Exponential identity of functions

The exponential identity of functions: $$[X \times Y,Z]=[X,[Y,Z]],$$ where $[A,B]$ stands for the set of all functions from $A$ to $B$. The bijection is seen via the formula: $$f(x,y)=f(y)(x).$$ Another version of the identity is indeed exponential: $$Z^{X \times Y}=(Z^Y)^X.$$

So, every time one sees a function defined on a product, it can be interpreted as a collection of functions parametrized by one of the components.

Examples:

• homotopy,
• differential forms.

Compare to $$Hom (U \otimes V, W) \cong Hom (U, Hom(V, W)),$$ where $Hom (-,-)$ denotes the vector space of all linear maps and $U \otimes V$ is the tensor product.