Contribution
A point-by-point rebuttal of Bain’s (2013) argument that category theory supports radical ontic structural realism (ROSR). It sets four tasks that a categorial ROSR must discharge (§2). It argues that Bain’s strategy fails the first two at the formal level (§§3–5). It then argues that his physics cases, TQFT via nCob/Hilb and sheaves of Einstein algebras, fail the last two: they do not eliminate relata as opposed to spacetime points, and their physical relevance is unshown (§§6–7). It closes by holding that “balanced” (non-eliminative) OSR keeps the upper hand.
Key insight
In a category the relational data, the morphisms, relate C-objects, and C-objects stand for structures or models, not physical relata. The candidates for physical relata are elements of C-objects. Those are never eliminated: they generalize to arrows X → A, which exist in every category. So category theory changes what “structure” means without providing relations free of relata. And because morphisms fix C-objects only up to isomorphism, the categorial notion of structure does not map onto the ROSR/BOSR dispute at all (§5).
Assumptions
- ROSR is a scientific realism. Its relations must be exemplified in the physical world, so a coherent formalisation of “free-standing” relations is owed (§1, p. 2).
- Formalism and ontology are linked as Bain assumes (fn. 7, “we take it for granted as Bain does”).
- Bain’s identification: C-objects are structures or models, and ROSR targets elements of C-objects (§3, pp. 8–9).
- BOSR is available as a coherent alternative: relata with relational, non-intrinsic identity (§1, p. 3). This is assumed, not defended here.
- Four tasks for categorial ROSR (§2, p. 6): (i) category theory must allow free-standing relations without objects; (ii) these must not be isomorphically reformulable in set theory (no mere relabelling); (iii) fundamental physics must be reformulable in such categories; (iv) that reformulation must be superior, with physical significance the set-theoretic one lacks.
Key results
This is a philosophical argument; there are no theorems. What it argues:
- §2: the objections to ROSR divide into metaphysical (1), logical or set-theoretic (2) and physical (3). Objection (2) cannot be dismissed as prejudice. Even a “dual” set-theoretic presentation, with relations as the domain, stays two-sorted (pp. 5–6).
- §3: a category can be axiomatised with arrows only, identifying objects with identity morphisms. This does not eliminate objects: “a mere relabelling” (p. 8). The arrows-only data still needs ordered pairs of composable arrows and a composition operation (fn. 11). In any case C-objects are not the target; elements of C-objects are (pp. 8–9).
- §4, first problem: generalized elements X → A exist in any category, so “elements or constituents of a structure are not truly eliminated” even in categories unlike Set (p. 10).
- §4, second problem, “throwing out the relations with the relata”: set-style relations need Cartesian products, which not all categories have. Where elements vanish, relations may vanish too. Terminal objects and products are independent conditions, so theory choice decides whether elements, relations, or both exist. “BOSR … has the upper hand, except in the case the relevant category admits Cartesian products, but no terminal objects” (p. 10). Concretizing a category with a faithful U : C → Set lets n-ary relations be defined as subfunctors of U^n. But then elements can be introduced by the same route (p. 11).
- §5: “the structure of a C-object is given by the C-morphisms to and from this object”; it is “determined … only up to isomorphism”; “the morphisms only distinguish isomorphism classes of structures”. And “if relational at all, the morphisms are relations between objects, not elements of objects” (p. 11). If category theory is more fundamental, showing that a category has no elements would refute BOSR “but it wouldn’t be enough to vindicate ROSR” (p. 12). If the physically relevant categories are toposes, ROSR must retreat to a generality physics does not call for (pp. 12–13).
- §6 (TQFT): nCob and Hilb have elements (manifold points, vectors). Bain’s “not essential” rests on their morphisms not being structure-preserving functions. Categorical dissimilarity from Set “do[es] not entail that there are no physical objects and relata at all” (p. 14). Hilb’s non-unitary morphisms generalise quantum theory, and this “can ultimately only possibly underwrite ROSR if it is physically called for” (fn. 22).
- §7 (sheaves of Einstein algebras): EA models eliminate points in name only, since points correspond to maximal ideals (Gelfand, fn. 25). The physical significance of boundary constructions beyond conformal (Penrose) boundaries is doubtful (Curiel 1999), so task (iv) “is certainly not satisfied” (p. 16). Sections are not maximal ideals (fn. 32). Without global sections, local sections remain as “local elements” that can serve as local relata (p. 18). Eliminating spacetime points does not eliminate “any physical object or relata tout court” (pp. 18–19). And no categorial tools are needed to reach the non-fundamentality of points (p. 18).
- §8: category theory offers no “relevantly more hospitable environment than set theory for ROSR”. “C-morphisms among C-objects are not the right candidates—the right structures—for ROSR to be realist about” (p. 19).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Objects-as-identity-arrows is relabelling, not elimination | moderate | informal argument plus Mac Lane’s arrows-only data (§3, fn. 11) |
| C2 | Generalized elements X → A exist in every category, so elements are not eliminated | strong | definition (§4). Stronger still with Yoneda, which the paper does not cite |
| C3 | Where Cartesian products fail, set-style relations fail too; elements and relations are independent | moderate | category-theoretic definitions (§4, fn. 14) |
| C4 | Hilb and nCob have no products | false for Hilb | assertion (p. 10). Hilbert spaces have direct-sum biproducts |
| C5 | Morphisms relate C-objects, not elements of C-objects, so they are not ROSR’s relations | moderate | conceptual argument (§5). Yoneda partly blurs the line, since T → A is both a morphism and a generalized element |
| C6 | Morphisms determine C-objects only up to isomorphism | strong | the definition of isomorphism (fn. 13). This is Yoneda’s Corollary II, unnamed |
| C7 | Showing a category has no elements would refute BOSR but not establish ROSR | moderate | argument (§5) |
| C8 | Boundary constructions in sheaf-theoretic GR lack demonstrated physical significance | moderate | appeal to Curiel (1999) and the singularities literature (§7) |
| C9 | Sections of a sheaf of Einstein algebras are not maximal ideals (against Bain) | strong | definitions (fn. 32), credited to Zafiris |
| C10 | Absence of global sections leaves local sections as local relata | moderate | the definition of a sheaf (§7) |
| C11 | Category theory offers “little if any comfort” to ROSR | moderate | cumulative argument (§8). It depends on BOSR being coherent, which is assumed |
Concepts
- ROSR: eliminative OSR, with physical structures as networks of “free-standing” relations without relata (p. 1).
- BOSR (“balanced” OSR): keeps thin relata whose identity is fixed by their position in a relational structure (p. 3). The authors’ own term.
- C-object vs physical object: a technical object of a category (a structure or model) versus what “enters into physical relations” (fn. 12).
- Element of a C-object: 1 → A (global); generalized element X → A (§4).
- Concrete category: ⟨C, U⟩ with U : C → Set faithful. Subfunctor V ⊆ U: V(A) ⊆ U(A), with V(f) the restriction (fns. 16–17).
- Sheaf: a presheaf satisfying (M) locality and (G) gluing, following Tennison 1975 (§7).
Connections
The paper answers Bain 2013 (yo2) directly. It builds on Esfeld & Lam 2008/2011, Lam & Esfeld 2012 and Wüthrich 2009, 2012. It engages French & Ladyman 2003, French 2010, 2012, Muller 2010 (definition versus axiomatisation of structure) and Landry 2007 (“shared structure need not be shared set-structure”). Its category theory draws on Goldblatt, Mac Lane, Borceux and Lawvere & Schanuel. Its physics draws on Baez 2006, Heller & Sasin 1994/1995, Geroch 1972, Curiel 1999 and Huggett & Wüthrich. A parallel rebuttal is Lal & Teh (arXiv:1404.3049, 2014), which I checked for Yoneda (no hits) but did not read in full. Eva (2016, EJPS 6:231–246) replies to both on Bain’s side, but I could not reach it from a legitimate free source.
Bearing on the record
- Status of the Yoneda–OSR link in this literature. Neither the founding pro-paper (yo2) nor this rebuttal cites the Yoneda lemma. This paper nonetheless states the two Yoneda facts that decide the question: generalized elements are ineliminable (C2), and morphisms determine objects only up to isomorphism (C6). With yo1 it gives the record a precise answer. The determination-by-relations reading of Yoneda supports objects as positions (BOSR or moderate OSR). It does not support eliminating objects (ROSR), because the objects are presupposed and determined only up to isomorphism.
- Explicit Yoneda–OSR claims found elsewhere (not read in full, not filed): Adlam, “Relational Observables, Quiddities, and Structural Realism” (arXiv:2507.20313, 2025), has one sentence in its conclusion (p. 20). It says the Yoneda lemma “implies that objects may emerge from the ‘morphisms’”, citing Yokura 2017 (arXiv:1712.02064). I read Yokura’s 5-page note in full, and it says nothing of the kind. It proves a remark about a canonical natural transformation h_A → Sub F* and never discusses objects emerging from morphisms, so the citation does not support the claim. Hendren, Grasso, Ellia & Tononi (arXiv:2608.24928, 2026, consciousness science), Box 1, say OSR’s “commitment is made precise by its mathematical centerpiece, the Yoneda lemma”, citing Tsuchiya & Saigo 2021 (qualia structure). That is the most explicit statement found. It is an assertion in a box, it is about qualia, and no OSR author I read (LIT-217, yo2, this paper) calls Yoneda OSR’s centrepiece. The nLab “structuralism” page juxtaposes Yoneda and structural realism in adjacent sentences, citing Awodey for mathematical structuralism only.
- ML practice: none.
Limitations
- It assumes BOSR is coherent and never defends it against Chakravartty-style objections (LIT-196).
- It does not use Yoneda, so its §5 line between morphisms (relations between objects) and elements (of objects) is drawn more sharply than the mathematics supports.
- It contains category-theoretic slips: C4, the topos definition, the claim about noncommutative maximal ideals, and Cartesian closedness offered where finite products suffice.
- By their own statement (fn. 7) the authors set aside the relation between formalism and ontology, which is the premise ROSR most needs.
- The reading is of the preprint, not the published version.
Open questions
- Can a Yoneda-based “objects are their generalized elements” reading be developed as a precise BOSR, with relata identified up to isomorphism by their hom-functors, and does it meet LIT-154’s non-discernibility cases by adopting identity = isomorphism (univalence)?
- Is there a physically relevant category with products but no terminal object: the one case the authors leave open for ROSR (pp. 10–11)?
- Does Eva (2016) meet the generalized-element objection? This is unverified; the paper was not reachable.
Corrections to the seeded skim
- No Yoneda, but its content is here. The paper never mentions the Yoneda lemma. Two of its central moves are Yoneda consequences stated without the name. (a) §4 (p. 10): “a generalized notion of an element of an object A of any category C can be defined as a C-morphism X → A, where X is any C-object …, so that the notion of elements or constituents of a structure are not truly eliminated”. By Yoneda (yo1) these generalized elements, with their functoriality, determine A up to isomorphism, which makes the authors’ point stronger than they state it. (b) §5 (p. 11): “Given a category C, the structure of a C-object is given by the C-morphisms to and from this object … determined by these conditions only up to isomorphism … Thus, the morphisms only distinguish isomorphism classes of structures.” That is Corollary II of yo1 read philosophically. This paper therefore comes closest to what the owner asked, an explicit analysis of whether determination by relations up to isomorphism supports eliminating objects. It answers no: the determination is of C-objects, which are presupposed, and only up to isomorphism. It still does not cite the lemma.
- Answer to the owner’s question (does “determined up to isomorphism by its relations” support eliminating objects?). On this paper’s analysis combined with yo1: no, for two separate reasons. (1) The determination presupposes the C-objects. The hom-functor is indexed by them, the conclusion is an isomorphism between two of them, and identifying objects with identity arrows only relabels them (§3, fn. 11). (2) ROSR’s target is not C-objects at all but elements, the physical relata. By Yoneda, elements generalize to all probes T → A, which exist in every category (§4). The paper does miss one nuance. Yoneda also blurs its sharp line (§5, p. 11) between morphisms as “relations between objects” and elements “of objects”: a generalized element T → A is both. A ROSRer could exploit that identification, so that an object’s “elements” just are the morphisms into it. But the objects T and A remain, so the result is positions in a structure (moderate/balanced OSR), not relations without relata.
- Against yo2 (Bain): the paper correctly flags Bain’s error that sections of a sheaf of Einstein algebras are maximal ideals (fn. 32). It also correctly flags his reliance on terminal-object elements (§4) and his undefined “essential” (§4). It repeats one of his errors: Hilb and nCob “do not have products” (p. 10). Finite-dimensional Hilbert spaces with bounded linear maps have direct-sum (bi)products. What fails is that ⊗ is not Cartesian.
- Other slips in the paper (my reading): (1) §5 (p. 12): “one adds axioms demanding, e.g., that the categories have terminal objects and Cartesian products. A category satisfying these additional axioms is called topos”. A category with those is merely one with finite products. A topos further needs all finite limits, exponentials and a subobject classifier. The “e.g.” softens this, but the definition as given is wrong. (2) §7 (p. 18): “non-commutative algebras in general have no maximal ideals”. Every non-zero unital ring has a maximal two-sided ideal (Zorn). What noncommutative algebras lack in general is enough maximal ideals or characters to serve as points: a simple algebra such as a matrix algebra has only {0}. (3) §4 (p. 11) attributes the existence of U(A)^n to Set being Cartesian closed; finite products suffice.
- Against LIT-045 (SEP): agrees with the entry’s framing of the incoherence objection, and separates it into metaphysical, set-theoretic and physical versions (§2). That is a useful refinement for the record.
- Against LIT-196 (Chakravartty), NOTE-149: the paper’s “balanced OSR” (BOSR) is the non-eliminative OSR that LIT-196 argues collapses. Lam & Wüthrich simply assume BOSR is available, and never meet Chakravartty’s priority argument. The two works bear on different horns: this paper closes the ROSR horn against category theory, and LIT-196 presses the BOSR horn.
- Against LIT-154 (Leitgeb & Ladyman): the paper invokes the PII embarrassments (FLRW spacetimes, bosons) as a reason one might want ROSR (§1, p. 3). Its “only up to isomorphism” point (§5) is the categorial form of the same limit that LIT-154 shows for weak discernibility.
- Against LIT-115 (Corfield): no conflict. The paper’s §5 point that the categorial notion of structure differs deeply from the set-theoretic one, so that the ROSR/BOSR debate is “at odds with the way structures get characterized in category theory”, fits Corfield’s case for category theory as the language of sameness questions. It draws a deflationary moral where Corfield draws a programmatic one.