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Concept in homotopy theory
mapping between topological spaces i : A → X {\displaystyle i:A\to X} is a cofibration if it has the homotopy extension property with respect to all topological
Cofibration
Continuous, position-preserving mapping from a topological space into a subspace
(Hurewicz) cofibration if it has the homotopy extension property for maps to any space. This is one of the central concepts of homotopy theory. A cofibration f
Retraction_(topology)
Branch of mathematics
and the notion of a cofibration there is then often implicit. A fibration in the sense of Hurewicz is the dual notion of a cofibration: that is, a map p
Homotopy_theory
Mathematical category with weak equivalences, fibrations and cofibrations
of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category
Model_category
with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms. The associated homotopy category of a model
Weak equivalence (homotopy theory)
Weak_equivalence_(homotopy_theory)
Topological construction
map of topological spaces by a homotopy equivalent cofibration. Note that pointwise, a cofibration is a closed inclusion. Mapping cylinders are quite
Mapping_cylinder
Property in algebraic topology
homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations
Homotopy_extension_property
Theory in algebraic topology
rise to cohomology theories. We can also directly relate fibrations and cofibrations: a fibration p : E → B {\displaystyle p\colon E\to B} is defined by having
Eckmann–Hilton_duality
equivalences, cofibrations and fibrations, respectively, are the C-local equivalences the original cofibrations of M and (necessarily, since cofibrations and weak
Bousfield_localization
Category theory
Waldhausen, who introduced this notion (under the term category with cofibrations and weak equivalences) to extend the methods of algebraic K-theory to
Waldhausen_category
Concept in algebraic topology
E\to B.} Due to the duality of fibration and cofibration, there also exists a sequence of cofibrations. These two sequences are known as the Puppe sequences
Fibration
Concept in algebra
with cofibrations and weak equivalences; such a category is called a Waldhausen category and generalizes Quillen's exact category. A cofibration can be
K-theory_of_a_category
Model structure on the category of simplicial sets
three classes of morphisms between simplicial sets called fibrations, cofibrations and weak equivalences, which fulfill the properties of a model structure
Kan–Quillen_model_structure
Model structure on the category of simplicial sets
three classes of morphisms between simplicial sets called fibrations, cofibrations and weak equivalences, which fulfill the properties of a model structure
Joyal_model_structure
Mathematical concept
characterized by having a right lifting property with respect to any trivial cofibration in the category. This property makes fibrant objects the "correct" objects
Fibrant_object
and a long coexact sequence, built from the mapping cone (which is a cofibration). Intuitively, the Puppe sequence allows us to think of homology theory
Puppe_sequence
Set-theoretic function
commutative ring and I {\displaystyle I} is an ideal of R . {\displaystyle R.} Cofibration – Concept in homotopy theory Identity function – Function that returns
Inclusion_map
Type of group in mathematics
_{k=0}^{\infty }\operatorname {O} (k)} Since the inclusions are all closed, hence cofibrations, this can also be interpreted as a union. On the other hand, Sn is a
Orthogonal_group
Continuous deformation between two continuous functions
subset of some set to the set itself. It is useful when dealing with cofibrations. Since the relation of two functions f , g : X → Y {\displaystyle f,g\colon
Homotopy
Topological construction on a map between spaces
a homology theory, and i : A → X {\displaystyle i\colon A\to X} is a cofibration, then E ∗ ( X , A ) = E ∗ ( X / A , ∗ ) = E ~ ∗ ( X / A ) {\displaystyle
Mapping_cone_(topology)
Theory of algebraic structures in general
Some authors also require the identity map to be a closed inclusion (a cofibration). Most algebraic structures are examples of universal algebras. Rings
Universal_algebra
Topological subject
the suspension functor becomes invertible. For example, the notion of cofibration sequence and fibration sequence are equivalent. Adams filtration Adams
Stable_homotopy_theory
Category theory generalization of fumction factorization
category C and classes of (so-called) weak equivalences W, fibrations F and cofibrations C so that C has all limits and colimits, ( C ∩ W , F ) {\displaystyle
Factorization_system
last property, an anodyne extension is also known as an acyclic cofibration (a cofibration that is a weak equivalence). Also, the weak equivalences between
Fibration_of_simplicial_sets
Concept in algebraic topology
A\hookrightarrow X} . Sometimes i {\displaystyle i} is assumed to be a cofibration. A morphism from ( X , A ) {\displaystyle (X,A)} to ( X ′ , A ′ ) {\displaystyle
Topological_pair
{\displaystyle \{*\}\hookrightarrow [0,1]} be a cofibration, a weak equivalence, both (trivial cofibration) or none. For example, if we suppose { 0 } ↪ [
Directed_algebraic_topology
Transforming a function in such a way that it only takes a single argument
the dual. The duality between the mapping cone and the mapping fiber (cofibration and fibration) can be understood as a form of currying, which in turn
Currying
Special kind of model structure
object argument can be applied, so that they generate all cofibrations and trivial cofibrations using the lifting property: Cofib = ⊥ ( I ⊥ ) ; {\displaystyle
Cisinski_model_structure
Collection of maps which give the same result
also used for injections, surjections, and bijections, as well as the cofibrations, fibrations, and weak equivalences in a model category. Commutativity
Commutative_diagram
Application of homotopy to algebraic varieties
equivalence. f is a cofibration if it is a monomorphism. f is a fibration if it has the right lifting property with respect to any cofibration which is a weak
A¹_homotopy_theory
Mathematical construction used in homotopy theory
structure on the category of simplicial sets, one has to define fibrations, cofibrations and weak equivalences. One can define fibrations to be Kan fibrations
Simplicial_set
Algebraic construct classifying topological spaces
of the fiber. When the fibration is the mapping fibre, or dually, the cofibration is the mapping cone, then the resulting exact (or dually, coexact) sequence
Homotopy_group
Mathematical concept
{\displaystyle \operatorname {Hom} } ". Terms like "cohomology" and "cofibration" all have a slightly stronger association with the first variable, i
Limit_(category_theory)
Topological space with only one nontrivial homotopy group
) {\displaystyle K(G,n)\to *\to K(G,n+1)} . Note that this is not a cofibration sequence ― the space K ( G , n + 1 ) {\displaystyle K(G,n+1)} is not
Eilenberg–MacLane_space
Concepts in algebraic topology
of homotopy pushouts, such as the mapping cylinder used to define a cofibration. This notion is motivated by the following observation: the (ordinary)
Homotopy_colimit_and_limit
Tool in homological algebra
spectral sequence converging to the homotopy of the initial space of a cofibration. Bousfield–Kan spectral sequence converging to the homotopy colimit of
Spectral_sequence
Subject area in mathematics
simplicial category S⋅C (the S is for Segal) defined in terms of chains of cofibrations in C. This freed the foundations of K-theory from the need to invoke
Algebraic_K-theory
Special kind of adjunction between categories named after Daniel Quillen
functors with F left adjoint to G such that F preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that G
Quillen_adjunction
General concept and operation in mathematics
projective and injective modules in homological algebra, fibrations and cofibrations in topology and more generally model categories. Two functors F: C →
Duality_(mathematics)
Mathematics glossary
cofiber of ƒ). cofibrant approximation cofibration A map i : A → B {\displaystyle i:A\to B} is a cofibration if it satisfies the property: given h 0
Glossary of algebraic topology
Glossary_of_algebraic_topology
Correspondence between properties of a category and its opposite
applied to lattices. Limits and colimits are dual notions. Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory
Dual_(category_theory)
Algebraic topology uses abstract algebra to study topological spaces
Vector bundle Associated bundle Fibration Hopf bundle Classifying space Cofibration Homotopy groups of spheres Plus construction Whitehead theorem Weak equivalence
List of algebraic topology topics
List_of_algebraic_topology_topics
for G if G is a topological group such that the inclusion {1} → G is a cofibration. Morton, H. R. (1967). "Symmetric Products of the Circle". Mathematical
Symmetric_product_(topology)
History of maths
spaces with homotopy equivalences as weak equivalences, Hurewicz cofibrations as cofibrations and Hurewicz fibrations as fibrations form an ABC model category
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Injective cofibrations and injective weak equivalences are the natural transformations, which componentswise only consist of cofibrations and weak equivalences
Injective and projective model structure
Injective_and_projective_model_structure
Endofunctor on the category of simplicial sets
}(X)} is a monomorphism and a weak homotopy equivalence, hence a trivial cofibration of the Kan–Quillen model structure. Ex ∞ ( X ) {\displaystyle \operatorname
Extension_(simplicial_set)
distinction; for example, an op-fibration is not the same thing as a cofibration. codensity monad Codensity monad. coend The coend of a functor F : C
Glossary_of_category_theory
Special kind of model structure
fibrations, called right proper, and pushouts (cofiber product) along cofibrations, called left proper. It is helpful to construct weak equivalences and
Proper_model_structure
Proposition— Let i : A → X , j : A → Y {\displaystyle i:A\to X,j:A\to Y} be cofibrations. Then a map f : X → Y {\displaystyle f:X\to Y} under A is a homotopy
Fiber-homotopy_equivalence
Four mathematical theorems
Waldhausen Localization Theorem—Let A {\displaystyle A} be the category with cofibrations, equipped with two categories of weak equivalences, v ( A ) ⊂ w ( A )
Basic theorems in algebraic K-theory
Basic_theorems_in_algebraic_K-theory
Homological construction in category theory
categories, which give an abstract category-theoretic system of fibrations, cofibrations and weak equivalences. Typically one is interested in the underlying
Derived_functor
Concept category theory (mathematics)
C_{0}^{\perp \ell r}} are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations. Let sSet be the category of simplicial sets. Let
Lifting_property
injective model structure is similar, but with weak equivalences and cofibrations instead. A simplicial presheaf F on a site is called a stack if, for
Simplicial_presheaf
Concept in math
category C with three distinguished types of morphisms called fibrations, cofibrations and weak equivalences, satisfying several axioms. The associated homotopy
Homotopy_category
Endofunctor on the category of simplicial sets
property) as well as anodyne extensions in combination, hence cofibrations and trivial cofibrations of the Kan–Quillen model structure. This makes the adjunction
Subdivision_(simplicial_set)
structure on s S e t / A {\displaystyle \mathbf {sSet} /A} . Covariant cofibrations are monomorphisms. Covariant fibrant objects are the left fibrant objects
Co- and contravariant model structure
Co-_and_contravariant_model_structure
Similarly, if C is a hereditary ∞-category with weak fibrations and cofibrations, then L ( Hom _ ( I , C ) ) → ∼ Hom _ ( I , L ( C ) ) {\displaystyle
Localization_of_an_∞-category
COFIBRATION
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Boy/Male
Arabic
Long
Boy/Male
Indian, Punjabi, Sikh
Elixir of the Truth
Boy/Male
Tamil
Akendra | அகேநà¯à®¤à¯à®°
Name of a God
Male
English
English short form of Latin Hector, HECK means "defend; hold fast."
Girl/Female
German
Shining; Brilliant
Girl/Female
Indian
Birth
Girl/Female
Hindu
Light
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : of uncertain origin, probably from Middle English metecalf ‘food calf’, i.e. a calf being fattened up for eating at the end of the summer. It is thus either an occupational name for a herdsman or slaughterer, or a nickname for a sleek and plump individual, from the same word in a transferred sense. The variants in med- appear early, and suggest that the first element was associated by folk etymology with Middle English mead ‘meadow’, ‘pasture’.
Surname or Lastname
English
English : habitational name from Wingrave in Buckinghamshire, probably named in Old English as ‘grove (Old English grÄf) of the family or followers of (-inga-) of a man named WÄ«ga’.
Girl/Female
Tamil
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