Talk:Functor

Covariance and contravariance in a coordinate co/basis
I'm having trouble with the paragraph beginning "There is a convention which refers to "vectors"...

I think it's easily shown to be wrong with a scalar example. Suppose the change of basis is $$x'=\lambda x$$. Then the dual transformation is $$\omega'=\lambda^{-1}\omega$$. And in higher dimensions, a change of basis $$\mathbf x' = \boldsymbol\Lambda \mathbf x$$ induces the dual transformation $$\boldsymbol\omega'=\boldsymbol\omega\boldsymbol\Lambda^{-1}$$. (Note: this verifies the invariance of the contraction $$\boldsymbol\omega'\mathbf x'=\boldsymbol\omega\mathbf x$$ under a change of basis) This correction can remediate the third sentence of the paragraph, as well.

The fourth/final sentence is correct, although the terminological confusion might be explained by the fact that, while the pull-back of the cobasis $$(\mathbf{e}^j)$$ transforms contravariantly, the components of a covector (with respect to its cobasis) transform covariantly, and vice versa for the basis and components of a vector.--ScriboErgoSum (talk) 08:17, 21 November 2021 (UTC)