Synonyms for cofibration or Related words with cofibration
cofibrations birational functorial epimorphism pushforward antilinear etale unital semilinear homeomorphism surjective bimodule equivariant coalgebra biholomorphic diffeomorphism bialgebra metrizable homeomorphisms surjection isomorphisms cocycle irrationals morphism ultralimit cobordisms monomorphism seminorm involutive functoriality monomorphisms fibrations monodromy paracompact unramified epimorphisms codimension groupoid involutory subalgebras tychonoff cochain isometries homotopic nilpotent koszul morphisms subalgebra cobordism cobordantExamples of "cofibration" |
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The homotopy colimit generalizes the notion of a cofibration. |
Note that pointwise, a cofibration is a closed inclusion. |
and given a cofibration formula_19 we get the sequence |
functor, and the cofibration sequences are the distinguished triangles. |
In any model category, a fibration which is also a weak equivalence is called a trivial (or acyclic) fibration. A cofibration which is also a weak equivalence is called a trivial (or acyclic) cofibration. |
If formula_28 has the homotopy extension property, then the simple inclusion map formula_29 is a cofibration. |
For example, the inclusion of a subcomplex in a CW complex is a cofibration. |
The construction serves to replace any map of topological spaces by a homotopy equivalent cofibration. |
A map "f": "A" → "X" of topological spaces is a (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" is always injective, in fact a homeomorphism to its image. If "X" is Hausdorff (or a compactly generated weak Hausdorff space), then the image of a cofibration "f" is closed in "X". |
A more general notion of cofibration is developed in the theory of model categories. |
and a cofibration formula_16 is defined by having the dual homotopy extension property, represented by dualising the previous diagram: |
In fact, if you consider any cofibration formula_30, then we have that formula_31 is homeomorphic to its image under formula_32. This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property. |
If "E" is a homology theory, and formula_31 is a cofibration, then formula_32, which follows by applying excision to the mapping cone. |
In fact any attempt to define a model structure over some category of directed spaces has to face the following question: should an inclusion map formula_19 be a cofibration, a weak equivalence, both (trivial cofibration) or none. For example, if we suppose formula_20 is a trivial cofibration, then formula_21 (as a subpospace of the directed plane) is equivalent to a point since the collection of trivial cofibrations is stable under pushout. This fact is prohibitive for computer science application though it is a trivial fact from homotopy theory if we drop the direction feature. |
The mapping cylinder may be viewed as a way to replace an arbitrary map by an equivalent cofibration, in the following sense: |
Dually is the notion of cofibrant object, defined to be an object formula_1 such that the unique morphism formula_2 from the initial object to formula_1 is a cofibration. |
The above considerations also apply when looking at the sequences associated to a fibration or a cofibration, as given a fibration formula_17 we get the sequence |
A model structure on a category "C" consists of three distinguished classes of morphisms (equivalently subcategories): weak equivalences, fibrations, and cofibrations, and two functorial factorizations formula_1 and formula_2 subject to the following axioms. Note that a fibration that is also a weak equivalence is called an acyclic (or trivial) fibration and a cofibration that is also a weak equivalence is called an acyclic (or trivial) cofibration (or sometimes called an anodyne morphism). |
In mathematics, more specifically algebraic topology, a pair formula_1 is shorthand for an inclusion of topological spaces formula_2. Sometimes formula_3 is assumed to be a cofibration. A morphism from formula_1 to formula_5 is given by two maps formula_6 and |
The concept of homotopy colimit is a generalization of "homotopy pushouts", such as the mapping cylinder used to define a cofibration. This notion is motivated by the following observation: the (ordinary) pushout |