Category Theory as Structuralist. Part Metaphysic, Part Mathematic. (1)

What are categories good for? Elementary category theory is mostly concerned with universal properties. These define certain patterns of morphisms that uniquely characterize (up to isomorphism) a certain mathematical structure. An example that we will be concerned with is the notion of a ‘terminal object’. Given a category C, a terminal object is an object I such that, for any object A in C, there is a unique morphism of type f : A → I. So for instance, on Set the singleton {∗} is the terminal object, and so we obtain a characterization of the singleton set in terms of the morphisms in Set. Other standard constructions, e.g. the cartesian product, disjoint union etc. can be characterized as universal.

Just as morphisms in a category preserve the structure of the objects, we can also define maps between categories that preserve the composition law. Let C and D be categories. A functor F : C → D is a mapping that:

(i) assigns an object F (A) in D to each object A in C; and

(ii) assigns a morphism F(f) : F(A) → F(B) to each morphism f : A → B in C, subject to the conditions F(g ◦ f) = F(g) ◦ F(f) and F(1_A) = 1_F(A) for all A in C.

Examples abound: we can define a powerset functor P : Set → Set that assigns the powerset P (X ) to each set X , and assigns the function P(f)::X →f[X] to each function f :X →Y, where f[X] ⊆ Y is the image of f.

Functors ‘compare’ categories, and we can once again increase the level of abstraction: we can compare functors as follows. Let F : C → D and G : C → D be a pair of functors. A natural transformation η : F ⇒ G is a family of functions {η_A : F (A) → G(A)} A ∈|C| indexed by the objects in C, such that for all morphisms f : A → B in C we have η_B ◦ F (f ) = G(f ) ◦ η_A.

Category theory can be thought of as ‘structuralist’ in the following simple sense: it de-emphasizes the role played by the objects of a category, and tries to spell out as many statements as possible in terms of the morphisms between those objects. These formal features of category theory have developed into a vision of how to do mathematics. This has for instance been explicitly articulated by Awodey, who says that a category-theoretic ‘structuralist’ perspective of mathematics, is based on specifying :

…for a given theorem or theory only the required or relevant degree of information or structure, the essential features of a given situation, for the purpose at hand, without assuming some ultimate knowledge, specification, or “determination” of the objects involved.

Awodey presents one reasonable methodological sense of ‘category-theoretic struc- turalism’: a view about how to do mathematics that is guided by the features of category theory.

Let us now contrast Awodey’s sense of structuralism with a position in the philosophy of science known as Ontic Structural Realism (OSR). Roughly speaking, OSR is the view that the ontology of the theory under consideration is given only by structures and not by objects (where ‘object’ is here being used in a metaphysical, and not a purely mathematical sense). Indeed, some OSR-ers would claim that:

(Objectless) It is coherent to have an ontology of (physical) relations without admitting an ontology of (physical) relata between which these rela- tions hold.

On the face of it, the ‘simple structuralism’ that is evident in the practice of category theory is very different from that envisaged by OSR; and in particular, it is hardly obvious how this simple structuralism could be applied to yield (Objectless). On the other hand, one might venture that applying (some form of) this simple structuralism to physical theories will serve the purposes of OSR.

Let us consider which forms of OSR have an interest in such an category-theoretic argument for (Objectless). According to Frigg and Votsis’ detailed taxonomy of structural realist positions, the most radical form of OSR insists on an extensional (in the logical sense of being ‘uninterpreted’) treatment of physical relations, i.e. physical relations are nothing but relations defined as sets of ordered tuples on appropriate formal objects. This view is faced not only with the problem of defending (Objectless) but with the further implausibility of implying that the concrete physical world is nothing but a structured set.

More plausible is a slightly weaker form of ontic structural realism, which Frigg calls Eliminative OSR (EOSR). Like OSR, EOSR maintains that relations are ontologically fundamental, but unlike OSR, it allows for relations that have intensions. Defenders of EOSR have typically responded to the charge of (Objectless)’s incoherence in various ways. For example, some claim that our ontology is ‘structure all the way down’ without a fundamental level (Ladyman and Ross)

, or that the EOSR position should be interpreted as reconceptualizing objects as bundles of relations.

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