extent.
n.
To arrange or adjust the spaces in or between; as, to space words, lines, or letters.