Cofibration

In mathematics, in particular homotopy theory, a continuous mapping

$i\colon A \to X$,

where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines fibrations. For a more general notion of cofibration see the article about model categories.

Basic theorems

• For Hausdorff spaces a cofibration is a closed inclusion (injective with closed image); for suitable spaces, a converse holds
• Every map can be replaced by a cofibration via the mapping cylinder construction
• There is a cofibration (A, X), if and only if there is a retraction from

$X \times I$

to

$(A \times I) \cup (X \times \{0\})$,

since this is the pushout and thus induces maps to every space sensible in the diagram.

Examples

• Cofibrations are preserved under push-outs and composition, as one sees from the definition via diagram-chasing.
• A frequently used fact is that a cellular inclusion is a cofibration (so, for instance, if (X,A) is a CW pair, then $A \to X$ is a cofibration). This follows from the previous fact since $S^{n-1} \to D^n$ is a cofibration for every n.

References

• Peter May, "A Concise Course in Algebraic Topology" : chapter 6 defines and discusses cofibrations, and they are used throughout

This topology-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Topology stubs Homotopy theory Wikimedia Foundation. 2010. Игры ⚽ Поможем сделать НИР Camp Crame Battle of the Vistula River Look at other dictionaries: Cofibration — En mathématiques, une cofibration est une application qui satisfait la propriété d extension des homotopies, ce qui est le cas pour les inclusions de CW complexes. Le quotient de l espace but par l espace source est alors appelé cofibre de l… …   Wikipédia en Français cofibration — noun A particular form of mapping See Also: fibration …   Wiktionary Model category — In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ( arrows ) called weak equivalences , fibrations and cofibrations . These abstract from a conventional homotopy category, of… …   Wikipedia Mapping cylinder — In mathematics, specifically algebraic topology, the mapping cylinder of a function f between topological spaces X and Y is the quotient where the union is disjoint, and ∼ is the equivalence relation That is, the mapping cylinder Mf …   Wikipedia Fibration — In mathematics, especially algebraic topology, a fibration is a continuous mapping:p:E o B,satisfying the homotopy lifting property with respect to any space. Fiber bundles (over paracompact bases) constitute important examples. In homotopy… …   Wikipedia Weak equivalence — In mathematics, a weak equivalence is a notion from homotopy theory which in some sense identifies objects that have the same basic shape . This notion is formalized in the axiomatic definition of a closed model category.Formal definitionA closed …   Wikipedia Homotopy extension property — In mathematics, in the area of algebraic topology, the homotopy extension property indicates when a homotopy can be extended to another one, so that the original homotopy is simply the restriction of the extended homotopy.DefinitionGiven A subset …   Wikipedia Waldhausen category — In mathematics a Waldhausen category is a category C equipped with cofibrations co( C ) and weak equivalences we( C ), both containing all isomorphisms, both compatible with pushout, and co( C ) containing the unique morphisms :scriptstyle 0,… …   Wikipedia A¹ homotopy theory — In algebraic geometry and algebraic topology, a branch of mathematics, A1 homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to… …   Wikipedia Limit (category theory) — In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”