construction: the mapping fibre The dual to the mapping cone is the mapping fibre . Given the pointed map one defines the mapping fiber as . Here, I is the unit interval and is a continuous path in the space (the exponential object) . The mapping fiber is sometimes denoted as ; however this conflicts with the same notation for the mapping cylinder. It is dual to the mapping cone in the sense that the product above is essentially the fibered product or pullback which is dual to the pushout used to construct the mapping cone. In this particular case, the duality is essentially that of currying, in that the mapping cone has the curried form where is simply an alternate notation for the space of all continuous maps from the unit interval to . The two variants are related by an adjoint functor. Observe that the currying preserves the reduced nature of the maps: in the one case, to the tip of the cone, and in the other case, paths to the basepoint. Applications CW-complexes Attaching a cell. Effect on fundamental group Given a space X and a loop representing an element of the fundamental group of X, we can form the mapping cone . The effect of this is to make the loop contractible in , and therefore the equivalence class of in the fundamental group of will be simply the identity element. Given a group presentation by generators and relations, one gets a 2-complex with that fundamental group. Homology of a pair The mapping cone lets one interpret the homology of a pair as the reduced homology of the quotient. Namely, if E is a homology theory, and is a cofibration, then , which follows by applying excision to the mapping cone. Relation to homotopy (homology) equivalences A map between simply-connected CW complexes is a homotopy equivalence if and only if its mapping cone is contractible. More generally, a map is called n-connected (as a map) if its mapping cone is n-connected (as a space), plus a little more. Let be a fixed homology theory. The map induces isomorphisms on , if and only if the map induces an isomorphism on , i.e., . Mapping cones are famously used to construct the long coexact Puppe sequences, from which long exact sequences of homotopy and relative homotopy groups can be obtained. See also Cofibration Mapping cone (homological algebra)