The following questions are germane to our understanding of gauge-(in)variant
quantities and physical possibility: how are gauge transformations and
spacetime diffeomorphisms understood as symmetries, in which ways are they
similar, and in which are they different? To what extent are we justified in
endorsing different attitudes — nowadays called sophistication, haecceitism,
and eliminativism — towards each? This is the second of four papers taking up
this question, and it is the one that most engages with the metaphysical
debates surrounding our understanding of symmetry and equivalence.

In this paper, I will provide two desiderata for the application of a
treatment of symmetries known as `sophistication’ and show that both general
relativity and Yang-Mills theory satisfy these desiderata. The first
desideratum for symmetries is mathematical, and was shown to hold for general
relativity and Yang-Mills in the first paper: (i) that they correspond to the
automorphisms of the structured base sets for the models of the two theories.
Here I will extend the desideratum to more general theories. The second
desideratum is mathematical-physical: (ii) that the general type of structure
to which Desideratum (i) refers is axiomatizable and that this axiomatization
can be phrased in terms of basic physical predicates of the theory. In the
third paper in the series I will provide yet a third desideratum, to deal with
an issue that goes beyond the standard debate about sophistication.

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