Definition. Exponent / exponentiable object [mde-D16E]
Definition. Exponent / exponentiable object [mde-D16E]
An object \(X\) of a category with finite products is an exponent, powerful or an exponentiable object if \((- \times X)\) has a left adjoint, that is, \(Y^X\) exists for all \(Y\).
In the absence of products, we can ask that for all \(Y\) there is an object \(Y^X\) and a natural transformation \[\mathbb {C}(-, Y^X) \times \bc (-,X) \to \mathbb {C}(-,Y)\] in natural bijection with morphisms \(Z \to Y^X\).