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)