## Pullback Map

A pullback is a general Categorical operation appearing in a number of mathematical contexts, sometimes going under a different name. If is a linear transformation between Vector Spaces, then (usually called Transpose Map or Dual Map because its associated matrix is the Matrix Transpose of ) is an example of a pullback map.

In the case of a Diffeomorphism and Differentiable Manifold, a very explicit definition can be formulated. Given an -form on a Manifold , define the -form on by its action on an -tuple of tangent vectors as the number . This defines a map on -forms and is the pullback map.