This may ultimately be a silly question, but a pedantic mind like mine gets tied into knots over differing notation.
Let $\mathbb{W}$ be a complex two-dimensional vector space which carries the fundamental representation of $SL(2,\mathbb{C})$ by letting $M\in SL(2,\mathbb{C})$ act by matrix multiplication. The elements of $\mathbb{W}$ are called $\textit{left-handed Weyl spinors}$, if I understand the physics terminology correctly.
My issue is the following: After picking a basis of $\mathbb{W}$, we can write elements in component notation. Some authors have the convention that $\mathbb{W}\ni w=w^{\alpha} s_{\alpha}$ where the $w^{\alpha}$ are the numerical components. (For an example, see Appendix A in this link: http://www.maths.ed.ac.uk/~jmf/Teaching/Lectures/BUSSTEPP.pdf) Consequently, $M\cdot w=M^{\alpha}\,_{\beta} w^{\beta}$. This makes good sense to me on the grounds that spinors should be analogous to vectors and thus have upper indices. But certain deviant authors (see section 5.4 of http://www.mth.kcl.ac.uk/~jbg34/Site/Resources/lectnotes%28master%29.pdf, for example) insist on $w=w_{\alpha} s^{\alpha}$ so $M\cdot w=M_{\alpha}\,^{\beta} w_{\beta}$. So there seems to be two notational camps about upper and lower index spinors.
Is there any good reason for this disagreement in notation? Is one convention superior to the other? Is this issue totally inconsequential? I'll be grateful for any perspective.