Exploring Categorical Structuralism

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Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-mathematics (2005, background): “Such a proof is effective whether the background set theory is formulated in traditional terms or in the idiom of category theory, as suggested in McLarty [ 20 ].”
Set theory for category theory (2008, background): “Xthere exists a object S x unique up to isomorphism with ϕ(x,S x), there exists a morphism S→Xsuch that for any xthere is a pullback square S x // S 1 x //X. This version of replacement is from [McL04]; other similar ones can be found in [Osi74, Law05]. These references also prove that etcs plus r”