Objects Do Not Determine Structure: A Symmetry Rigidity Theorem for Mathematical Structuralism

Mathematical structuralism holds that mathematical objects are determined by their structural roles — the pattern of relations they bear to other objects — rather than by intrinsic properties (Benacerraf, 1965; Shapiro, 1997). We prove the converse side of this thesis in the categorical setting: bare objects do not canonically determine structure. Our central result is the Symmetry Rigidity Theorem. Formalizing "bare objects" as the image of the object functor Ob: Cat_sm → Set and "canonical rec