# 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(1A) = 1F(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 {ηA : F (A) → G(A)} A ∈|C| indexed by the objects in C, such that for all morphisms f : A → B in C we have ηB ◦ F (f ) = G(f ) ◦ ηA.

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.

# Speculative Bubbles and Excess Demand Functions. Now,  we shall indicate how local and global properties of the deterministic dynamics vary when the parameter α is allowed to vary. α measures the strength of the non-linearity of the excess demand functions

pt+1 = pt + θn[(1−κt−ξ)(exp(αν(p − pt))−1) + κt(exp(α(1−µ)(pct − pt)) − 1) + ξγεt],
pft+1 = pt + ν(p − pt), pct+1 = pct + µ(pt − pct)  —– (1)

where κt = 1/(1 + exp(ψ((pt − pct)2 − (pt − pft)2)) .

The first of the equations above represents the adjustment of the price performed by the market maker. The second and third equations show the formation of the expectations of fundamentalists and chartists, respectively. The fourth equation represents the movement of the chartist fraction. Throughout this section, we assume that there exists no noise traders (ξ = 0). Thus, the dynamics of the system is deterministic. As can be checked easily, the dynamic system has as an unique fixed point: P ≡ (p¯, p¯f, p¯c) = (p, p, p).

The local stability conditions:

To investigate the dynamics of the model, we shall first determine the local stability region of the unique equilibrium point P. The local stability analysis of the equilibrium point P is performed via evaluation of the three eigenvalues of the Jacobian matrix at P. Let us denote by

c(λ) = λ3 − Tλ2 + Dλ —– (2)

the associated characteristic polynomial of the Jacobian matrix at P, where

D = (2−µ−0.5θnα(1−µ+ν))

and

T = (1−µ−0.5θnα(1+ν)(1−µ)).

An eigenvalue of the Jacobian is 0, and the other roots λ1 and λ2, satisfy the relation

λ2 − Tλ + D = 0 —– (3)

Thus, the stability of the equilibrium point is determined by the absolute values of λ1 and λ2. The eigenvalues λ1, λ2 are λ1,2 = T/2 + ± √ ∆/2 where ∆ ≡ T2 − 4D. As is well known, a sufficient condition for local stability consists of the following inequalities:

(i) 1−T +D > 0,

(ii) 1+T +D > 0, and

(iii) D < 1,

giving necessary and sufficient conditions for the two eigenvalues to be inside the unit circle of the complex plane. Elementary computations lead to the condition:

α < (3 − 2µ) / (θn(1 + ν − µ(1 + 0.5ν))) —– (4)

The above local conditions demonstrate that while increasing α starting from a sufficiently low value inside the stability region, a loss of local stability of the equilibrium point P may occur via a flip bifurcation, when crossing the curve

α = (3 − 2µ) / (nθ(1 + ν − µ(1 + 0.5ν))) —– (5)

Global bifurcations:

What happens as α is further increased? We will see that first, a stable 2-cycle appears; secondly, the time path of the price diverges; and thirdly, the time evolution of pt displays a remarkable transition from regular to chaotic behavior around an upward time trend. It is important that the nonstationary chaos can be transformed to a stationary series by differentiation once ∆pt = pt−pt−1. The price series fluctuates irregularly around an upward price trend, and the series of the price change fluctuates chaotically within a finite interval. Below, this non-stationary chaotic pricing will be referred to as a speculative bubble.

Speculative bubbles:

Why does a speculative bubble or non-stationary chaos appear when α increases? Unfortunately, it is difficult to prove mathematically the existence of the strange attractor. However, as we shall see, it is possible to give an intuitive interpretation of speculative bubbles.  On a phase plot, the stretching and folding is the root of the chaotic price behavior. The next question is why speculative bubbles occur when α is large. The price adjustment equation suggests that a price trend can occur only when excess demand aggregated over two types of traders is positive on average. Why would average excess demand be positive as α become large? For an answer, let us recall the traders’ excess-demand functions. xt denotes the traders’ excess demand, and pet+1 denotes the expected price. The partial derivative of the excess-demand function with respect to the parameter α is obtained as

∂xt/∂α = (pet+1 − pt) exp(α(pet+1 − pt)), —– (7)

where we assume β = 1. This equation suggests a rise in α increasing excess demand exponentially, given the expected price change (pet+1 − pt) is positive. As shown in the preceding paragraph, the maximum value of the excess demand increases freely when α increases, while the maximum value of the excess supply never can be lower than −1/β, regardless of the value of α. Assume that, in the initial state, the price pt exceeds the fundamental price p in period t, and that the chartists forecast a rise over the next period. The fundamentalists predict a falling price. Then, chartists try to buy and fundamentalists try to sell stock. When α is sufficiently large, one can safely state that chartist excess demand will exceed fundamentalist excess supply. The market maker observes excess demand, and raises the price for the following period. If the rise is strong, chartists may predict a new rise, which provokes yet another reaction of the market maker. Once the price deviates strongly from the fundamental price, the fundamentalists become perpetual sellers of stock, and when α is sufficiently large, the chartists’ excess demand may begin to exceed the fundamentalists’ excess supply on average. Furthermore, fundamentalists may be driven out of the market by evolutionary competition. It follows that when the value of α is sufficiently large, average excess demand is positive and speculative bubbles occur.

# Theories of Fields: Gravitational Field as “the More Equal Among Equals” Descartes, in Le Monde, gave a fully relational definition of localization (space) and motion. According to Descartes, there is no “empty space”. There are only objects, and it makes sense to say that an object A is contiguous to an object B. The “location” of an object A is the set of the objects to which A is contiguous. “Motion” is change in location. That is, when we say that A moves we mean that A goes from the contiguity of an object B to the contiguity of an object C3. A consequence of this relationalism is that there is no meaning in saying “A moves”, except if we specify with respect to which other objects (B, C,. . . ) it is moving. Thus, there is no “absolute” motion. This is the same definition of space, location, and motion, that we find in Aristotle. Aristotle insists on this point, using the example of the river that moves with respect to the ground, in which there is a boat that moves with respect to the water, on which there is a man that walks with respect to the boat . . . . Aristotle’s relationalism is tempered by the fact that there is, after all, a preferred set of objects that we can use as universal reference: the Earth at the center of the universe, the celestial spheres, the fixed stars. Thus, we can say, if we desire so, that something is moving “in absolute terms”, if it moves with respect to the Earth. Of course, there are two preferred frames in ancient cosmology: the one of the Earth and the one of the fixed stars; the two rotates with respect to each other. It is interesting to notice that the thinkers of the middle ages did not miss this point, and discussed whether we can say that the stars rotate around the Earth, rather than being the Earth that rotates under the fixed stars. Buridan concluded that, on ground of reason, in no way one view is more defensible than the other. For Descartes, who writes, of course, after the great Copernican divide, the Earth is not anymore the center of the Universe and cannot offer a naturally preferred definition of stillness. According to malignants, Descartes, fearing the Church and scared by what happened to Galileo’s stubborn defense of the idea that “the Earth moves”, resorted to relationalism, in Le Monde, precisely to be able to hold Copernicanism without having to commit himself to the absolute motion of the Earth!

Relationalism, namely the idea that motion can be defined only in relation to other objects, should not be confused with Galilean relativity. Galilean relativity is the statement that “rectilinear uniform motion” is a priori indistinguishable from stasis. Namely that velocity (but just velocity!), is relative to other bodies. Relationalism holds that any motion (however zigzagging) is a priori indistinguishable from stasis. The very formulation of Galilean relativity requires a nonrelational definition of motion (“rectilinear and uniform” with respect to what?).

Newton took a fully different course. He devotes much energy to criticise Descartes’ relationalism, and to introduce a different view. According to him, space exists. It exists even if there are no bodies in it. Location of an object is the part of space that the object occupies. Motion is change of location. Thus, we can say whether an object moves or not, irrespectively from surrounding objects. Newton argues that the notion of absolute motion is necessary for constructing mechanics. His famous discussion of the experiment of the rotating bucket in the Principia is one of the arguments to prove that motion is absolute.

This point has often raised confusion because one of the corollaries of Newtonian mechanics is that there is no detectable preferred referential frame. Therefore the notion of absolute velocity is, actually, meaningless, in Newtonian mechanics. The important point, however, is that in Newtonian mechanics velocity is relative, but any other feature of motion is not relative: it is absolute. In particular, acceleration is absolute. It is acceleration that Newton needs to construct his mechanics; it is acceleration that the bucket experiment is supposed to prove to be absolute, against Descartes. In a sense, Newton overdid a bit, introducing the notion of absolute position and velocity (perhaps even just for explanatory purposes?). Many people have later criticised Newton for his unnecessary use of absolute position. But this is irrelevant for the present discussion. The important point here is that Newtonian mechanics requires absolute acceleration, against Aristotle and against Descartes. Precisely the same does special relativistic mechanics.

Similarly, Newton introduced absolute time. Newtonian space and time or, in modern terms, spacetime, are like a stage over which the action of physics takes place, the various dynamical entities being the actors. The key feature of this stage, Newtonian spacetime, is its metrical structure. Curves have length, surfaces have area, regions of spacetime have volume. Spacetime points are at fixed distance the one from the other. Revealing, or measuring, this distance, is very simple. It is sufficient to take a rod and put it between two points. Any two points which are one rod apart are at the same distance. Using modern terminology, physical space is a linear three-dimensional (3d) space, with a preferred metric. On this space there exist preferred coordinates xi, i = 1,2,3, in terms of which the metric is just δij. Time is described by a single variable t. The metric δij determines lengths, areas and volumes and defines what we mean by straight lines in space. If a particle deviates with respect to this straight line, it is, according to Newton, accelerating. It is not accelerating with respect to this or that dynamical object: it is accelerating in absolute terms.

Special relativity changes this picture only marginally, loosing up the strict distinction between the “space” and the “time” components of spacetime. In Newtonian spacetime, space is given by fixed 3d planes. In special relativistic spacetime, which 3d plane you call space depends on your state of motion. Spacetime is now a 4d manifold M with a flat Lorentzian metric ημν. Again, there are preferred coordinates xμ, μ = 0, 1, 2, 3, in terms of which ημν = diag[1, −1, −1, −1]. This tensor, ημν , enters all physical equations, representing the determinant influence of the stage and of its metrical properties on the motion of anything. Absolute acceleration is deviation of the world line of a particle from the straight lines defined by ημν. The only essential novelty with special relativity is that the “dynamical objects”, or “bodies” moving over spacetime now include the fields as well. Example: a violent burst of electromagnetic waves coming from a distant supernova has traveled across space and has reached our instruments. For the rest, the Newtonian construct of a fixed background stage over which physics happen is not altered by special relativity.

The profound change comes with general relativity (GTR). The central discovery of GR, can be enunciated in three points. One of these is conceptually simple, the other two are tremendous. First, the gravitational force is mediated by a field, very much like the electromagnetic field: the gravitational field. Second, Newton’s spacetime, the background stage that Newton introduced introduced, against most of the earlier European tradition, and the gravitational field, are the same thing. Third, the dynamics of the gravitational field, of the other fields such as the electromagnetic field, and any other dynamical object, is fully relational, in the Aristotelian-Cartesian sense. Let me illustrate these three points.

First, the gravitational field is represented by a field on spacetime, gμν(x), just like the electromagnetic field Aμ(x). They are both very concrete entities: a strong electromagnetic wave can hit you and knock you down; and so can a strong gravitational wave. The gravitational field has independent degrees of freedom, and is governed by dynamical equations, the Einstein equations.

Second, the spacetime metric ημν disappears from all equations of physics (recall it was ubiquitous). At its place – we are instructed by GTR – we must insert the gravitational field gμν(x). This is a spectacular step: Newton’s background spacetime was nothing but the gravitational field! The stage is promoted to be one of the actors. Thus, in all physical equations one now sees the direct influence of the gravitational field. How can the gravitational field determine the metrical properties of things, which are revealed, say, by rods and clocks? Simply, the inter-atomic separation of the rods’ atoms, and the frequency of the clock’s pendulum are determined by explicit couplings of the rod’s and clock’s variables with the gravitational field gμν(x), which enters the equations of motion of these variables. Thus, any measurement of length, area or volume is, in reality, a measurement of features of the gravitational field.

But what is really formidable in GTR, the truly momentous novelty, is the third point: the Einstein equations, as well as all other equations of physics appropriately modified according to GTR instructions, are fully relational in the Aristotelian-Cartesian sense. This point is independent from the previous one. Let me give first a conceptual, then a technical account of it.

The point is that the only physically meaningful definition of location that makes physical sense within GTR is relational. GTR describes the world as a set of interacting fields and, possibly, other objects. One of these interacting fields is gμν(x). Motion can be defined only as positioning and displacements of these dynamical objects relative to each other.

To describe the motion of a dynamical object, Newton had to assume that acceleration is absolute, namely it is not relative to this or that other dynamical object. Rather, it is relative to a background space. Faraday, Maxwell and Einstein extended the notion of “dynamical object”: the stuff of the world is fields, not just bodies. Finally, GTR tells us that the background space is itself one of these fields. Thus, the circle is closed, and we are back to relationalism: Newton’s motion with respect to space is indeed motion with respect to a dynamical object: the gravitational field.

All this is coded in the active diffeomorphism invariance (diff invariance) of GR. Active diff invariance should not be confused with passive diff invariance, or invariance under change of coordinates. GTR can be formulated in a coordinate free manner, where there are no coordinates, and no changes of coordinates. In this formulation, there field equations are still invariant under active diffs. Passive diff invariance is a property of a formulation of a dynamical theory, while active diff invariance is a property of the dynamical theory itself. A field theory is formulated in manner invariant under passive diffs (or change of coordinates), if we can change the coordinates of the manifold, re-express all the geometric quantities (dynamical and non-dynamical) in the new coordinates, and the form of the equations of motion does not change. A theory is invariant under active diffs, when a smooth displacement of the dynamical fields (the dynamical fields alone) over the manifold, sends solutions of the equations of motion into solutions of the equations of motion. Distinguishing a truly dynamical field, namely a field with independent degrees of freedom, from a nondynamical filed disguised as dynamical (such as a metric field g with the equations of motion Riemann[g]=0) might require a detailed analysis (for instance, Hamiltonian) of the theory. Because active diff invariance is a gauge, the physical content of GTR is expressed only by those quantities, derived from the basic dynamical variables, which are fully independent from the points of the manifold.

In introducing the background stage, Newton introduced two structures: a spacetime manifold, and its non-dynamical metric structure. GTR gets rid of the non-dynamical metric, by replacing it with the gravitational field. More importantly, it gets rid of the manifold, by means of active diff invariance. In GTR, the objects of which the world is made do not live over a stage and do not live on spacetime: they live, so to say, over each other’s shoulders.

Of course, nothing prevents us, if we wish to do so, from singling out the gravitational field as “the more equal among equals”, and declaring that location is absolute in GTR, because it can be defined with respect to it. But this can be done within any relationalism: we can always single out a set of objects, and declare them as not-moving by definition. The problem with this attitude is that it fully misses the great Einsteinian insight: that Newtonian spacetime is just one field among the others. More seriously, this attitude sends us into a nightmare when we have to deal with the motion of the gravitational field itself (which certainly “moves”: we are spending millions for constructing gravity wave detectors to detect its tiny vibrations). There is no absolute referent of motion in GTR: the dynamical fields “move” with respect to each other.

Notice that the third step was not easy for Einstein, and came later than the previous two. Having well understood the first two, but still missing the third, Einstein actively searched for non-generally covariant equations of motion for the gravitational field between 1912 and 1915. With his famous “hole argument” he had convinced himself that generally covariant equations of motion (and therefore, in this context, active diffeomorphism invariance) would imply a truly dramatic revolution with respect to the Newtonian notions of space and time. In 1912 he was not able to take this profoundly revolutionary step, but in 1915 he took this step, and found what Landau calls “the most beautiful of the physical theories”.