Object as Category-Theoretic or Object as Ontological: The Inadequacy of Implicitly Quantifying Over Elements. (2)

It will be convenient to use the term ‘object’ in two senses. First, as an object of a category, i.e. in a purely mathematical sense. We shall call this a C- object (‘C’ for category-theoretic). Second, in the sense commonly used in structural realist debates, and which was already introduced above, viz. an object is a physical entity which is a relatum in physical relations. We shall call this an O-object (‘O’ for ‘ontological’).

We will also need to clarify our use of the term ‘element’. We use ‘element’ to mean an element of a set, or as it is also often called, a ‘point’ of a set (indeed it will be natural for us to switch to the language of points when discussing manifolds, i.e. spacetimes.) This familiar use of element should be distinguished from the category-theoretic concepts of ‘global element’ and ‘generalized element’, which is introduced below.

Jonathan Bain’s first strategy for defending (Objectless) draws on the following idea: the usual set-theoretic representations of O-objects and relations can be translated into category-theoretic terms, whence these objects can be eliminated. In fact, the argument can be seen as consisting of two different parts.

In the first part, Bain attempts to give a highly general argument, in the sense that it turns only on the notion of universal properties and the translatability of statements about certain mathematical representations (i.e. elements of sets) of O-objects into statements about morphisms between C-objects. As Bain himself notes, the general argument fails, and he thus introduces a more specific argument, which is what he wishes to endorse. The specific argument turns on the idea of obtaining a translation scheme from a ‘categorical equivalence’ between a geometric category and an algebraic category, which in turn allows one to generalize the original C-objects. The argument is ‘specific’ because such equivalences only hold between rather special sorts of categories.

The details of Bain’s general argument can be reconstructed as follows:

G1: Physical objects and the structures they bear are typically identified with the elements of a set X and relations on X respectively.

G2: The set-theoretic entities of G1 are to be represented in category-theoretic language by considering the category whose objects are the relevant structured sets, and whose morphisms are functions that preserve ‘structure’.

G3: Set-theoretic statements about an object of a category (of the type in G2) can often be expressed without making reference to the elements of that object. For instance:

1. In any category with a terminal object any element of an object X can be expressed as a morphism from the terminal object to X. (So for instance, since the singleton {∗} is the terminal object in the category Set, an element of a set X can be described by a morphism {∗} → X.)

2. In a category with some universal property, this property can be described purely in terms of morphisms, i.e. without making any reference to elements of an object.

To sum up, G1 links O-objects with a standard mathematical representation, viz. elements of a set. And G2 and G3 are meant to establish the possibility that, in certain cases, category theory allows us to translate statements about elements of sets into statements about the structure of morphisms between C-objects.

Thus, Bain takes G1–G3 to suggest that: 

C: Category theory allows for the possibility of coherently describing physical structures without making any reference to physical objects.

Indeed, Bain thinks the argument suggests that the mathematical representatives of O- objects, i.e. the elements of sets, are surplus, and that category theory succeeds in removing this surplus structure. Note that even if there is surplus structure here, it is not of the same kind as, e.g. gauge-equivalent descriptions of fields in Yang-Mills theory. The latter has to do with various equivalent ways in which one can describe the dynamical objects of a theory, viz. field. By contrast, Bain’s strategy involves various equivalent descriptions of the entire theory.

Bain himself thinks that the inference from G1–G3 to C fails, but he does give it serious consideration, and it is easy to see why: its premises based on the most natural and general translation scheme in category theory, viz. redescribing the properties of C-objects in terms of morphisms, and indeed – if one is lucky – in terms of universal properties. 

First, the premise G1. Structural realist doctrines are typically formalized by modeling O-objects as elements of a set and structures as relations on that set. However, this is seldom the result of reasoned deliberation about whether standard set theory is the best expressive resource from some class of such resources, but rather the product of a deeply entrenched set-theoretic viewpoint within philosophy. Were philosophers familiar with an alternative to set theory that was at least as powerful, e.g. category theory, then O-objects and structures might well have been modeled directly in the alternative formalism. Of course, it is also a reasonable viewpoint to say that it is most ‘natural’ to do the philosophy/foundations of physics in terms of set theory – what is ‘natural’ depends on how one conceives of such foundational investigations.

So we maintain that there is no reason for the defender of O-objects to accept G1. For instance, he might try to construct a category such that O-objects are modeled by C-objects and structures are modeled by morphisms. For example, there are examples of categories whose C-objects might coincide with the mathematical representatives of O-objects. For instance, in a path homotopy category, the C-objects are just points of the relevant space, and one might in turn take the points of a space to be O-objects, as Bain does in his example of general relativity and Einstein algebras. Or he might take as his starting point a non-concrete category, whose objects have no underlying set and thus cannot be expressed in the terms of G1.

The premise G2, on the other hand, is ambiguous—it is unclear exactly how Bain wants us to understand ‘structure’ and thus ‘structure-preserving maps’. First, note that when mathematicians talk about ‘structure-preserving maps’ they usually have in mind morphisms that do not preserve all the features of a C-object, but rather the characteristic (albeit partial) features of that C-object. For instance, with respect to a group, a structure-preserving map is a homomorphism and not an isomorphism. Bain’s example of the category Set is of this type, because its morphisms are arbitrary functions (and not bijective functions).

However, Bain wants to introduce a different notion of ‘structure’ that contrasts with this standard usage, for he says:

(Structure) …the intuitions of the ontic structural realist may be preserved by defining “structure” in this context to be “object in a category”.

If we take this claim seriously, then a structure-preserving map will turn out to be an isomorphism in the relevant category – for only isomorphisms preserve the complete ‘structural essence’ of a structured set. For instance, Bain’s example of the category whose objects are smooth manifolds and whose morphisms are diffeomorphisms is of this type. If this is really what Bain has in mind, then one inevitably ends up with a very limited and dull class of categories. But even if one relaxes this notion of ‘structure’ to mean ‘the structure that is preserved by the morphisms of the category, whatever they happen to be’, one still runs into trouble with G3.

We now turn to the premise G3. First, note that G3 (i) is false, as we now explain. It will be convenient to introduce a piece of standard terminology: a morphism from a terminal object to some object X is called a global element of X. And the question of whether an element of X can be expressed as a global element in the relevant category turns on the structure of the category in question. For instance, in the category Man with smooth manifolds as objects and smooth maps as morphisms, this question receives a positive answer: global elements are indeed in bijective correspondence with elements of a manifold. This is because the terminal object is the 0-dimensional manifold {0}, and so an element of a manifold M is a morphism {0} → M. But in many other categories, e.g. the category Grp, the answer is negative. As an example, consider that Grp has the trivial group 1 as its terminal object and so a morphism from 1 to a group G only picks out its identity and not its other elements. In order to obtain the other elements, one has to introduce the notion of a generalized element of X, viz. a morphism from some ‘standard object’ U into X. For instance, in Grp, one takes Z as the standard object U, and the generalized elements Z → G allow us to recover the ordinary elements of a group G.

Second, while G3 (ii) is certainly true, i.e. universal properties can be expressed purely in terms of morphisms, it is a further – and significant – question for the scope and applicability of this premise whether all (or even most) physical properties can be articulated as universal properties.

Hence we have seen that the categorically-informed opponent of (Objectless) need not accept these premises – there is a lot of room for debate about how exactly one should use category theory to conceptualize the notion of physical structure. But supposing that one does: is there a valid inference from G1–G3 to C? Bain himself notes that the plausibility of this inference trades on an ambiguity in what one means by ‘reference’ in C. If one merely means that such constructions eliminate explicit but not implicit reference to objects, then the argument is indeed valid. On the other hand, a defense of OSR requires the elimination of implicit reference to objects, and this is what the general argument fails to offer – it merely provides a translation scheme from statements involving elements (of sets) to statements involving morphisms between C-objects. So, the defender of objects can maintain that one is still implicitly quantifying over elements. 

Advertisement