Ontological-Objects Categoric-Theoretic Physics. A Case of Dagger Functor. Note Quote.

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Jonathan Bain’s examples in support of the second strategy are:

(i) the category Hilb of complex Hilbert spaces and linear maps; and

(ii) the category nCob, which has (n−1)-dimensional oriented closed manifolds as objects, and n-dimensional oriented manifolds as morphisms.

These examples purportedly represent ‘purely’ category-theoretic physics. This means that formal statements about the physical theory, e.g. quantum mechanics using Hilb, are derived using the category-theoretic rules of morphisms in Hilb.

Now, prima facie, both of these examples look like good candidates for doing purely category-theoretic physics. First, each category is potentially useful for studying the properties of quantum theory and general relativity respectively. Second, each possesses categorical properties which are promising for describing physical properties. More ambitiously, they suggest that one could use categorical tools to develop an approach for integrating parts of quantum theory and general relativity.

Let us pause to explain this second point, which rests on the fact that, qua categories, Hilb and nCob share some important properties. For example, both of these categories are monoidal, meaning that both categories carry a generalisation of the tensor product V ⊗ W of vector spaces V and W. In nCob the monoidal structure is given by the disjoint union of manifolds; whereas in Hilb, the monoidal structure is given by the usual linear-algebraic tensor product of Hilbert spaces.

A second formal property shared by both categories is that they each possess a contravariant involutive endofunctor (·)called the dagger functor. Recall that a contravariant functor is a functor F : C → D that reverses the direction of arrows, i.e. a morphism f : A → B is mapped to a morphism F (f ) : F (B) → F (A). Also recall that an endofunctor on a category C is a functor F : C → C, i.e. the domain and codomain of F are equal. This means that, given a cobordism f: A → B in nCob or a linear map L: A → B in Hilb, there exists a cobordism f: B → A and a linear adjoint L : B → A respectively, satisfying the involution laws f ◦ f = 1A and f ◦ f = 1B, and identically for L.

The formal analogy between Hilb and nCob has led to the definition of a type of quantum field theory, known as a topological quantum field theory (TQFT), first introduced by Atiyah and Witten. A TQFT is a (symmetric monoidal) functor:

T : nCob → Hilb,

and the conditions placed on this functor, e.g. that it preserve monoidal structure, reflect that its domain and target categories share formal categorical properties. To further flesh out the physical interpretation of TQFTs, we note that the justification for the term ‘quantum field theory’ arises from the fact that a TQFT assigns a state space (i.e. a Hilbert space) to each closed manifold in nCob, and it assigns a linear map representing evolution to each cobordism. This can be thought of as assigning an amplitude to each cobordism, and hence we obtain something like a quantum field theory.

Recall that the significance of these examples for Bain is their apparent status as purely category-theoretic formulations of physics which, in virtue of their generality, do not make any reference to O-objects (represented in the standard way, i.e. as elements of sets). We now turn to a criticism of this claim.

Bain’s key idea seems to be that this ‘generality’ consists of the fact that nCob and Hilb (and thus TQFTs) have very different properties (qua categories) from Set. In fact, he claims that three such differences count in favor of (Objectless):

(i) nCob and Hilb are non-concrete categories, but Set (and other categories based on it) are concrete.

(ii) nCob and Hilb are monoidal categories, but Set is not.

(iii) nCob and Hilb have a dagger functor, but Set does not.

We address these points and their implications for (Objectless) in turn. First, (i). Bain wants to argue that since nCob and Hilb ‘cannot be considered categories of structured sets’, nor can these categories be interpreted as having O-objects. If one is talking about categorical properties, this claim is best couched in the standard terminology as the claim that these are not concrete categories. But this inference is faulty for two reasons. First, his point about non-concreteness is not altogether accurate, i.e. point (i) is false as stated. On the one hand, it is true that nCob is not a concrete category: in particular, while the objects of nCob are structured sets, its morphisms are not functions, but manifolds, i.e. sets equipped with the structure of a manifold. But on the other hand, Hilb is certainly a concrete category, since the objects are Hilbert spaces, which are sets with extra conditions; and the morphisms are just functions with linearity conditions. In other words, the morphisms are structure-preserving functions. Thus, Bain’s examples of category-theoretic physics are based in part on concrete categories. Second and more importantly, it is doubtful that the standard mathematical notion of concreteness will aid Bain in defending (Objectless). Bain wants to hold that the non-concreteness of a category is a sufficient condition for its not referring to O-objects. But nCob is an example of a non-concrete category that apparently contain O-objects—indeed the same O-objects (viz. space-time points) that Bain takes to be present in geometric models of GTR. We thus see that, by Bain’s own lights, non-concreteness cannot be a sufficient condition of evading O-objects.

So the example of nCob still has C-objects that are based on sets, albeit morphisms which are more general than functions. However, one can go further than this: the notion of a category is in fact defined in a schematic way, which leaves open the question of whether C-objects are sets or whether functions are morphisms. One might thus rhetorically ask whether this could this be the full version of ‘categorical generality’ that Bain needs in order to defend (Objectless). In fact, this is implausible, because of the way in which such a schematic generality ends up being deployed in physics.

On to (ii): unfortunately, this claim is straightforwardly false: the category Set is certainly monoidal, with the monoidal product being given by the cartesian product.

Finally, (iii). While it is true that Set does not have a dagger functor, and nCob and Hilb do, it is easy to construct an example of a category with a dagger functor, but which Bain would presumably agree has O-objects. Consider the category C with one object, namely a manifold M representing a relativistic spacetime; the morphisms of C are taken to be the automorphisms of M. As with nCob, this category has natural candidates for O-objects (as Bain assumes), viz. the points of the manifold. But the category C also has a dagger functor: given an automorphism f : M → M, the morphism f : M → M is given by the inverse automorphism f−1. In contrast, the category Set does not have a dagger functor: this follows from the observation that for any set A that is not the singleton set {∗}, there is a unique morphism f : A → {∗}, but the number of morphisms g : {∗} → A is just the cardinality |A| > 1. Hence there does not exist a bijection between the set of morphisms {f : A → {∗}} and the set of morphisms {g : {∗} → A}, which implies that there does not exist a dagger functor on Set. Thus, by Bain’s own criterion, it is reasonable to consider C to be structurally dissimilar to Set, despite the fact that it has O-objects.

More generally, i.e. putting aside the issue of (Objectless), it is quite unclear how one should interpret the physical significance of the fact that nCob/Hilb, but not Set has a dagger functor. For instance, it turns out that by an easy extension of Set, one can construct a category that does have a dagger functor. This easy extension is the category Rel, whose objects are sets and whose morphisms are relations between objects (i.e. subsets of the Cartesian product of a pair of objects). Note first that Set is a subcategory of Rel because Set and Rel have same objects, and every morphism in Set is a morphism in Rel. This can be seen by noting that every function f : A → B can be written as a relation f ⊆ A × B, consisting of the pairs (a, b) defined by f(a) = b. Second, note that – unlike Set – Rel does have a non-trivial involution endofunctor, i.e. a dagger functor, since given a relation R : A → B, the relation

Frobenius Algebras

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Theorem:

To give an open string theory is equivalent to giving a Frobenius algebra A inside Vect. To give a closed string theory is equivalent to giving a commutative Frobenius algebra B inside Vect.

The algebra A (B) is defined on the vector space which is the image under Z of the interval I (circle S1). To prove that a open/closed string theory defines a Frobenius algebra on these vector spaces is easy, especially after one reformulates the definition of a Frobenius algebra in a categorical or ‘topological’ way. To prove the converse, that every Frobenius algebra arises as Z(I) or Z(S1) for some open/closed Topological Quantum Field Theory (TQFT) Z is the more interesting result. There are three different ways of proving this fact.

The first and perhaps most modern way (elegantly set forth in Kock’s work) is to express 2Cob and OCob using generators and relations, and to use a result of Abrams work, which formulates the axioms for a Frobenius algebra in exactly the same way. The second way is to use the Atiyah-style definition of a TQFT, where the burden of proof is to show that, given a Frobenius algebra A, one can define the vectors Z(M) ∈ Z(∂M) in a consistent way, i.e. the definition is independent of the cutting of M into smaller pieces (this is called consistency of the sewing in conformal field theory). The third way has been implicitly suggested by Moore is to take advantage of the fact that it is relatively harmless to consider 2d cobordisms as embedded inside R3.

Definition:

Frobenius algebras are classical algebras that were once, shamefully, called ‘Frobeniusean algebras’ in honour of the Prussian mathematician Georg Frobenius. They have many equivalent definitions; but before we list them it is worthwhile to record the following fact.

Lemma:

Suppose A is an arbitrary vector space equipped with a bilinear pairing ( , ) : A ⊗ A → C. Then the following are equivalent:

  1. (a)  A is finite dimensional and the pairing is nondegenerate; i.e. A is finite dimensional and the map A → A∗ which sends v → (v, ·) is an isomorphism.
  2. (b)  A is self dual in the rigid monoidal sense; i.e. there exists a copairing i : C → A ⊗ A which is dual to the pairing e : A ⊗ A → C given by e(a, b) = ε(ab).

Proof:

(a) ⇒ (b). Choose a basis (e1, . . . , en) of A. Then by assumption the functionals (ei, ·) are a basis for A∗. Then there exist vectors e1,…en in A such that (ei, ej) = δji. Define the copairing i by setting

1 →  ∑i ei ⊗ ei

Then a general vector v = λiei goes through the composite V →i⊗id V ⊗ V ⊗ V →e⊗id V – as:

v = λiei → λiej ⊗ ej ⊗ ei → λiej(ej, ei) = λiei = v —– (1)

Similarly, w = λiei goes through the composite V →id⊗i V ⊗ V ⊗ V →e⊗id V as:

w = λiei → λiei ⊗ ej ⊗ ej → λi(ei, ej)ej = λiei = w —– (2)

(b) ⇒ (a) . The  copairing  i  singles out a vector in A ⊗ A b y 1 → ∑ni ei ⊗ ei for some vectors ei, ei ∈ A and some number n (note that we have not used finite dimensionality here). Now take an arbitrary v ∈ A and send it through the composite V →i⊗id V ⊗ V ⊗ V →e⊗id V:

v → ei ⊗ ei ⊗ v → ei(ei, v) —– (3)

By assumption this must be equal to v. This shows that (e1, . . . , en) spans A, so A is finite dimensional. Now we show that v → (v, ·) is injective, and hence an isomorphism. Suppose (v, ·) is the zero functional. Then in particular (v, ei) = 0 ∀ i. But these scalars are exactly the coordinates in the ‘basis’ (e1, . . . , en), so that v = 0.

This lemma translates the algebraic notion of nondegeneracy into category language, and from now on we shall use the two meanings interchangeably. It also makes explicit that a nondegenerate pairing allows one to construct, from a basis (e1, . . . , en) for A, a corresponding dual basis (e1, . . . , en), which satisfies e(ei, ej) = δij,and which can be recovered from the decomposition  i(1) = ∑iei ⊗ ei

A Frobenius algebra is

(a)  A finite dimensional algebra A equipped with a nondegenerate form (also called trace) ε : A → C.

(b)  A finite dimensional algebra (A, β) equipped with a pairing β : A ⊗ A → C which is nondegenerate and associative.

(c)  A finite dimensional algebra (A, γ) equipped with a left algebra isomorphism to its dual γ : A → A∗.

Observe that if A is an algebra, then there is a one-to-one correspondence between forms ε : A → C and associative bilinear pairings (·, ·) : A ⊗ A → C. Given a form, define the pairing by (a, b) = ε(ab), this is obviously associative. Given the pairing, define a form by ε(a) = (1,a) = (a,1); these are equal since the pairing is associative. This establishes the equivalence of (a) and (b).