Source
Corfield (2017), LIT-115, “Duality as a category-theoretic concept”, Studies in History and Philosophy of Modern Physics 59: 55–61, read in full in NOTE-141. Page references are the journal’s.
What was actually shown
The core construction (§3, p. 58). An adjunction between C and D^op, restricted to the objects isomorphic to their round-trip image, gives an equivalence C′ ≃ D′^op, a dual equivalence. Pontrjagin duality is the example.
The family around it (§3, pp. 57–59). Representational duality from a pairing A × B → C; enrichment in truth values, which gives six classic Galois-connection dualities, and in extended reals, which gives Fenchel–Legendre duality; Isbell conjugation; dualising objects that “keep summer and winter homes” in two categories (2 for Stone duality, R/Z for Pontrjagin); profunctors; Lawvere and Rosebrugh’s formal against concrete duality. All “revolve around” adjunction.
Sameness (p. 60). The right notion of sameness for categories is equivalence. C and C^op are not equivalent unless C is self-dual, and C ↦ C^op is the one nontrivial autoequivalence of the category of categories. Set is not self-dual (∅ and 1 behave asymmetrically), so making Set the default for concrete categories breaks a symmetry; Rel is self-dual.
The physicists’ cases (pp. 57, 60). Homological mirror symmetry can be read as an equivalence of A∞-categories. Mirror symmetry is a Z2 action on the Hodge diamond, and a Z2 action alone need not be called a duality. S-duality is a vestige of an SL(2,Z) action. T-duality involves “extra choices”, so a background may have none, one or several T-duals.
What could have come out otherwise. The standard dualities could have resisted any common categorical form. In Corfield’s survey they do not, though the survey is exposition drawn from the literature, not a proof.
What this does not say
- That duality has one account. “There is unlikely to be a single monolithic account of duality” (pp. 59–60).
- That set theory cannot express dualities. It can “in principle”; the claim is that it could not do so as “naturally” or systematically (C4).
- That physicists are wrong to say “duality”. Reserving the word for “involution with some form of structural reversal” is a preference (“In my view it is preferable”, p. 57).
- That predicate logic fails for these questions. That it is “far from optimal” is an expectation, with one worked contrast (quotient against action groupoid); no failure is exhibited (NOTE-141, Corrections).
- Anything about gauge–gravity duality. AdS/CFT is not mentioned.
Connections
- De Haro and Butterfield (LIT-199) define a duality as an isomorphism of model roots, which is an equivalence in Corfield’s vocabulary. Their accounts in the record, THEORY-459 and THEORY-690, use the word for a different relation from this one, and a reader moving between them has to translate. The two do not disagree on any fact.
- Wallace’s open question of what notion of mathematical equivalence a structural realism needs (LIT-151) gets a direction from this paper, equivalence and groupoids, not a criterion.
- Stone duality, the dualising object 2, is the record’s THEORY-020.
Note of 2026-10-19: the core construction filed apart
The construction in §3 is now THEORY-729, Active, with the proof written out: an adjunction between C and D^op restricts to a dual equivalence between the full subcategories on which its unit and counit are isomorphisms. What keeps this THEORY Proposed is the claim that most mathematical dualities have that form, which the source shows by worked exposition and not by proof. Also kept are the verdicts on the dualities physicists name, which rest on a stated preference. The promote_when says what would settle both.