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...

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...

of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category...

map of topological spaces by a homotopy equivalent cofibration. Note that pointwise, a cofibration is a closed inclusion. Mapping cylinders are quite...

(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...

with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms. The associated homotopy category of a model...

commutative ring and I {\displaystyle I} is an ideal of R . {\displaystyle R.} Cofibration Identity function – Function that returns its argument unchanged MacLane...

characterized by having a right lifting property with respect to any trivial cofibration in the category. This property makes fibrant objects the "correct" objects...

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...

Waldhausen, who introduced this notion (under the term category with cofibrations and weak equivalences) to extend the methods of algebraic K-theory to...

three classes of morphisms between simplicial sets called fibrations, cofibrations and weak equivalences, which fulfill the properties of a model structure...

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...

applied to lattices. Limits and colimits are dual notions. Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory...

with cofibrations and weak equivalences; such a category is called a Waldhausen category and generalizes Quillen's exact category. A cofibration can be...

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...

object argument can be applied, so that they generate all cofibrations and trivial cofibrations using the lifting property: Cofib = ⊥ ( I ⊥ ) ; {\displaystyle...

_{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...

three classes of morphisms between simplicial sets called fibrations, cofibrations and weak equivalences, which fulfill the properties of a model structure...

equivalences, cofibrations and fibrations, respectively, are the C-local equivalences the original cofibrations of M and (necessarily, since cofibrations and weak...

also used for injections, surjections, and bijections, as well as the cofibrations, fibrations, and weak equivalences in a model category. Commutativity...

) {\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...

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...

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...

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...

the dual. The duality between the mapping cone and the mapping fiber (cofibration and fibration): chapters 6,7  can be understood as a form of currying...

Vector bundle Associated bundle Fibration Hopf bundle Classifying space Cofibration Homotopy groups of spheres Plus construction Whitehead theorem Weak equivalence...

{\displaystyle \operatorname {Hom} } ". Terms like "cohomology" and "cofibration" all have a slightly stronger association with the first variable, i...

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...

last property, an anodyne extension is also known as an acyclic cofibration (a cofibration that is a weak equivalence). Also, the weak equivalences between...

Some authors also require the identity map to be a closed inclusion (a cofibration). Most algebraic structures are examples of universal algebras. Rings...