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**Cofibration**— In mathematics, in particular homotopy theory, a continuous mapping , 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 … Wikipedia**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**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