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I removed this banner. It's a formal mathematical definition of an object in category theory. Things are bound to get technical, that is the nature of the topic. Nothing is lost if a layperson with no background in mathematics does not understand it. — Preceding unsigned comment added by Spacebusdriver (talk • contribs) 21:37, 9 March 2025 (UTC)[reply]

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 ${\displaystyle x'=\lambda x}$. Then the dual transformation is ${\displaystyle \omega '=\lambda ^{-1}\omega }$. And in higher dimensions, a change of basis ${\displaystyle \mathbf {x} '={\boldsymbol {\Lambda }}\mathbf {x} }$ induces the dual transformation ${\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}{\boldsymbol {\Lambda }}^{-1}}$. (Note: this verifies the invariance of the contraction ${\displaystyle {\boldsymbol {\omega }}'\mathbf {x} '={\boldsymbol {\omega }}\mathbf {x} }$ 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 ${\displaystyle (\mathbf {e} ^{j})}$ 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)[reply]