How to write left superscripts

Question

I can find nothing in the Mathjax tutorial about left superscripts, which are used for the set of functions with a given domain and codomain.

Answer 1

As pointed out in the comments (by lulu and Xander), it is possible to put the superscript on a blank space. And an alternative is to put it on a vertical (=zero width) phantom box created by TeX's \vphantom-command, or empty braces ${}$. I experiment with the differences in what follows.

This is what it looks like if you simply put the superscript on a blank space: $$\,^2\Gamma^i_{jk},\qquad \,^2\bigcup_{n=1}^{17}U_i,\qquad\,^2\alpha_i$$ You see that the superscript is always at the same height irrespective of the height of the following character. This makes sense because the elevation of a superscript depends on the height of the box it is a superscript of. Here the box contains that blank space.

Next I test what changes, if instead of a blank space we place the superscript on a vertical phantom box that gets its height calculated from the character immediately following the superscript: $$\vphantom{\Gamma}^2\Gamma^i_{jk},\qquad \vphantom{\bigcup}^2\bigcup_{n=1}^{17}U_i,\qquad\vphantom{\alpha}^2\alpha_i$$ You see that this time the left superscript on a tall character is placed higher. MathJax-Source for the above displayed line: $$\vphantom{\Gamma}^2\Gamma^i_{jk},\qquad \vphantom{\bigcup}^2\bigcup_{n=1}^{17}U_i,\qquad\vphantom{\alpha}^2\alpha_i$$. You see that you need to write the superscripted character twice – once as a phantom and then for real.

Using empty braces it looks like $${}^2\Gamma^i_{jk},\qquad {}^2\bigcup_{n=1}^{17}U_i,\qquad{}^2\alpha_i.$$ As expected, the elevation of the left superscript is then determined only by the height of the box created by the empty braces.

Having the superscript on a blank space may also affect horizontal spacing. After all, the vertical phantom box should have zero width, whereas the box containing any amount of blank space has a strictly positive width. So the output may look different. Let's test.

Here are $\,^j\Gamma_{ik}$ symbols $\,^i\alpha_j$ where the left superscript is on a blank space.

Here are $\vphantom{\Gamma}^j\Gamma_{ik}$ symbols $\vphantom{\alpha}^i\alpha_j$ where the left superscript is on a vertical phantom.

Here are ${}^j\Gamma_{ik}$ symbols ${}^i\alpha_j$ where the left superscript is on a box created by empty braces.

As expected, there is a bit less white space in front of the symbol, when we use either a phantom or empty braces.

Your pick!

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However, neither of these suggestions logically ties the superscript to the character on its right. Therefore something strange or undesirable may happen. The phantoms are just a hack (as is, of course, the use of a blank space). Anyway, my recommendation is to use \vphantom because it creates a box of the appropriate height and zero width. If, instead, you need a box of zero height and the width of a known character, there is the analogous command \hphantom, 'h' for horizontal.