Source
Lam & Wüthrich, LIT-222 (preprint, §§3–8), read in NOTE-194. The target is Bain, LIT-223 (§§2–6), read in NOTE-193. The Yoneda lemma, LIT-221 (read in NOTE-195), supplies the mathematics. The synthesis is in the curation entry of 2026-09-26, “The Yoneda lemma, and whether it supports ontic structural realism”.
What was actually shown
Bain’s argument. In set theory a relation is a subset of X × X, so it presupposes its relata. Bain proposes defining “structure” as “object in a category”, taking elements to be arrows 1 → A. He argues that these are “not essential” when the objects are not structured sets (LIT-223 pp. 1623–1625). He offers two cases: sheaves of Einstein algebras for GR with asymptotic boundaries, which may have no global sections, and nCob/Hilb for TQFT. He concludes that radical OSR “avoids the charge that it rests on an incoherent claim” (p. 1634). “Essential” is never defined.
The rebuttal, point by point (LIT-222).
- Arrows-only presentations only relabel. Identifying objects with identity arrows still needs composable pairs and a composition operation (§3, fn. 11).
- Elements are not eliminated. Generalized elements X → A exist in every category, so terminal-object elements were never the only kind (§4 p. 10). By Yoneda these probes, with their functoriality, fix A up to isomorphism (LIT-221, Cor. II). The rebuttal is therefore stronger than its authors state it. Bain’s claim that elements “only make sense” in categories with terminal objects is false.
- Morphisms relate C-objects. Those objects are structures or models, not the physical relata. The morphisms “only distinguish isomorphism classes of structures” (§5 p. 11). The relations the categorial reading supplies are therefore between presupposed objects.
- Where elements vanish, relations can vanish too. Set-style relations need products, and a category can lack both terminal objects and products (§4).
- The physics eliminates points only. Einstein-algebra models drop manifold points “in name only”, because points correspond to maximal ideals. Without global sections, local sections remain as local relata. nCob and Hilb have manifold points and vectors (§§6–7). Bain’s own closing claim, “structure devoid of objects”, goes beyond his §3 (NOTE-193).
The readings also found errors on both sides. Sections are not maximal ideals (Bain p. 1629). Hilb has direct-sum products, so both papers are wrong that it lacks them. Lam & Wüthrich mis-define a topos. None of these errors bears on the conclusion.
What this does not say
- That radical OSR is incoherent. Lam & Wüthrich close one route to it, and Chakravartty’s argument leaves eliminative OSR “untouched” (LIT-196, p. 200).
- That balanced (non-eliminative) OSR is vindicated. Lam & Wüthrich assume it is coherent and do not defend it (NOTE-194). That horn is the subject of THEORY-034, on Chakravartty’s dilemma.
- That category theory is idle for structuralism. It changes what “structure” means, and Corfield’s programmatic use (LIT-115) is untouched.
- That Bain, Lam or Wüthrich invoke the Yoneda lemma. None of them does.
Connections
- THEORY-032, on the Yoneda lemma: the formal side of the same finding.
- LIT-154 (Leitgeb & Ladyman): “up to isomorphism” is the categorial form of the limit on weak discernibility.
- LIT-217 (Ladyman 2001): the non-eliminative position that the categorial reading does support.
Note of 2026-10-10: Eva cited again, still unread
Eva 2016 was cited again in the exchange, at A187 §2 (“critically examines
whether category theory can substantiate radical OSR”) and at A191 §4. Its
identifier is doi:10.1007/s13194-015-0129-6 (Category Theory and Physical
Structuralism, EJPS, 2016), verified against Crossref on 2026-10-10. It is
still unread and not registered. A187 reads it as support for its own caution,
which is not what this entry’s promote_when expects of “the one reply on
Bain’s side”. Neither reading has been checked. It remains the promote_when
reading.