Editing Currying (section)

=== Algebraic topology ===
In [[algebraic topology]], currying serves as an example of [[Eckmann–Hilton duality]], and, as such, plays an important role in a variety of different settings. For example, [[loop space]] is adjoint to [[reduced suspension]]s; this is commonly written as
:<math>[\Sigma X,Z] \approxeq [X, \Omega Z]</math>
where <math>[A,B]</math> is the set of [[homotopy class]]es of maps <math>A \rightarrow B</math>, and <math>\Sigma A</math> is the [[Suspension (topology)|suspension]] of ''A'', and <math>\Omega A</math> is the [[loop space]] of ''A''. In essence, the suspension <math>\Sigma X</math> can be seen as the cartesian product of <math>X</math> with the unit interval, modulo an equivalence relation to turn the interval into a loop. The curried form then maps the space <math>X</math> to the space of functions from loops into <math>Z</math>, that is, from <math>X</math> into <math>\Omega Z</math>.<ref name=rotman/> Then <math>\text{curry}</math> is the [[adjoint functor]] that maps suspensions to loop spaces, and uncurrying is the dual.<ref name=rotman/>

The duality between the [[mapping cone (topology)|mapping cone]] and the mapping fiber ([[cofibration]] and [[fibration]])<ref name=may/>{{rp|at=chapters 6,7}} can be understood as a form of currying, which in turn leads to the duality of the [[long exact sequence|long exact]] and coexact [[Puppe sequence]]s.

In [[homological algebra]], the relationship between currying and uncurrying is known as [[tensor-hom adjunction]]. Here, an interesting twist arises: the [[Hom functor]] and the [[tensor product]] functor might not [[lift (mathematics)|lift]] to an [[exact sequence]]; this leads to the definition of the [[Ext functor]] and the [[Tor functor]].