 On the side of eliminative structuralism, further sub-divisions should be made as well. Again, there are fully eliminative positions, which avoid commitment to any kind of abstract objects.  Hellman’s modal structuralism is designed to be of this sort. But there are also semi-eliminative positions, which avoid commitment to abstract structures in addition to more usual, relatively concrete mathematical objects, which are accepted.  Set-theoretic structuralism is a good illustration; universalist structuralism is another, at least when backed by set theory.  Should relativist structuralism in general, and set-theoretic structuralism in particular, be seen as cases of in re structuralism, in the sense that abstract structure exist “in”  their more concrete instantiations? Perhaps; but this thesis does not seem to be forced on us (cf. Leitgeb forthcoming for more on this issue). Note also that we would end up with a form of non-eliminative structuralism then instead. The same question arises for in re forms of structuralism  more generally; the details will matter. (For more on such  approaches—sometimes labeled  “Aristotelian” as opposed to  “Platonist”—see Pettigrew 2008 and Franklin 2014.) 