Examples of "cofibration"
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