Traces in monoidal categories

The main result of this paper is the construction of a trace and a trace pairing for endomorphisms satisfying suitable conditions in a monoidal category. This construction is a common generalization of the trace for endomorphisms of dualizable ob jects in a balanced monoidal category and the trace of nuclear operators on a locally convex topological vector space with the approximation property.

This categorical trace depends on the choice of a natural family of isomorphisms s = {s X,Y : X ⊗ Y → Y ⊗ X} for X, Y ∈ C. We don't assume that s satisfies the relations (5.8) required for the braiding isomorphism of a braided monoidal category. Apparently lacking an established name, we will refer to s as switching isomorphisms. We would like to thank Mike Shulman for this suggestion.
For the monoidal category Vect, equipped with the standard switching isomorphism s X,Y : X ⊗ Y → Y ⊗ X, x ⊗ y → y ⊗ x, the categorical trace of a finite rank (i.e. thick) endomorphism f : X → X agrees with its classical trace (see Theorem 4.1). More generally, if X is a dualizable object of a monoidal category C, the above definition agrees with the classical definition of the trace in that situation (Theorem 4.22). In general, the above trace is not well-defined, since it might depend on the factorization of f given by the triple (Z, b, t) rather than just the morphism f . As we will see in Section 4.4, this happens for example in the category of Banach spaces.
To understand the problem with defining tr(f ), let us write tr(Z, t, b) ∈ C(I, I) for the composition (1. 3), and Ψ(Z, t, b) ∈ C(X, Y ) for the composition (1.2). There is an equivalence relation on these triples (see Definition 3.3) such that tr(Z, t, b) and Ψ(Z, t, b) depend only on the equivalence class [Z, t, b]. In other words, there are welldefined maps tr : C(X, X) −→ C(I, I) where C(X, Y ) denotes the equivalence classes of triples (Z, t, b) for fixed X, Y ∈ C. We note that by construction the image of Ψ consists of the thick morphisms from X to Y . We will call elements of C(X, Y ) thickened morphisms. If f ∈ C(X, Y ) with Ψ( f ) = f ∈ C(X, Y ), we say that f is a thickener of f . Using the notation C tk (X, Y ) for the set of thick morphisms from X to Y , it is clear that there is a well-defined trace map tr : C tk (X, X) → C(I, I) making the diagram (1. 4) commutative if and only if X has the following property: Definition 1.5. An object X in a monoidal category C with switching isomorphisms has the trace property if the map tr is constant on the fibers of Ψ.
For the category Ban of Banach spaces and continuous maps, we will show in Section 4.4 that the map Ψ can be identified with the homomorphism where X ′ is the Banach space of continuous linear maps f : X → C equipped with the operator norm and ⊗ is the projective tensor product. Operators in the image of Φ are referred to as nuclear operators, and hence a morphism in Ban is thick if and only if it is nuclear. It is a classical result that the trace property for a Banach space X is equivalent to the injectivity of the map Φ which in turn is equivalent to the approximation property for X: the identity operator of X can be approximated by finite rank operators in the compact-open topology, see e.g. [Ko,§43.2(7)]. Every Hilbert space has the approximation property, but deciding whether a Banach space has this property is surprisingly difficult. Grothendieck asked this question in the fifties, but the first example of a Banach space without the approximation property was found by Enflo only in 1973 [En]. Building on Enflo's work, Szankowski showed in 1981 that the Banach space of bounded operators on an (infinite dimensional) Hilbert space does not have the approximation property [Sz].
Theorem 1.7. Let C be a monoidal category with switching isomorphisms, i.e. C comes equipped with a family of natural isomorphisms s X,Y : X ⊗ Y → Y ⊗ X. If X ∈ C is an object with the trace property, then the above categorical trace tr(f ) ∈ C(I, I) is welldefined for any thick endomorphism f : X → X. This compares to the two classical situations mentioned above as follows: (i) If X is a dualizable object, then X has the trace property and any endomorphism f of X is thick. Moreover, the categorical trace of f agrees with its classical trace.
(ii) In the category TV of topological vector spaces (locally convex, complete, Hausdorff ), a morphism is thick if and only if it is nuclear, and the approximation property of an object X ∈ TV implies the trace property. Moreover, if f : X → X is a nuclear endomorphism of an object with the approximation property, then the categorical trace of f agrees with its classical trace.
The first part sums up our discussion above. Statements (i) and (ii) appear as Theorem 4.22 respectively Theorem 4.27 below. It would be interesting to find an object in TV which has the trace property but not the approximation property.
To motivate our second main result, Theorem 1.10, we note that a monoidal functor F : C → D preserves thick and thickened morphisms and gives commutative diagrams for the map Ψ from (1.4). If F is compatible with the switching isomorphisms then it also commutes with tr. However, the trace property is not functorial in the sense that if some object X ∈ C has the trace property then it is not necessarily inherited by F (X) (unless F is essentially surjective and full or has some other special property). In particular, when the functor F is a field theory, then, as explained in the next section, this non-functoriality causes a problem for calculating the partition function of F . We circumvent this problem by replacing the trace by a closely related trace pairing tr : C tk (X, Y ) × C tk (Y, X) −→ C(I, I) (1.8) for objects X, Y of a monoidal category C with switching isomorphisms. Unlike the trace map tr : C tk (X, X) −→ C(I, I) discussed above, which is only defined if X has the trace property, no condition on X or Y is needed to define this trace pairing tr(f, g) as follows. Let f ∈ C(X, Y ), g ∈ C(Y, X) be thickeners of f respectively g (i.e., Ψ( f ) = f and Ψ( g) = g). We will show that elements of C(X, Y ) can be pre-composed or postcomposed with ordinary morphisms in C (see Lemma 3.9). This composition gives elements f • g and f • g in C(Y, Y ) which we will show to be equal in Lemma 3.11. Hence the trace pairing defined by is independent of the choice of f and g. We note that Ψ and hence if Y has the trace property, then In other words, the trace pairing tr(f, g) is a generalization of the categorical trace of f • g, defined in situations where this trace might not be well-defined. The trace pairing has the following properties that are analogous to properties one expects to hold for a trace. We note that the relationship (1.9) immediately implies these properties for our trace defined for thick endomorphism of objects satisfying the trace property.
Theorem 1.10. Let C be a monoidal category with switching isomorphisms. Then the trace pairing (1.8) is functorial and has the following properties: 1. tr(f, g) = tr(g, f ) for thick morphisms f ∈ C tk (X, Y ), g ∈ C tk (Y, X). If Y has the trace property then tr(f, g) = tr(f • g) and symmetrically for X.
2. If C is an additive category with distributive monoidal structure (see Definition 5.3), then the trace pairing is a bilinear map.
, provided s gives C the structure of a symmetric monoidal category. More generally, this property holds if C is a balanced monoidal category.
We recall that a balanced monoidal category is a braided monoidal category equipped with a natural family of isomorphisms θ = {θ X : X → X} called twists satisfying a compatibility condition (see Definition 5.12). Symmetric monoidal categories are balanced monoidal categories with θ ≡ id. For a balanced monoidal category C with braiding isomorphism c X,Y : X ⊗ Y → Y ⊗ X and twist θ X : X → X, one defines the switching isomorphism s X,Y : There are interesting examples of balanced monoidal categories that are not symmetric monoidal, e.g., categories of bimodules over a fixed von Neumann algebra (monoidal structure given by Connes fusion) or categories of modules over quantum groups. Traces in the latter are used to produce polynomial invariants for knots. Originally, we only proved the multiplicative property of our trace pairing for symmetric monoidal categories. We are grateful to Gregor Masbaum for pointing out to us the classical definition of the trace of an endomorphism of a dualizable object in a balanced monoidal category which involves using the twist (see [JSV]).
The rest of this paper is organized as follows. In section 2 we explain the motivating example: we consider the d-dimensional Riemannian bordism category, explain what a thick morphism in that category is, and show that the partition function of a 2-dimensional Riemannian field theory can be expressed as the relative trace of the thick operators that a field theory associates to annuli. Section 2 is motivational and can be skipped by a reader who wants to see the precise definition of C(X, Y ), the construction of tr and a statement of the properties of tr which are presented in section 3. In section 4 we discuss thick morphisms and their traces in various categories. In section 5 we prove the properties of tr and deduce the corresponding properties of the trace pairing stated as Theorem 1.10 above.
Both authors were partially supported by NSF grants. They would like to thank the referee for many valuable suggestions. The first author visited the second author at the Max-Planck-Institut in Bonn during the Fall of 2009 and in July 2010. He would like to thank the institute for the support and for the stimulating atmosphere.

Motivation via field theories
A well-known aximatization of field theory is due to Graeme Segal [Se] who defines a field theory as a monoidal functor from a bordism category to the category TV of topological vector spaces. The precise definition of the bordism category depends on the type of field theory considered; for a d-dimensional topological field theory, the objects are closed (d − 1)-dimensional manifolds and morphisms are d-dimensional bordisms (more precisely, equivalence classes of bordisms where we identify bordisms if they are diffeomorphic relative boundary). Composition is given by gluing of bordisms, and the monoidal structure is given by disjoint union.
For other types of field theories, the manifolds constituting the objects and morphisms in the bordism category come equipped with an appropriate geometric structure; e.g., a conformal structure for conformal field theories, a Riemannian metric for Riemannian field theories, or a Euclidean structure (= Riemannian metric with vanishing curvature tensor) for a Euclidean field theory. In these cases more care is needed in the definition of the bordism category to ensure the existence of a well-defined composition and the existence of identity morphisms.
Let us consider the Riemannian bordism category d-RBord. The objects of d-RBord are closed Riemannian (d − 1)-manifolds. A morphism from X to Y is a d-dimensional Riemannian bordism Σ from X to Y , that is, a Riemannian d-manifold Σ with boundary and an isometry X ∐ Y → ∂Σ. More precisely, a morphism is an equivalence class of Riemannian bordisms, where two bordisms Σ, Σ ′ are considered equivalent if there is an isometry Σ → Σ ′ compatible with the boundary identifications. In order to have a well-defined compositon by gluing Riemannian bordisms we require that all metrics are product metrics near the boundary. To ensure the existence of identity morphisms, we enlarge the set of morphisms from X to Y by also including all isometries X → Y . Pre-or post-composition of a bordism with an isometry is the given bordism with boundary identification modified by the isometry. In particular, the identity isometry Y → Y provides the identity morphism for Y as object of the Riemannian bordism category d-RBord.
A more sophisticated way to deal with the issues addressed above was developed in our paper [ST2]. There we don't require the metrics on the bordisms to be a product metric near the boundary; rather, we have more sophisticated objects consisting of a closed (d − 1)-manifold equipped with a Riemannian collar. Also, it is technically advantageous not to mix Riemannian bordisms and isometries. This is achieved in that paper by constructing a suitable double category (or equivalently, a category internal to categories), whose vertical morphisms are isometries and whose horizontal morphisms are bordisms between closed (d − 1)-manifolds equipped with Riemannian collars. The 2-morphisms are isometries of such bordisms, relative boundary. When using the results of the current paper in [ST1], we translate between the approach here using categories versus the approach via internal categories used in [ST2].
Let E be d-dimensional Riemannian field theory, that is, a symmetric monoidal functor For the bordism category d-RBord the symmetric monoidal structure is given by disjoint union; for the category TV it is given by the projective tensor product. Let X be a closed Riemannian (d − 1)-manifold and Σ be a Riemannian bordism from X to itself. Let Σ gl be the closed Riemannian manifold obtained by gluing the two boundary pieces (via the identity on X). Both Σ and Σ gl are morphisms in d-RBord: We note that ∅ is the monoidal unit in d-RBord, and hence the vector space E(∅) can be identified with C, the monoidal unit in TV. In particular, E(Σ gl ) ∈ Hom(E(∅), E(∅)) = Hom(C, C) = C is a complex number.
Question. How can we calculate E(Σ gl ) ∈ C in terms of the operator E(Σ) : E(X) → E(X)?
We would like to say that E(Σ gl ) is the trace of the operator E(Σ), but to do so we need to check that the conditions guaranteeing a well-defined trace are met. For a topological field theory E this is easy: In the topological bordism category every object X is dualizable (see Definition 4.17), hence E(X) is dualizable in TV which is equivalent to dim E(X) < ∞. By contrast, for a Euclidean field theory the vector space E(X) is typically infinite dimensional, and hence to make sense of the trace of the operator E(Σ) associated to a bordism Σ from X to itself, we need to check that the operator E(Σ) is thick and that the vector space E(X) has the trace property.
It is easy to prove (see Theorem 4.38) that every object X of the bordism category d-RBord has the trace property and that among the morphisms of d-RBord (consisting of Riemannian bordisms and isometries), exactly the bordisms are thick. The latter characterization motivated the adjective 'thick', since we think of isometries as 'infinitely thin' Riemannian bordisms. It is straightforward to check that being thick is a functorial property in the sense that the thickness of Σ implies that E(Σ) is thick. Unfortunately, as already mentioned in the introduction, the trace property is not functorial and we cannot conclude that E(X) has the trace property.
Replacing the problematical trace by the well-behaved trace pairing leads to the following result. It is applied in [ST1] to prove the modularity and integrality of the partition function of a supersymmetric Euclidean field theory of dimension 2.
Theorem 2.1. Suppose Σ 1 is a Riemannian bordism of dimension d from X to Y , and Σ 2 is a Riemannian bordism from Y to X. Let Σ = Σ 1 • Σ 2 be the bordism from Y to itself obtained by composing the bordisms Σ 1 and Σ 2 , and let Σ gl be the closed Riemannian d-manifold obtained from Σ by identifying the two copies of Y that make up its boundary. If E is d-dimensional Riemannian field theory, then E(Σ gl ) = tr(E(Σ 2 ), E(Σ 1 )).

Thickened morphisms and their traces
In this section we will define the thickened morphisms C(X, Y ) and the trace tr( f ) ∈ C(I, I) of thickened endomorphisms f ∈ C(X, X) for a monoidal category C equipped with a natural family of isomorphisms s X,Y : X ⊗ Y → Y ⊗ X.
We recall that a monoidal category is a category C equipped with a functor ⊗ : C × C → C called the tensor product, a distinguished element I ∈ C and natural isomorphisms for objects X, Y, Z ∈ C. These natural isomorphisms are required to make two diagrams (known as the associativity pentagon and the triangle for unit) commutative, see [McL]. It is common to use diagrams to represent morphisms in C (see for example [JS1]). The pictures represent a morphism f : U ⊗ V ⊗ W → X ⊗ Y and the tensor product of morphisms g : X → Y and g ′ : X ′ → Y ′ , respectively. The composition h • g of morphisms g : X → Y and h : Y → Z is represented by the picture With tensor products being represented by juxtaposition of pictures, the isomorphisms X ∼ = X ⊗ I ∼ = I ⊗ X suggest to delete edges labeled by the monoidal unit I from our picture. E.g., the pictures represent morphisms t : I → Y ⊗ Z and b : Z ⊗ X → I, respectively. Rephrasing Definition 1.1 of the introduction in our pictorial notation, a morphism f : Here t stands for top and b for bottom. We will use the notation C tk (X, Y ) ⊂ C(X, Y ) for the subset of thick morphisms.

Thickened morphisms
It will be convenient for us to characterize the thick morphisms as the image of a map the domain of which we refer to as thickened morphisms. Definition 3.3. Given objects X, Y ∈ C, a thickened morphism from X to Y is an equivalence class of triples (Z, t, b) consisting of an object Z ∈ C, and morphisms To describe the equivalence relation, it is useful to think of these triples as objects of a category, and to define a morphism from ( In other words, a thickened morphism is a path component of the category defined above. We write C(X, Y ) for the thickened morphisms from X to Y .
As suggested by the referee, it will be useful to regard C(X, Y ) as a coend; this will streamline the proofs of some results. Let us consider the functor We note that for any (t, b ′ ) ∈ S(Z ′ , Z) the two triples i.e., the quotient space of Z∈C S(Z, Z) obtained by identifying all image points of these two maps.
This coequalizer can be formed for any functor S : C op × C → Set; it is called the coend of S, and following [McL,Ch. IX,§6], we will use the integral notation Z∈C S(Z, Z) for the coend. Summarizing our discussion, we have the following way of expressing C(X, Y ) as a coend: Lemma 3.6. Given a triple (Z, t, b) as above, let Ψ(Z, t, b) ∈ C(X, Y ) be the composition on the right hand side of equation (3.2). Then Ψ only depends on the equivalence class [Z, t, b] of (Z, t, b), i.e. the following map is well-defined: We note that by construction the image of Ψ is equal to C tk (X, Y ), the set of thick morphisms from X to Y . As mentioned in the introduction, for f ∈ C(X, Y ) we call Remark 3.8. The difference between an orientable versus an oriented manifold is that the former is a property, whereas the latter is an additional structure on the manifold. In a similar vain, being thick is a property of a morphism f ∈ C(X, Y ), whereas a thickener f is an additional structure. To make the analogy between these situations perfect, we were tempted to introduce the words thickenable or thickable into mathematical English. However, we finally decided against it, particularly because the thick-thin distinction for morphisms in the bordism category is just perfectly suited for the purpose.
Thickened morphisms can be pre-composed or post-composed with ordinary morphisms to obtain again thickened morphisms as follows: The proof of the following result is straightforward and we leave it to the reader.
Lemma 3.9. The composition of morphisms with thickened morphisms is well-defined and compatible with the usual composition via the map Ψ in the sense that the following diagrams commutes: This shows that g is an equivalence from (Z 1 , t 1 , b 1 •(id Z 1 ⊗f 2 )) to (Z 2 , (f 1 ⊗id Z 2 )•t 2 , b 2 ) and hence these triples represent the same element of C(U, Y ) as claimed.

The trace of a thickened morphism
Our next goal is to show that for a monoidal category C with switching isomorphism s X,Z : is well-defined. We recall from equation 1.3 of the introduction that tr(Z, t, b) is defined by tr(Z, t, b) = b • s X,Z • t. In our pictorial notation, we write it as We note that the naturality of s X,Y is expressed pictorially as Lemma 3.15. tr(Z, t, b) depends only on the equivalence class of (Z, t, b). In particular, the map (3.12) is well-defined.
In terms of the coend description, this lemma is actually obvious because tr is the composition where the first map is given by switching and the second by composition.

Traces in various categories
The goal in this section is to describe thick morphisms in various categories in classical terms and relate our categorical trace to classical notions of trace. At a more technical level, we describe the map Ψ : C(X, Y ) → C(X, Y ) and its image C tk (X, Y ) in these categories.
In the first subsection this is done for the category of vector spaces (not necessarily finite dimensional). In the second subsection we show that Ψ is a bijection if X is dualizable and hence any endomorphism of a dualizable object has a well-defined trace. Traces in categories for which all objects are dualizable are well-studied [JS2]. In the third subsection we introduce semi-dualizable objects and describe the map Ψ in more explicit terms. In a closed monoidal category every object is semi-dualizable. In the following subsection we apply these considerations to the category of Banach spaces. Then we discuss the category of topological vector spaces, and finally the Riemannian bordism category.

The category of vector spaces
Theorem 4.1. Let C be the monoidal category Vect F of vector spaces (not necessarily finite dimensional) over a field F equipped with the usual tensor product and the usual switching isomorphism s X,Y : 3. For a finite rank endomorphism f , its categorical trace tr(f ) agrees with its usual (classical) trace which we denote by cl-tr(f ).
For a vector space X, let X * be the dual vector space, and define the morphism We note that the set C(F, Y ⊗ X * ) can be identified with Y ⊗ X * via the evaluation map t → t(1). Using this identification, Φ maps an elementary tensor y ⊗ g ∈ Y ⊗ X * to the linear map x → yg(x) which is a rank ≤ 1 operator. Since finite rank operators are finite sums of rank one operators, we see that the image Φ consists of the finite rank operators. The classical trace of the rank one operator Φ(y ⊗ g) is defined to be g(y); by linearity this extends to a well-defined trace for all finite rank operators. The proof of Theorem 4.1 is based on the following lemma that shows that Ψ is equivalent to the map Φ.
is a bijection and Ψ•α = Φ. Here ev : X * ⊗X → F is the evaluation map g ⊗x → g(x).
In subsection 4.2, we will construct the map Φ and prove this lemma in the more general context where X is a semi-dualizable object of a monoidal category.
Proof of Theorem 4.1. Statement (1) follows immediately from the Lemma. To prove part (2), it suffices to show that Φ is injective which is well known and elementary.
For the proof of part (3) let f : X → X be a thick morphism. We recall that its categorical trace tr(f ) is defined by tr(f ) = tr( f ) for a thickener f ∈ C(X, X). So Ψ( f ) = f and this is well-defined thanks to the injectivity of Ψ. Using that α is a bijection, there is a unique t ∈ C(F, and hence tr(f ) = g(y) = cl-tr(f ) is the classical trace of the rank one operator Φ(t) = f given by x → yg(x). Since both the categorical trace of α(t) and the classical trace of Φ(t) depend linearly on t, this finishes the proof.
Remark 4.5. We can replace the monoidal category of vector spaces by the monoidal category SVect of super vector spaces. The objects of this category are just Z/2-graded vector spaces with the usual tensor product ⊗. The grading involution on X ⊗ Y is the tensor product ǫ X ⊗ ǫ Y of the grading involutions on X respectively Y . The switching isomorphism X ⊗ Y ∼ = Y ⊗ X is given by x ⊗ y → (−1) |x||y| y ⊗ x for homogeneous elements x ∈ X, y ∈ Y of degree |x|, |y| ∈ Z/2. Then the statements above and their proof work for SVect as well, except that the categorical trace of a finite rank endomorphism T : X → X is its super trace str(T ) := cl-tr(ǫ X T ).

Thick morphisms with semi-dualizable domain
The goal of this subsection is to generalize Lemma 4.11 from the category of vector spaces to general monoidal categories C, provided the object X ∈ C satisfies the following condition.
Definition 4.6. An object X of a monoidal category C is semi-dualizable if the functor By Yoneda's Lemma, the object X ∨ is unique up to isomorphism. It is usually referred to as the (left) internal hom and denoted by C(X, I).
To put this definition in context, we recall that a monoidal category C is closed if for any X ∈ C the functor C → C, Z → Z ⊗ X has a right adjoint; i.e., if there is a functor C(X, ) : C −→ C and natural bijections In particular, in a closed monoidal category every object X is semi-dualizable with X ∨ = C(X, I). The category C = Vect F is an example of a closed monoidal category; for vector spaces X, Y the internal hom C(X, Y ) is the vector space of linear maps from X to Y . Hence every vector space is semi-dualizable with semi-dual X ∨ = C(X, I) = X * . Other examples of closed monoidal categories is the category of Banach spaces (see subsection 4.4) and the category of bornological vector spaces [Me]. As also discussed in [Me,p. 9], the symmetric monoidal category TV of topological vector spaces with the projective tensor product is not an example of a closed monoidal category (no matter which topology on the space of continuous linear maps is used). Still, some topological vector spaces are semi-dualizable (e.g., Banach spaces), and so it seems preferable to state our results for semi-dualizable objects rather than objects in closed monoidal categories.
In Lemma 4.11 we will generalize a statement about the category Vect F to a statement about a general monoidal category C. To do so, we need to construct the maps Φ and α in the context of a general monoidal category. This is straightforward by using the same definitions as above, just making the following replacements: 1. Replace F, the monoidal unit in Vect F , by the monoidal unit I ∈ C.
2. Replace X * , the vector space dual to X, by X ∨ , the semi-dual of X ∈ C. Here we need to assume that X ∈ C is semi-dualizable which is automatic for any object of a closed category like Vect F .
3. For a semi-dualizable object X ∈ C, the evaluation map ev : is by definition the morphism that corresponds to the identity morphism X ∨ → X ∨ under the bijection (4.7) for Z = X ∨ . It is easy to check that this agrees with the usual evaluation map X * ⊗ X → F for C = Vect F .
The following Lemma is a generalization of Lemma 4.3.
Lemma 4.11. Let X be a semi-dualizable object of a monoidal category C. Then for any object Y ∈ C the map is a natural bijection which makes the diagram Proof. The commutativity of the diagram is clear by comparing the definitions of the maps Φ (see Equation (4.10)), α (Equation (4.12)), and Ψ (Equation (3.7)). To see that α is a bijection we factor it in the following form Here the first map sends t ∈ C(I, . This map is a bijection by the coend form of the Yoneda Lemma according to which for any functor F : C → Set the map is a bijection (see [Ke,Equation 3.72]. The second map of Equation (4.13) is induced by the bijection C(Z, X ∨ ) ∼ = C(Z ⊗ X, I). By construction the identity map id X ∨ corresponds to the evaluation map via this bijection, and hence the composition of these two maps is α. The monoidal structure ⊗ on C induces a monoidal structure * on F, the convolution tensor product [Day], defined by for an object S ∈ C. Equipped with the convolution product, the functor category F is a closed monoidal category (see Equation ( We can regard C as a full monoidal subcategory of F via the Yoneda embedding. In particular, every object X ∈ C has a left semi-dual X ∨ = F(X, −) ∈ F and hence by the previous lemma we have a bijection F(X, Y ) ∼ = F(I, Y * X ∨ ). Explicitly, the semi-dual X ∨ is given by The referee observed that the Yoneda embedding induces a bijection In particular, morphism f ∈ C(X, Y ) is thick if and only if its image under the Yoneda embedding is thick. To see that the above map is a bijection, we evaluate the right hand side which we recognize as the coend description of C(X, Y ) (Equation (3.5)). The second equality is a consequence of (the coend form of) the Yoneda lemma (equation (4.14)).

Thick morphisms with dualizable domain
As mentioned in the introduction, there is a well-known trace for endomophisms of dualizable objects in a monoidal category (see e.g. [JSV,§3]). After recalling the definition of dualizable and the construction of that classical trace, we show in Theorem 4.22 that our categorical trace is a generalization.
Definition 4.17. [JS2,Def. 7.1] An object X of a monoidal category C is (left) dualizable if there is an object X ∨ ∈ C (called the (left) dual of X) and morphisms ev : X ∨ ⊗ X → I (called evaluation map) and coev : I → X ⊗ X ∨ (coevaluation map) such that the following equations hold.
If X is dualizable with dual X ∨ , then there is a family of bijections In particular, a dualizable object is semi-dualizable in the sense of Definition 4.6.
Example 4.20. A finite dimensional vector space X is a dualizable object in the category Vect F : we take X ∨ to be the vector space dual to X, ev to be the usual evaluation map, and define coev : where {e i } is a basis of X and {e i } is the dual basis of X ∨ . It is not hard to show that a vector space X is dualizable if and only if it is finite dimensional. Definition 4.21. Let C be a monoidal category with switching isomorphisms. Let f : X → X be an endomorphism of a dualizable object X ∈ C. Then the classical trace cl-tr(f ) ∈ C(I, I) is defined by This definition can be found for example in section 3 of [JSV] for balanced monoidal categories (see Definition 5.12 for the definition of a balanced monoidal category and how the switching isomorphism is determined by the braiding and the twist of the balanced monoidal category). In fact, the construction in [JSV] is more general: they associate to a morphism f : A ⊗ X → B ⊗ X a trace in C(A, B) if X is dualizable; specializing to A = B = I gives the classical trace described above.
Theorem 4.22. Let C be a monoidal category with switching isomorphisms, and let X be a dualizable object of C. Then 1. The map Ψ : C(X, Y ) → C(X, Y ) is a bijection. In particular, any morphism with domain X is thick, and any endomorphism f : X → X has a well-defined categorical trace tr(f ) ∈ C(I, I).
2. The categorical trace of f is equal to its classical trace cl-tr(f ) defined above.
Remark 4.23. Part (1) of the Theorem implies in particular that if X is (left) dualizable, then the identity id X is thick. The referee observed that the converse holds as well. To see this, assume that id X is thick. Then id X is in the image of Ψ : C(X, X) → C(X, X) and hence by Lemma 4.11 in the image of Φ : C(I, X ⊗ X ∨ ) → C(X, X). If coev ∈ C(I, X ⊗ X ∨ ) belongs to the preimage Φ −1 (id X ), then it is straightforward to check that Equations (4.10) hold and hence X is dualizable. The first equation holds by construction of Φ; to check the second equation, we apply the bijection (4.7) (for Z = X ∨ ) to both sides and obtain ev for both.
Proof. To prove part (1), it suffices by Lemma 4.11 to show that the map Φ : C(I, Y ⊗ X ∨ ) → C(X, Y ) (see Equation (4.10)) is a bijection. Comparing Φ with the natural bijection (4.19) for dualizable objects X ∈ C, we see that this bijection is equal to Φ in the special case Z = I.
To prove part (2) we recall that by definition tr(f ) = tr( f ) ∈ C(I, I) for any f ∈ Ψ −1 (f ) ⊂ C(X, X) (see Equation (1.4)). In the situation at hand, Ψ is invertible by part (1), and using the factorization Ψ = Φ • α −1 provided by Lemma 4.11, we have Here the second equality follows from the explicit form of the inverse of Φ (which agrees with the map (4.19) for Z = I).

The Category Ban of Banach spaces
Let X be a Banach space and f : X → X a continuous linear map. There are classical conditions (f is nuclear and X has the approximation property, see Definition 4.25) which guarantee that f has a well-defined (classical) trace which we again denote by cl-tr(f ) ∈ C. For example any Hilbert space H has the approximation property and a continuous linear map f : H → H is nuclear if and only if it is trace class. The main result of this subsection is Theorem 4.26 which shows that these classical conditions imply that f has a well-defined categorical trace and that the categorical trace of f agrees with its classical trace. Before stating this result, we review the (projective) tensor product of Banach spaces and define the notions nuclear and approximation property.
The category Ban of Banach spaces is a monoidal category whose monoidal structure is given by the projective tensor product, defined as follows. For Banach spaces X, Y , the projective norm on the algebraic tensor product X ⊗ alg Y is given by where the infimum is taken over all ways of expressing z ∈ X ⊗ alg Y as a finite sum of elementary tensors. Then the projective tensor product X ⊗ Y is defined to be the completion of X ⊗ alg Y with respect to the projective norm. It is well-known that Ban is a closed monoidal category (see Equation (4.8)). For Banach spaces X, Y the internal hom space Ban(X, Y ) is the Banach space of continuous linear maps T : X → Y equipped with the operator norm ||T || := sup x ∈ X, ||x|| = 1||T (x)||. In particular, every Banach space has a left semi-dual X ∨ = Ban(X, I) in the sense of Definition 4.17 which is just the Banach space of continuous linear maps X → C. The categorically defined evaluation map ev : X ∨ ⊗X → C (see Equation(4.9)) agrees with the usual evaluation map defined by f ⊗ (4.10)) is determined by sending y ∈ f ∈ Y ⊗ X ∨ to the map x → yf (x) (see the discussion after Theorem 4.1).
We note that the morphism set Ban(X ⊗ Y, Z) is in bijective correspondence to the continuous bilinear maps X ×Y → Z. This bijection is given by sending g : X ⊗Y → Z to the composition g • χ, where χ : X × Y → X ⊗ Y is given by (x, y) → x ⊗ y. In particular, if X ∨ is the Banach space dual to X, we have a morphism ev : called the evaluation map.
A Banach space X has the approximation property if the identity of X can be approximated by finite rank operators with respect to the compact-open topology.
1. A morphism f : X → Y is thick if and only if it is nuclear.
2. If X has the approximation property, it has the trace property.
3. If X has the approximation property and f : X → X is nuclear, then the categorical trace of f agrees with its classical trace.
Proof. This result holds more generally in the category TV which we will prove in the following subsection (see Theorem 4.27). For the proofs of statements (2) and (3) we refer to that section. For the proof of statement (1) we recall that a morphism f : X → Y in the category C = Ban is thick if and only if if is in the image of the map Ψ : (1) is a corollary of Lemma 4.11 according to which these two maps are equivalent for any semi-dualizable object X of a monoidal category C. In particular, since Ban is a closed monoidal category, every Banach space is semi-dualizable.

A Category of topological vector spaces
In this subsection we extend Theorem 4.26 from Banach spaces to the category TV whose objects are locally convex topological vector spaces which are Hausdorff and complete. We recall that the topology on a vector space X is required to be invariant under translations and dilations. In particular, it determines a uniform structure on X which in turn allows us to speak of Cauchy nets and hence completeness; see [Sch,section I.1] for details. The morphisms of TV are continuous linear maps, and the projective tensor product described below gives TV the structure of a symmetric monoidal category. It contains the category Ban of Banach spaces as a full subcategory. Theorem 4.27. Let X, Y be objects in the category TV.
1. A morphism f : X → Y is thick if and only if it is nuclear.
2. If X has the approximation property, then it has the trace property.
3. If X has the approximation property, and f : X → X is nuclear, then the categorical trace of f agrees with its classical trace.
Before proving this theorem, we define nuclear morphisms in TV and the approximation property. Then we'll recall the classical trace of a nuclear endomorphism of a topological vector space with the approximation property as well as the projective tensor product. 4.29) where f 0 is a nuclear map between Banach spaces (see Definition 4.25) and p, j are continuous linear maps.
The definition of nuclearity in Schaefer's book ( [Sch,p. 98]) is phrased differently. We give his more technical definition at the end of this section and show that a continuous linear map is nuclear in his sense if and only if it is nuclear in the sense of the above definition.
Approximation property. An object X ∈ TV has the approximation property if the identity of X is in the closure of the subspace of finite rank operators with respect to the compact-open topology [Sch, Chapter III, section 9] (our completeness assumption for topological vector spaces implies that uniform convergence on compact subsets is the same as uniform convergence on pre-compact subsets).
The classical trace for nuclear endomorphisms. Let f be a nuclear endomorphism of X ∈ TV, and let I ν : X → X be a net of finite rank morphisms which converges to the identity on X in the compact-open topology. Then f •I ν is a finite rank operator which has a classical trace cl-tr(f • I ν ). It can be proved that the limit lim ν cl-tr(f • I ν ) exists [Li,Proof of Theorem 1] and is independent of the choice of the net I ν (this also follows from our proof of Theorem 4.27). The classical trace of f is defined by Projective tensor product. The projective tensor product of Banach spaces defined in the previous section extends to topological vector spaces as follows. For X, Y ∈ TV the projective topology on the algebraic tensor product X ⊗ alg Y is the finest locally convex topology such that the canonical bilinear map is continuous [Sch,p. 93]. The projective tensor product X ⊗ Y is the topological vector space obtained as the completion of X ⊗ alg Y with respect to the projective topology. It can be shown that it is locally convex and Hausdorff and that the morphisms TV(X ⊗ Y, Z) are in bijective correspondence to continuous bilinear maps X × Y → Z; this bijection is given by sending f : X ⊗ Y → Z to f • χ [Sch, Chapter III, section 6.2].
Semi-norms. For checking the convergence of a sequence or continuity of a map between locally convex topological vector spaces, it is convenient to work with seminorms. For X ∈ TV and U ⊂ X a convex circled 0-neighborhood (U is circled if λU ⊂ U for every λ ∈ C with |λ| ≤ 1) one gets a semi-norm Conversely, a collection of semi-norms determines a topology, namely the coarsest locally convex topology such that the given semi-norms are continuous maps. For example, if X is a Banach space with norm || ||, we obtain the usual topology on X.
As another example, the projective topology on the algebraic tensor product X ⊗ alg Y is the topology determined by the family of semi-norms || || U,V parametrized by convex circled 0-neighborhoods U ⊂ X, V ⊂ Y defined by where the infimum is taken over all ways of writing z ∈ X ⊗ alg Y as a finite sum of elementary tensors. It follows from this description that the projective tensor product defined above is compatible with the projective tensor product of Banach spaces defined earlier (see [Sch,Chapter III,section 6.3]). Our next goal is the proof of Theorem 4.27, for which we will use the following lemma.
For the proof of this lemma, we will need the following two ways to construct Banach spaces from a topological vector space X: 1. Let U be a convex, circled neighborhood of 0 ∈ X. Let X U be the Banach space obtained from X by quotiening out the null space of the semi-norm || || U and by completing the resulting normed vector space. Let p U : X → X U be the evident map.
2. Let B be a convex, circled bounded subset of X. We recall that B is bounded if for each neighborhood U of 0 ∈ X there is some λ ∈ C such that B ⊂ λU . Let X B be the vector space X B := ∞ n=1 nB equipped with the norm ||x|| B := inf{λ ∈ R >0 | x ∈ λB}. If B is closed in X, then X B is complete (by our assumption that X is complete), and hence X B is a Banach space ([Sch, Ch. III, §7; p. 97]). The inclusion map j B : X B → X is continuous thanks to the assumption that B is bounded.
Proof of Lemma 4.31. To prove part (1) we use the fact (see e.g. Theorem 6.4 in Chapter III of [Sch]) that any element of the completed projective tensor product Y ⊗ Z, in particular the element t(1), can be written in the form containing B ′ ). We note that B ′ is bounded, hence its convex, circled hull is bounded, and hence B is bounded. We define j : Y 0 → Y to be the map j B : Y B → Y . To finish the proof of part (1), it suffices to show that t(1) is in the image of the inclusion map j B ⊗id Z : Y B ⊗Z ֒→ Y ⊗Z. It is clear that each partial sum n i=1 λ i y i ⊗ z i belongs to the algebraic tensor product Y B ⊗ alg Z, and hence we need to show that the sequence of partial sums is a Cauchy sequence with respect to the semi-norms || || B,V on Y B ⊗ alg Z that define the projective topology (here V runs through all convex circled 0-neighborhoods V ⊂ Z). Since y i ∈ B, it follows ||y i || B ≤ 1 and hence we have the estimate Since z i → 0, we have ||z i || V ≤ 1 for all but finitely many i, and this implies that the partial sums form a Cauchy sequence.
To prove part (2) we recall that the morphism b : Z ⊗ X → C corresponds to a continuous bilinear map b ′ : is clear by construction. Defining the map p : X → X 0 to be p U : X → X U , we obtained the desired factorization of b.
Remark 4.33. The proofs above and below imply that for fixed objects X, Y ∈ TV, the thickened morphisms TV(X, Y ) actually form a set. Any triple (Z, t, b) can be factored into Banach spaces X 0 , Y 0 , Z 0 as explained in the 3 pictures below. By the argument above, we may actually choose X 0 = X U where U runs over certain subsets of X. Since X is fixed, it follows that the arising Banach spaces X U range over a certain set. Finally, by Lemma 4.11, the Banach space Z 0 may be replaced by X ∨ U without changing the equivalence class of the triple. Therefore, the given triple (Z, t, b) is equivalent to a triple of the form (X ∨ U , t ′ , b ′ ). Since the collection of objects X ∨ U ∈ TV forms a set, we see that TV(X, Y ) is a set as well.
In this paper, we have not addressed the issue whether C(X, Y ) is a set because this problem does not arise in the examples we discuss: The argument above for C = TV is the hardest one, in all other examples we actually identify C(X, Y ) with some very concrete set.
This problem is similar to the fact that presheaves on a given category do not always form a category (because natural transformations do not always form a set). So we are following the tradition of treating this problem only if forced to.
Proof of Theorem 4.27. The factorization (4.29) shows that a nuclear morphism f : X → Y factors through a nuclear map f 0 : X 0 → Y 0 of Banach spaces. Then f 0 is thick by Theorem 4.26 and hence f is thick, since pre-or post-composition of a thick morphism with an any morphism is thick. To prove the converse, assume that f is thick, i.e., that it can be factored in the form Then using Lemma 4.31 to factorize t and b, we see that f can be further factored in the form It remains to show that f 0 is a nuclear map between Banach spaces. In the category of Banach spaces a morphisms is nuclear if and only if it is thick by Theorem 4.26. At first glance, it seems that the factorization of f 0 above shows that f 0 is thick. However, on second thought one realizes that we need to replace Z by a Banach space to make that argument. This can be done by using again our Lemma 4.31 to factorize t 0 and hence f 0 further in the form This shows that f 0 is a thick morphism in the category Ban and hence nuclear by Theorem 4.26. The key for the proof of parts (2) and (3) of Theorem 4.27 will be the following lemma. To prove part (2) of Theorem 4.27, assume that X has the approximation property and let I ν : X → X be a net of finite rank operators converging to the identity of X in the compact open topology. Then by the lemma, for any f ∈ TV(X, X), the net converges to tr( f • id X ) = tr( f ).
Here I ν ∈ TV(X, X) are thickeners of I ν . They exist since every finite rank morphisms is nuclear and hence thick by part (1). This implies the trace condition for X, since tr( f ) = lim ν tr(f • I ν ) depends only on f . To prove part (3) let f ∈ C(X, X) be a nuclear endomorphism of X ∈ TV and let I ν be a net of finite rank operators converging to the identity of X in the compact open topology. By part (1) f is thick, i.e., there is a thickener f ∈ C(X, X). Then as discussed above, we have and For the finite rank operator f • I ν its classical trace cl-tr(f • I ν ) and its categorical trace tr(f • I ν ) agree by part (3) of Theorem 4.1.
Proof of Lemma 4.34 and hence tr( f • g) is given by the composition As in Equation (4.32) we write t(1) ∈ X ⊗ Z in the form To show that the map g → tr( f • g) is continuous, we will construct for given ǫ > 0 a compact subset K ⊂ X and an open subset U ⊂ Y such that for To construct U , we note that the continuity of b ′ implies that there are 0-neighborhoods U ⊂ Y , V ⊂ Z such that y ∈ U , z ∈ V implies |b ′ (z, y)| < ǫ. Without loss of generality we can assume z i ∈ V for all i (by replacing z i by cz i and x i by x i /c for a sufficiently small number c) and |λ i | = 1 (by replacing λ i by λ i /s and x i by sx i for s = |λ i |). We define K : Finally we compare our definition of a nuclear map between topological vector spaces (see Definition 4.25) with the more classical definition which can be found e.g. in [Sch,p. 98]. A continuous linear map f ∈ TV(X, Y ) is nuclear in the classical sense if there is a convex circled 0-neighborhood U ⊂ X, a closed, convex, circled, bounded subset B ⊂ Y such that f (U ) ⊂ B and the induced map of Banach spaces X U → Y B is nuclear. Proof. If f : X → Y is nuclear in the classical sense, it factors in the form where f 0 is nuclear. Hence f is nuclear.
Conversely, let us assume that f is nuclear, i.e., that it factors in the form where f 0 is a nuclear map between Banach spaces. To show that f is nuclear in the classical sense, we will construct a convex circled 0-neighborhood U ⊂ X, and a closed, convex, circled, bounded subset B ⊂ Y such that f (U ) ⊂ B and we have a commutative diagram Then the induced map f ′ factors through the nuclear map f 0 , hence f ′ is nuclear and f is nuclear in the classical sense. We define U := p −1 (B δ ), whereB δ ⊂ X 0 is the open ball of radius δ around the origin in the Banach space X 0 , and δ > 0 is chosen such thatB δ ⊂ f −1 0 (B 1 ). The continuity of p implies that U is open. Moreover, p(U ) ⊂B δ def = δB 1 implies ||p(x)|| < δ for x ∈ U which in turn implies the estimate ||p(x)|| < δ||x|| U for all x ∈ X. It follows that the map p factors through p U .
We define B := j(B 1 ) ⊂ Y , where B 1 is the closed unit ball in Y 0 . This is a bounded subset of Y , since for any open subset U ⊂ Y the preimage j −1 (U ) is an open 0-neighborhood of Y 0 and hence there is some ǫ > 0 such that B ǫ ⊂ j −1 (U ). Then ǫj(B 1 ) = j(B ǫ ) is a subset of U . This implies j(B 1 ) ⊂ 1 ǫ U and hence B is a bounded subset of Y (it is clear that B is closed, convex and circled). By construction of B we have the inequality ||j(y)|| B ≤ ||y|| which implies that j factors through j B . Also by construction, f (U ) is contained in B and hence f induces a morphism f ′ : X U → Y B . It follows that the outer edges of the diagram form a commutative square. The fact that j B is a monomorphism and that the image of p U is dense in X U then imply that the middle square of the diagram above is commutative.

The Riemannian bordism category
We recall from section 2 that the objects of the d-dimensional Riemannian bordism category d-RBord are closed (d − 1)-dimensional Riemannian manifolds. Given X, Y ∈ d-RBord, the set d-RBord(X, Y ) of morphisms from X to Y is the disjoint union of the set of isometries from X to Y and the set of Riemannian bordisms from X to Y (modulo isometry relative boundary), except that for X = Y = ∅, the identity isometry equals the empty bordism. 2. For any X, Y ∈ d-RBord the map is injective. In particular, every object X ∈ d-RBord has the trace property and every bordism Σ from X to X has a well-defined trace tr(Σ) ∈ d-RBord(∅, ∅).
3. If Σ is a Riemannian bordism from X to X, then tr(Σ) = Σ gl , the closed Riemannian manifold obtained by gluing the two copies of X in the boundary of Σ.
Proof. To prove part (1) suppose that f : X → Y is a thick morphism, i.e., it can be factored in the form We note that the morphisms t : ∅ → Y ∐ Z and b : Z ∐ X → ∅ must both be bordisms (the only case where say t could possibly be an isometry is Y = Z = ∅; however that isometry is the same morphism as the empty bordism). Hence the composition f is a bordism. Conversely, assume that Σ is a bordism from X to Y . Then Σ can be decomposed as in the following picture: ) are bordisms, where ǫ > 0 is chosen suitably so that Y ⊂ Σ has a neighborhood isometric to Y × [0, 2ǫ) equipped with the product metric. Regarding t as a Riemannian bordism from ∅ to Y ∐ Z, and similarly b as a Riemannian bordism from Z ∐ X to ∅, it is clear from the construction that the composition (4.39) is Σ.
To show that Ψ is injective, let [Z ′ , t ′ , b ′ ] ∈ d-RBord(X, Y ), and let Σ = Ψ([Z ′ , t ′ , b ′ ]) ∈ d-RBord be the composition (1.2). In other words, we have a decomposition of the bordism Σ into two pieces t ′ , b ′ which intersect along Z ′ . Now let (Z, t, b) be the triple constructed in the proof of part (1) above. By choosing ǫ small enough, we can assume that Z = Y × ǫ ⊂ Σ is in the interior of the bordism t ′ , and we obtain a decomposition of the bordism Σ as shown in the picture below.
Regarding g as a bordism from Z to Z ′ we see that which implies that the triple (Z, t, b) and (Z ′ , t ′ , b ′ ) are equivalent in the sense of Def- For the proof of part (3), we decompose as in the proof of (1) the bordism Σ into two pieces t and b by cutting it along the one-codimensional submanifold Z = X × {ǫ} (here Y = X since Σ is an endomorphism). We regard t as a bordism from ∅ to X ∐ Z and b as a bordism from Z ∐ X to ∅. Then tr(Σ) is given by the composition which geometrically means to glue the two bordisms along X and Z. Gluing first along Z we obtain the Riemannian manifold Σ, then gluing along X we get the closed Riemannian manifold Σ gl .

Properties of the trace pairing
The goal of this section is the proof of our main theorem 1.10 according to which our trace pairing is symmetric, additive and multiplicative. There are three subsections devoted to the proof of these three properties, plus a subsection on braided monoidal and balanced monoidal categories needed for the multiplicative property. Each proof will be based on first proving the following analogous properties for the trace tr( f ) of thickened endomorphisms f ∈ C(X, X): Theorem 5.1. Let C be a monoidal category, equipped with a natural family of isomorphisms s = s X,Y : X ⊗ Y → Y ⊗ X. Then the trace map tr : C(X, X) −→ C(I, I) has the following properties: 1. (symmetry) tr( f • g) = tr(g • f ) for f ∈ C(X, Y ), g ∈ C(Y, X); 2. (additivity) If C is an additive category with distributive monoidal structure (see Definition 5.3), then tr is a linear map; 3. (multiplicativity) tr( f 1 ⊗ f 2 ) = tr( f 1 ) ⊗ tr( f 2 ) for f 1 ∈ C(X 1 , X 1 ), f 2 ∈ C(X 2 , X 2 ), provided C is a symmetric monoidal category with braiding s. More generally, this property holds if C is a balanced monoidal category (see Definition 5.12).
For the tensor product f 1 ⊗ f 2 ∈ C(X 1 ⊗ X 2 , X 1 ⊗ X 2 ) of the thickened morphisms f i see Definition 5.15.

The symmetry property of the trace pairing
Proof of part (1) of Theorem 5.1. Let f = [Z, t, b] ∈ C(X, Y ). Then and and hence Here the second equality follows from the naturality of the switching isomorphism (see Picture (3.14)).
Here the second equation is part (1) of Theorem 5.1, while the third is a consequence of Lemma 3.11.

Additivity of the trace pairing
Throughout this subsection we will assume that the category C is an additive category with distributive monoidal structure (see Definitions 5.2 and 5.3 below). Often an additive category is defined as a category enriched over abelian groups with finite products (or equally coproducts). However, the abelian group structure on the morphism sets is actually determined by the underlying category C, and hence a better point of view is to think of 'additive' as a property of a category C, rather than specifying additional data.
Definition 5.2. A category C is additive if 1. There is a zero object 0 ∈ C (an object which is terminal and initial); 2. finite products and coproducts exist; To show that the additivity of tr implies that the pairing tr(f, g) is linear in f and g, we will need the following fact.
The proof of this lemma is analogous to the previous two proofs, and so we leave it to the reader.

Braided and balanced monoidal categories
We recall from [JS2, Definition 2.1] that a braided monoidal category is a monoidal category equipped with a natural family of isomorphisms c = c X,Y : X ⊗ Y → Y ⊗ X such that There is a close relationship between braided monoidal categories and the braid groups B n . We recall that the elements of B n are braids with n strands, consisting of isotopy classes of n non-intersecting piecewise smooth curves γ i : [0, 1] → R 3 , i = 1, . . . , n with endpoints at {1, . . . , n} × {0} × {0, 1}. One requires that the z-coordinate of each curve is strictly decreasing (so that strands are "going down"). Composition of braids is defined by concatenation of their strands; e.g., in B 2 we have the composition The last equality follows from the obvious isotopy in R 3 , also known as the second Reidemeister move. The three Reidemeister moves are used to understand isotopies of arcs in R 3 via their projections to the plane R 2 . Generically, the isotopy projects to an isotopy in the plane but there are three codimension one singularities where this does not happen: a cusp, a tangency and a triple point (of immersed arcs in the plane). The corresponding 'moves' on the planar projection are the Reidemeister moves; for example, in the above figure one can see a tangency in the middle of moving the braid in the center to the (trivial) braid on the right. A cusp singularity cannot arise for braids but a triple point can: the resulting (third Reidemeister move) isotopies are also known as braid relations. A typical example would be the following isotopy (with a triple point in the middle): = (5.9) Thinking of the groups B n as categories with one object, we can form their disjoint union to obtain the braid category B := ∞ n=0 B n . The category B can be equipped with the structure of a braided monoidal category [JS2, Example 2.1]; in fact, B is equivalent to the free braided monoidal category generated by one object (this is a special case of Theorem 2.5 of [JS2]).
More generally, if C k is the free braided monoidal category generated by k objects, there is a braided monoidal functor F : C k → B which sends n-fold tensor products of the generating objects to the object n ∈ B (whose automorphism group is B n ); on morphisms, F sends the structure maps α X,Y,Z , ℓ X and r X (see beginning of Section 3) to the identity, and the braiding isomorphisms c X i ,X j for generating objects X i , X j to the braid ∈ B 2 This suggests to represent c X,Y by an overcrossing and c −1 Y,X by an undercrossing in the pictorial representations of morphisms in braided monoidal categories, a convention broadly used in the literature [JS1] that we will adopt. In other words, one defines: The key result, [JS2,Cor. 2.6], is that for any objects X, Y ∈ C k the map is a bijection. One step in the argument is to realize that the Yang-Baxter relations 5.9 follow from the relations 5.8 together with the naturality of the braiding isomorphism c. An immediate consequence is the following statement which we will refer to below.
Proposition 5.10. (Joyal-Street) Let f, g : X 1 ⊗ · · · ⊗ X k → X σ(1) ⊗ · · · ⊗ X σ(k) be morphisms of a braided monoidal category C which are in the image of the tautological functor T : C k → C which sends the i-th generating object of the free braided monoidal category C k to X i . If f = T ( f ), g = T ( g) and the associated braids F ( f ), F ( g) ∈ B k agree, then f = g.
We note that a morphism f is in the image of T if and only if it can be written as a composition of the isomorphisms α, ℓ, r, c and their inverses. For such a composition it is easy to read off the braid F ( f ): in the pictorial representation of f we simply ignore all associators and units and replace each occurrence of the braiding isomorphism c (respectively its inverse) by an overcrossing (respectively and undercrossing); then F ( f ) is the resulting braid. For example, for Another result that we will need for the proof of the multiplicativity property are the following isotopy relations. Roughly speaking, they say that if the unit I ∈ C is involved, then the isotopy does not need to be 'relative boundary' as in the previous pictures.
Lemma 5.11. Let V , W be objects of a braided monoidal category C, and let f : V → I and g : I → V be morphisms in C. Then there are the following relations: Here the first and last equality come from interpreting compositions involving the box f with no output (= morphism with range I). The second equality is the naturality of the braiding isomorphism, and the third equality is a compatibility between the unit constraints r W , ℓ W and the braiding isomorphism which is a consequence of the relations (5.8) [JS2,Prop. 2.1].
This proves the first equality; the proofs of the other equalities are analogous.