The chromatic Brauer category and its linear representations

The Brauer category is a symmetric strict monoidal category that arises as a categorification of the Brauer algebras in the context of Banagl's framework of positive topological field theories (TFTs). We introduce the chromatic Brauer category as an enrichment of the Brauer category in which the morphisms are component-wise labeled. Linear representations of the (chromatic) Brauer category are symmetric strict monoidal functors into the category of real vector spaces and linear maps equipped with the Schauenburg tensor product. We study representation theory of the (chromatic) Brauer category, and classify all its faithful linear representations. As an application, we use indices of fold lines to construct a refinement of Banagl's concrete positive TFT based on fold maps into the plane.


Introduction
The Brauer algebras D m have first appeared in work of Brauer [6] on representation theory of the orthogonal group Opnq. In view of Schur-Weyl duality, they replace the role played by the group algebras of symmetric groups in representation theory of the general linear group. Generators of D m are the diagrams consisting of 2m vertices and m edges, where the vertices are arranged in two parallel rows of m vertices, and each vertex lies in the boundary of exactly one edge. Given a commutative ground ring k with unit, D m is the kpxq-algebra freely generated as kpxq-module by those diagrams. Multiplication is induced by concatenation of diagrams, where each arising free loop component gives rise to an additional multiplication with the indeterminate x. A signed variant of Brauer algebras has been studied in [15]. Brauer algebras play an important role in knot theory, where, for instance, Birman-Murakami-Wenzl algebras [5,14,18], which are the quantized version of Brauer algebras, have been used to construct generalizations of the Jones polynomial.
We are concerned with a natural categorification Br of Brauer's algebras that has been constructed by Banagl [3,4] in search of new topological invariants in the context of his framework of positive topological field theory (TFT). A similar categorification has been considered independently by Lehrer-Zhang [11] in a modern categorical approach to the invariant theory of the orthogonal and symplectic groups. Roughly speaking, morphisms in the so-called Brauer category Br are represented by 1-dimensional unoriented tangles in a high-dimensional Euclidean space. In particular, generators and relations for the strict monoidal category Br have been found in [4] (compare also [11]) by adapting the methods that are used by Turaev [17] for deriving a presentation for the category of tangle diagrams.
Let us discuss the main ideas behind Banagl's notion of positive TFT, and the role of the Brauer category Br and its representation theory in this context. By definition, the axioms for positive TFT [3] differ from Atiyah's original axioms for TFT [1] in that they are formulated over semirings instead of rings. Recall that semirings are not required to have additive inverse ("negative") elements. In computer science, semirings and related structures have been studied by Eilenberg [7] in the context of automata theory and formal languages. The essential advantage of positive TFTs over usual TFTs is that so-called Eilenberg completeness of certain semirings can be used to give a rigorous construction of positive TFTs of arbitrary dimension. This construction is implemented by Banagl in a process of quantization that requires so-called fields and an action functional as input. Inspiration comes from theoretical quantum physics, where the state sum is expressed by fields and an action functional via the Feynman path integral.
In [4], Banagl applies his framework to produce in arbitrary dimension an explicit positive TFT for smooth manifolds. The construction uses singularity theory of so-called fold maps, and the resulting state sum invariants can distinguish exotic smooth spheres from the standard sphere. Now, in this concrete setting, the role of fields is played by certain fold maps into the plane, and the action functional assigns to such fields morphisms in the Brauer category Br by extracting the 1-dimensional patterns that arise from the singular locus of fold maps. However, as pointed out in Section 8 of [3], it is desirable to compose such a category-valued action functional with a symmetric strict monoidal functor from Br to the category Vect of real vector spaces and linear maps. Note that one requires the category Vect to be equipped with a symmetric strict monoidal structure, which is provided by using the Schauenburg tensor product [16]. In this way, the Brauer category serves only as an intermediary structure, and the state sum of the resulting positive TFT will become accessible through linear algebra. Of course, the loss of information should be kept at a minimum during this linearization process, which is motivation for studying faithfulness of such linear representations Br Ñ Vect. This knowledge is required when it comes to the explicit computation of state sum invariants (compare Section 6.3 and Section 10.5 in [19] as well as Remark 9.5 in [20]).
In this paper we determine not only the faithful representations of Br, but also those of the chromatic Brauer category cBr which will be introduced in Section 3.1 as an enrichment of Br in which morphisms are component-wise labeled ("colored") by elements of a countable index set. Hence, in contrast to the Brauer category, isomorphic objects of the chromatic Brauer category need not be equal. Our reason for considering cBr is that it can be used to construct a refinement of Banagl's positive TFT based on fold maps in the following way (see Section 5). In analogy with the index of non-degenerate critical points in Morse theory, one can associate a (reduced) index to the singularities of a fold map. Those fold indices are intrinsically defined, locally constant along the singular set, and carry topological information about the source manifold. For fold maps from n-dimensional source manifolds into the plane, the set of possible fold indices is t0, . . . , tpn´1q{2uu. We will modify Banagl's original construction by defining a cBr-valued action functional which additionally remembers indices of fold lines as labels from the set N " t0, 1, 2, ...u of natural numbers.
Concerning linear representations of Br, Banagl has shown in Proposition 2.22 of [4] that there exist linear representations that are faithful on loops. This suffices for his purpose to show that state sum invariants of the positive TFT are able to detect exotic smooth structures on spheres. As a much more general result, we have the following Theorem 1.1 ( [13,19]). Let Y : Br Ñ Vect be a symmetric strict monoidal functor from the Brauer category into the category of real vector spaces and linear maps (equipped with the Schauenburg tensor product). Then the vector space Y pr1sq has finite dimension d. Moreover the functor Y is faithful if and only if d ě 2.
What is more, we will show Theorem 1.2 below. Since the Brauer category is naturally (monochromatically) embedded in the chromatic Brauer category, Theorem 1.1 is implied by Theorem 1.2. (To conclude this, one has to use that any linear representation of Br can be extended to one of cBr by means of our structure results Theorem 3.6 and the corresponding result for Br.) In preparation of the statement of our result on linear representations of cBr, note that the objects of cBr that are mapped to the object r1s of Br under the forgetful functor cBr Ñ Br are parametrized by the labels k P N, say pr1s, kq. Theorem 1.2. Let Y : cBr Ñ Vect be a symmetric strict monoidal functor from the chromatic Brauer category into the category of real vector spaces and linear maps (equipped with the Schauenburg tensor product). Then, for each k P N, the vector space Y ppr1s, kqq is of finite dimension, say d k . Suppose that d k ą 0 for all k P N. Then, the functor Y is faithful if and only if the sequence d 0 , d 1 , . . . satisfies for all pl k q kPN P À 8 k"0 Z the implication 8 ź k"0 d l k k " 1 ñ l k " 0 for all k P N. (1. 1) In particular, faithful linear representations of cBr exist because one can take d to be the sequence of prime numbers, and then apply Theorem 3.6 to construct a strict monoidal functor Y : cBr Ñ Vect which realizes d k for k P N as the dimension of the real vector space Y ppr1s, kqq.
The paper is structured as follows. In Section 2 we recall fundamental facts about monoidal categories in general, and the Schauenburg tensor product in particular. The chromatic Brauer category is introduced in Section 3, where its linear representations are classified by Theorem 3.6. The proof of our main result Theorem 1.2 will be given in Section 4. Finally, in Section 5 we discuss our application to Banagl's positive TFT based on fold maps.
Notation. Throughout the paper, the natural numbers will be meant to be the set N " t0, 1, 2, . . . u (including zero).
Acknowledgment. The authors are grateful to Prof. Banagl for providing the initial motivation for this work. The second author was partially supported by a scholarship of the German National Merit Foundation.

Preliminaries on strict monoidal categories
In this section, we introduce the definitions and notational conventions that will be used throughout the paper. We refer to [9] for the basic definitions in this section.
2.1. Strict monoidal categories. A monoidal category pC, b, I, α, λ, ρq is a category C equipped with a bifunctor b : CˆC Ñ C, an object I P ObpCq, called unit with respect to the tensor product b, and three isomorphisms which are functorial in X, Y, Z P ObpCq. Furthermore, α, λ and ρ satisfy the coherence conditions given by the pentagon axiom and triangle axiom. These are for all W, X, Y, Z P ObpCq. Here, α is called associativity constraint, and λ and ρ are called left and right unit constraints, respectively. A monoidal category C is called strict if the associativity and unit constraints α, λ, ρ are given by identity morphisms of the category. If pC, b, I, α, λ, ρq is a (strict) monoidal category, then we will call the data pb, I, α, λ, ρq a (strict) monoidal structure on C.
Let C and D be monoidal categories. A monoidal functor is a functor F : C Ñ D which respects the monoidal structure. To be more precise, it is a functor and an isomorphism ξ 0 : I D Ñ F pI C q such that the diagrams F pXq bD pF pY q bD F pZqq F pXq bD F pY bC Zq F pX bC pY bC Zqq commute for all X, Y, Z P ObpCq. The monoidal functor pF, ξ, ξ 0 q is called strict if the isomorphisms ξ 0 and ξ are identity morphisms of D.
Let pC, b, I, α, λ, ρq be a monoidal category. A symmetric braiding b on C is a monoidal functorial isomorphism b : b Ñ b˝τ , where the map τ : ObpCqÔ bpCq Ñ ObpCqˆObpCq is given by τ pX, Y q " pY, Xq for any pair pX, Y q of objects of the category C, satisfying the hexagon axiom, the unity coherence, and the inverse law which are given by the commutativity of the diagrams respectively, for all X, Y, Z P ObpCq.
A monoidal category C together with a symmetric braiding b is called symmetric monoidal category. For most of the time, we will work with strict monoidal categories. In that case, we will omit the associativity and unit constraints in the notation of a monoidal category, i.e. we will write pC, b, Iq instead of pC, b, I, α, λ, ρq.
Let pC, b, I C , b C q and pD, d, I D , b D q be symmetric strict monoidal categories. A monoidal functor pF, ξ, ξ 0 q : C Ñ D is called symmetric if F is compatible with the symmetric structures, i.e. if F pb CX,Y q " ξ F pY q,F pXq˝bDF pXq,F pY q˝ξ´1 X,Y for all X, Y P ObpCq.
Let pC, b, I, α, λ, ρ, bq be a symmetric monoidal category. An object X P ObpCq is called dualizable if there exists a triple pX ‹ , i X , e X q consisting of an object X ‹ P ObpCq, called a dual of X and morphisms i X : I Ñ X ‹ b X, and e X : X b X ‹ Ñ I, called unit and counit respectively, such that the diagrams commute. A symmetric strict monoidal category pC, b, I, βq is called compact if every object is dualizable. Let i denote the family of unit morphisms and let e denote the family of counit morphisms. We then write pC, b, I, β, i, eq for a compact category.
2.2. The Schauenburg tensor product. The usual tensor product b in the category Vect of real vector spaces and linear maps determines a monoidal structure on Vect (with unit object I " R) which is certainly not strict. When studying linear representations of strict monoidal categories, it is appropriate to endow Vect with a strict monoidal structure. As described in Theorem XI.5.3 in [9, p. 291], there is a general method for turning any given monoidal category C into a monoidally equivalent strict monoidal category C str . However, this procedure changes the category even on the object level, which is probably not convenient for studying properties of linear representations. Banagl employs an explicit strict monoidal structure on Vect, namely the Schauenburg tensor product d introduced in [16]. The resulting strict monoidal category pVect, d, Iq is monoidally equivalent to the usual monoidal structure on Vect obtained by b. The advantage of Schauenburg's construction is that the monoidal equivalence pF, ξ, ξ 0 q can be chosen in such a way that F is the identity on Vect, and ξ 0 is the identity on I. Via the natural isomorphism ξ : b Ñ d we can define elements vdw P V dW , vdw :" ξ V,W pvbwq, for v P V and w P W . Given elements u P U , v P V and w P W the identity There is a standard symmetric braiding β V,W : V b W Ñ W b V on Vect for all V, W P ObpVectq (with respect to the standard tensor product b). By defining b V,W " ξ W,V˝βV,W˝ξ´1 V,W we obtain a symmetric braiding with respect to d. All in all, the following proposition holds.
Proposition 2.1. The data pVect, d, R, bq define a symmetric strict monoidal category.
For the rest of this paper, we will use the Schauenburg tensor product on Vect; thus, we will from now on write b instead of d.
Remark 2.2. If the vector spaces V and W are equipped with ordered bases tv i u i and tw j u j , respectively, then we let the tensor product V b W be equipped with the lexicographically ordered basis tv i b w j u i,j . Given two morphisms A : Suppose that each of the vector spaces V , W , V 1 and W 1 is equipped with an ordered basis. If V b W and V 1 b W 1 are finite-dimensional, then it is well-known that the matrix representation of A b B is given by the Kronecker product of the matrix representations of A and B with respect to the fixed bases.
Schauenburg's result applies more generally to categories of structured sets. Essentially, a category C is called a category of structured sets (see Definition 4.1 in [16]) if there is a faithful functor from C into the category of sets, such that for any object in C and any bijection of the underlying set into an other set there is a unique object in C and a unique morphism in C realizing the bijection. As pointed out by Schauenburg, all categories of algebraic structures are categories of structured sets by means of the forgetful functor to the category of sets and transfer of the algebraic structures. In particular, this applies to the (non-strict) monoidal category of real vector spaces with the standard monoidal structure pVect, b, R, α, λ, ρq. Theorem 2.3 (cf. [16], Theorem 4.3). Let C be a category of structured sets. Then for each monoidal structure pb, I, α, λ, ρq on C, there exists a strict monoidal structure pd, Iq with the same unit object I such that pId C , ξ, 1 I q : pC, d, Iq Ñ pC, b, I, α, λ, ρq is a monoidal category equivalence.
3. The chromatic Brauer category and its linear representations 3.1. The chromatic Brauer category. Recall that the Brauer category Br is considered in Section 10 of [3] and Section 2.3 of [4], where Banagl defines it as a natural categorification of Brauer algebras. Roughly speaking, isomorphism classes of finite sets serve as objects, and morphisms are isotopy classes of unoriented tangles in Euclidean 4-space. We shall next introduce the chromatic Brauer category cBr as a certain enrichment of the Brauer category. Namely, we equip the components of objects and morphisms with colorings using a countable number of colors. The structure of our discussion will closely follow the above references to facilitate comparisons. Note that both categories Br and cBr have the structures of strict monoidal categories that are compact and symmetric. However, in contrast to the Brauer category, isomorphic objects of the chromatic Brauer category will not necessarily be equal.
Let us formally introduce the category cBr. Given m " 1, 2, ..., let rms denote the set t1, ..., mu. Let r0s denote the empty set. We will also use the notation M rms " t1, ..., mu Ă R 1 if we want to consider the set rms as a 0-submanifold of R 1 . The objects of cBr are pairs of the form prms, cq, where c is some map c : rms Ñ N. For m " 0, c " c H denotes the unique map H Ñ N. Morphisms prms, cq Ñ prm 1 s, c 1 q in cBr are represented by pairs pW, Ωq, where ‚ W is a 1-cobordism from rms to rm 1 s together with a smooth embedding W ãÑ r0, 1sˆR 3 , where the boundary satisfies BW " W X pt0, 1uˆR 3 q and BW X t0uˆR 3 " t0uˆM rmsˆt0uˆt0u, BW X t1uˆR 3 " t1uˆM rm 1 sˆt0uˆt0u, such that near the boundary, W is embedded as the product r0, εsˆM rmsˆt0uˆt0u \ r1´ε, 1sˆM rm 1 sˆt0uˆt0u for some ε ą 0, and ‚ Ω is a locally constant map Ω : W Ñ N such that Ω| rms " c and Ω| rm 1 s " c 1 . Given prms, cq and prm 1 s, c 1 q, two such pairs pW 0 , Ω 0 q and pW 1 , Ω 1 q determine the same morphism in cBr if there is a diffeomorphism α : W 0 Ñ W 1 of cobordisms such that the embedding of W 0 is smoothly isotopic to the composition of α with the embedding of W 1 , by an isotopy that is the identity near t0, 1uˆR 3 , and Ω 0 " Ω 1˝α .
The composition of two morphisms ϕ : prms, cq Ñ prm 1 s, c 1 q and ψ : prm 1 s, c 1 q Ñ prm 2 s, c 2 q is defined by composing representatives of ϕ and ψ followed by rescaling. That is, if ϕ is represented by the pair pW 1 , Ω 1 q and ψ by the pair pW 2 , Ω 2 q, then ψ˝ϕ is represented by the pair pW, Ωq, where W " W 1 Y rm 1 s W 2 and Ω| W 1 " Ω 1 , Ω| W 2 " Ω 2 . The smooth embedding W ãÑ r0, 1sˆR 3 is given by translating the embedding W 2 ãÑ r0, 1sˆR 3 to r1, 2sˆR 3 , gluing the embeddings W 1 ãÑ r0, 1sˆR 3 and W 2 ãÑ r1, 2sˆR 3 at M rm 1 s, and then reparametrizing the interval r0, 2s to r0, 1s. The identity morphism 1 pr0s,c H q : pr0s, c H q Ñ pr0s, c H q is represented by the empty 1-cobordism H together with the unique map H Ñ N. For m ą 0, the identity morphism 1 prms,cq on prms, cq is represented by the product cobordism r0, 1sˆM rmsˆt0uˆt0u and the map Ω " c˝proj rms . This completes the definition of the category cBr. Note that Hom cBr pprms, cq, prm 1 s, c 1 qq is empty if and only if there is a positive integer k P N for which the number |c´1pkq|`|c 1´1 pkq| is odd. Notationally, for k P N we denote by k the constant map k : rms Ñ N, kpiq " k for all i P rms.
For later reference, we record the following.
Lemma 3.1. Let ι : prms, cq Ñ prm 1 s, c 1 q be an isomorphism in cBr. Then, m " m 1 , and ι is uniquely determined by the object prms, c 1 q and a permutation σ ι of the set t1, ..., mu.
Proof. If ι is represented by pW, Ωq and an embedding W ãÑ r0, 1sˆR 3 , then every component of W is diffeomorphic to r0, 1s with one endpoint being mapped to t0uˆR 3 and the other to t1uˆR 3 . Hence, we have m " m 1 , and W induces a permutation σ ι of the set t1, . . . , mu. Hence, ι is uniquely determined by the object prms, cq and the underlying permutation σ ι . In particular, observe that c 1 " c˝σ´1 ι .
We equip cBr with the structure pcBr, b, Iq of a strict monoidal category as follows. The tensor product b : cBrˆcBr Ñ cBr is defined by "stacking" of objects and morphisms. More precisely, given objects prms, cq and prm 1 s, c 1 q, we let prms, cq b prm 1 s, c 1 q " prm`m 1 s, c 2 q, where c 2 pkq " cpkq whenever k P rms and c 2 pkq " c 1 pk´mq whenever k P rm`m 1 szrms. Given two morphisms ϕ : prms, cq Ñ prm 1 s, c 1 q and ψ : prns, bq Ñ prn 1 s, b 1 q represented by pairs pW ϕ , Ω ϕ q and pW ψ , Ω ψ q, respectively, the tensor product ϕ b ψ is represented by the pair pW ϕ \ W ψ , Ω ϕ \ Ω ψ q. More precisely, the embedding W ϕ \ W ψ ãÑ r0, 1sˆR 3 is an extension of W ϕ ãÑ r0, 1sˆR 3 by an embedding of W ψ , which is obtained by first applying the translation px, y, z, tq Þ Ñ px, y`m, z, tq to the original embedding W 1 ãÑ r0, 1sˆR 3 , then modifying the embedding near the rn 1 s-endpoints of W ψ appropriately to connect to the correct points in t1uˆM rm 1`n1 sˆt0uˆt0u, and finally making the resulting embedding disjoint to the embedding of W ϕ by means of a small isotopy. Let the unit object I be pr0s, c H q. Then it is easy to check that pcBr, b, Iq defines a strict monoidal category (compare Section 2.1). Note that, in contrast to the original Brauer category, the tensor product of cBr is not even commutative on objects.
Next we equip pcBr, b, Iq with a compact, symmetric structure pcBr, b, I, b, i, eq by introducing families of morphisms called braiding b, unit i and counit e. Given two objects prms, cq and prm 1 s, c 1 q in cBr we define the braiding b prms,cq,prm 1 s,c 1 q : prms, cq b prm 1 s, c 1 q Ñ prm 1 s, c 1 q b prms, cq to be the unique isomorphism whose underlying cobordism is represented by the loop-free diagram Given an object prms, cq in cBr we define the unit i prms,cq : pr0s, c H q Ñ prms, cq b prms, cq to be the unique morphism whose underlying cobordism is represented by the loop-free diagram .
Given an object prms, cq in cBr we define the counit e prms,cq : prms, cq b prms, cq Ñ pr0s, c H q to be the unique morphism whose underlying cobordism is represented by the loop-free diagram Then it is easy to check that pcBr, b, I, b, i, eq defines a compact, symmetric, strict monoidal category. Hereafter we write i pkq , e pkq and b pk,lq for i pr1s,kq , e pr1s,kq and b pr1s,kq,pr1s,lq , respectively. The morphisms i pkq , e pkq and b pk,lq , where k, l P N are called the elementary morphisms of cBr.
For every k P N, there is precisely one endomorphism λ pkq : pr0s, c H q Ñ pr0s, c H q, such that λ pkq is represented by pS 1 , Ω k q, where S 1 is the circle and Ω k " k. We will call this endomorphism λ pkq the k-loop. The endomorphisms in cBr of the identity object I are then given by End cBr pIq " tλ bn0 p0q b¨¨¨b λ bnm pmq | Dm P N : n 0 , ..., n m P Nu, where λ bn pkq denotes the n-fold tensor product λ pkq b¨¨¨b λ pkq and λ b0 pkq " 1 H . The k-loop endomorphism λ pkq can be factored as λ pkq " e pkq˝ipkq . Every such k-loop commutes with every other morphism ϕ P MorpcBrq, that is λ pkq b ϕ " ϕ b λ pkq . In particular, any isomorphism in cBr can never contain a k-loop. Loops are persistent, that is, if ϕ : prms, cq Ñ prm 1 s, c 1 q has ν k-loops and ψ : prm 1 s, c 1 q Ñ prm 2 s, c 2 q has µ k-loops, then ψ˝ϕ has at least ν`µ k-loops. Also, loops are cancellative: if λ pkq b ϕ " λ pkq b ψ, then ϕ " ψ.
The usual Brauer category Br introduced in [3, Section 10] can be obtained by cBr by forgetting about the coloring maps c and Ω as a data of objects and morphisms. Naturally, we find the forgetful functor ? : cBr Ñ Br by forgetting the coloring maps c and Ω. Then it is easy to check that the functor ? is strict monoidal and symmetric. Note that the functor ? is full but not faithful. However, if ϕ, ψ : prms, cq Ñ prm 1 s, c 1 q are loop-free morphisms then ϕ " ψ if and only if ?pϕq " ?pψq. We will also write ?pe pkq q " e 1 , ?pi pkq q " i 1 , ?pb pk,lq q " b 1,1 , ?pλ pkq q " λ and ?p1 pr1s,kq q " 1 for all k, l P N.

3.2.
Presentation of the chromatic Brauer category. Next we will discuss a presentation of the chromatic Brauer category cBr. For that purpose we will recall the notation of presenting monoidal categories from Section 2, page 14, of [4]. Let pC, b, Iq be a strict monoidal category and G be a collection of morphisms of C. Interpreting G as an alphabet, we may form words as follows: For all g P G and X P ObpCq, rgs and r1 X s are words. If w 1 and w 2 are words, then the string pw 1 b w 2 q is a word and the string pw 2˝w1 q whenever cod w 1 " dom w 2 . Every word w determines a morphism |w| of C by the rules Two words are called freely equivalent (we write "), if they can be obtained from each other by a finite sequence of subword substitutions implementing associativity for˝and b, identity cancellation for˝and b and compatibility between˝and b. Note that if w 1 and w 2 are freely equivalent, then |w 1 | " |w 2 |. ). Any word in G is freely equivalent to a word of the form r1 X s for some object X or to a word of the form with morphisms f 1 , ..., f k P G and objects X 1 , ..., X k , Y 1 , ..., Y k P ObpCq.
Also note the following useful fact which is used in the proof of Theorem 3.4 below. Lemma 3.3. Let pC, b, Iq be a strict monoidal category, then the group pEnd C pIq,˝q is abelian.
Proof. Let ϕ, ψ P End C pIq and write 1 for the map 1 I , then the statement follows from free equivalence.
Let F pGq denote the class of free equivalence classes of words in G. Then the realization |¨| is still well defined on F pGq. Let R be the collection of pairs pw 1 , w 2 q of words in G such that |w 1 | " |w 2 |. For two elements x, y P F pGq we define x " R y, and say that x, y are R-equivalent, if and only if one can obtain some representative of y from some representative of x by a finite sequence of subword substitutions, where an allowable substitution consists of replacing a subword w 1 by w 2 for pw 1 , w 2 q P R. We say that pC, b, Iq is generated by the generators G and the relations R, if ‚ any morphism in C can be obtained as |w| for some word w in G, and ‚ for any x, y P F pGq, we have x " R y if and only if |x| " |y| in C. The structure of morphisms in cBr is then elucidated by the following Theorem 3.4. Note that for the Brauer category Br the structure is obtained by forgetting the colorings of generators and relations via ?, see [11], Theorem 2.6. We will denote the set of generators of Br by G 0 " te 1 , i 1 , b 1,1 u and the set of relations of Br by R 0 .
(C4) Reidemeister 2 (double crossing): (C5) Reidemeister 3 (braiding relation, a.k.a. Yang-Baxter equation): (In the above diagrams, differently structured lines correspond to independent labels.) Proof. Let ϕ be a morphism in cBr and let pW, Ωq be a representative of ϕ. Then ?pϕq is a morphism in Br, therefore it can be expressed as a word in i 1 , e 1 and b 1,1 (i.e. having the form of the expression (3.1)). Then W is a gluing of cobordisms representing e 1 , i 1 and b 1,1 in Br. Finally, we extend those cobordisms to pairs in cBr by restriction of Ω. Now, let ϕ : prms, cq Ñ prm 1 s, c 1 q be a morphism in cBr and let x and y be free equivalence classes in F pGq having ϕ as their realizations. Let x 0 and y 0 be the induced free equivalence classes in F pG 0 q (G 0 " ti 1 , e 1 , b 1,1 u Ă MorpBrq) having ϕ 0 " ?pϕq as their realizations. Therefore, x 0 " R0 y 0 . On the other hand, ϕ 0 can be written as can be obtained as |w 0 | for some word w 0 in G 0 of the form (3.1). Thus, ϕ 0 can be obtained by the expression w " rp1 rm 1 s b e 1 q˝p1 rm 1 s b i 1 qs˝ν˝w 0 of the form (3.1). We choose a word w x of the form (3.1) representing x 0 . There exists a word w 1 x in G of the form (3.1) representing x such that w x is the induced free equivalence class in F pG 0 q. Indeed, the choice colorings is made as in the first half of the proof. Since |w x | " |w| one can obtain w from w x by a finite sequence of subword substitutions by relations of R 0 . In each of these steps we can easily extend the colorings to the next word. All in all, we obtain a lifting w 1 of w in cBr such that w 1 is obtained from w 1 x by a finite sequence of subword substitutions using relations from R. Analogously, there exists w 1 y and a lifting w 2 of w such that w 2 is obtained from w 1 y by a finite sequence of subword substitutions using relations from R. It remains to show that w 2 is obtained from w 1 in this way. Note that both of them colorings of w, the induced colorings of w 0 must be identical because on components with nonempty boundary the coloring is determined by the coloring of the boundary points. Therefore it suffices to show that loops of different colors commute, but this is precisely the statement of Lemma 3.3 for C " cBr.

3.3.
Linear representations of the chromatic Brauer category. For the purpose of constructing strict monoidal functors Y : cBr Ñ Vect, where Vect is equipped with the Schauenburg tensor product, we recall the notion of a duality structure on a finite dimensional real vector space V from Definition 2.5 in [4]. Namely, a duality structure on V is a pair pi, eq whose components are a symmetric copairing i : R Ñ V b V and a symmetric pairing e : V b V Ñ R, also called unit and counit, respectively, satisfying the zig-zag equation Now let d be the dimension of V and let tv 1 , ..., v d u be a basis of V . Then the set of all duality structures on V stands in an 1-1 correspondence to the symmetric and invertible pdˆdq-matrices Sympd, Rq X Glpd, Rq. Indeed, let e jk " epv j b v k q and write X :" Matpeq " pe jk q d j,k"1 , then X is symmetric due to the symmetry of e, and -by the zig-zag equation -X is invertible with inverse X´1 " Matpiq " pi jk q d j,k"1 , where i jk are given by ip1q " Conversely, let X P Sympd, RqXGlpd, Rq be symmetric and invertible. Then the matrices vecpX´1q and vecpXq T define a symmetric copairing and pairing such that the zig-zag equation is satisfied, where p´q T denotes the transposition of a matrix and vecp´q denotes the vectorization of a matrix formed by stacking the columns of the matrix into a single column vector.
The trace of the duality structure pi, eq on V is defined by Trpi, eq " e˝i. By the above description of duality structures we easily compute Trpi, eqp1q " d " dimpV q.
If Y : cBr Ñ Vect is a linear representation of cBr, then Y ppr1s, kqq " V k is for all k P N a finite dimensional vector space (cf. Proposition 2.7 in [4]) since the pair pY pi pkq q, Y pe pkq qq forms a duality structure on V k . Note that Y pλ pkq q " Y pe pkq q˝Y pi pkq q " d k .
Since Y is required to be symmetric, we also have Y pb pk,lq q Given a strict monoidal category C which is presented by generators and relations, the following result provides a construction of strict monoidal functors on C.
Proposition 3.5 ([9], Proposition XII.1.4). Let pC, b C , I C q and pD, b D , I D q be strict monoidal categories. Suppose that C is generated by morphisms G and relations R. Let F 0 : ObpCq Ñ ObpDq be a map such that F 0 pI C q " I D and be a map such that dompF 1 pgqq " F 0 pdompgqq and codpF 1 pgqq " F 0 pcodpgqq. Suppose that any pair pw 1 , w 2 q P R yields equal morphisms in D after replacing any symbol g P G of w 1 and w 2 by F 1 pgq and any symbol 1 X by 1 F0pXq . Then there exists a unique strict monoidal functor F : C Ñ D such that F pXq " F 0 pXq for all X P ObpCq and F pgq " F 1 pgq for all g P G.
Now, using Theorem 3.4 we may construct strict monoidal functors Y : cBr Ñ Vect by choosing duality structures.
Theorem 3.6 (Linear representations of cBr). Let V k be a finite dimensional real vector space and let the pair pi pkq , e pkq q be a duality structure on V k for all k P N. Then there exists a unique strict monoidal symmetric functor Y : pcBr, b, I, bq Ñ pVect, b, R, bq, which satisfies Y ppr1s, kqq " V k and preserves duality, i.e. Y pi pkq q " i pkq and Y pe pkq q " e pkq for all k P N.
Proof. Set Y 0 : ObpcBrq Ñ ObpVectq by fixing the images of pr1s, kq via Y 0 ppr1s, kqq " V k for all k P N and extend Y 0 to all of ObpcBrq while respecting the strict monoidal structure, in particular we have Y 0 pIq " R.
By Theorem 3.4 cBr is generated by the elementary morphisms e pkq i pkq and b pk,lq . For the relation (C1) to be satisfied as an expression in Vect, the image of pi pkq , e pkq q under Y need to be a duality structure on V k , therefore we define Y 1 pi pkq q " i pkq and Y 1 pe pkq q " e pkq . For Y becoming symmetric, we define To apply Proposition 3.5 we need to verify that the relations (C1)-(C5) are valid in Vect (as described in Proposition 3.5). The zig-zag relation (C1) is satisfied by the definition of the duality structure. Fix a basis tv where d k and d l denote the dimension of the vector space V k and the vector space V l , respectively. Then (C2) follows by the following computation.
, for all η, ν P t1, ..., d k u and µ P t1, ..., d l u. By definition of i being a symmetric copairing, (C3) is automatically satisfied in Vect. Also, the transposition b clearly satisfies (C4). Furthermore, b is a well-known solution of the Yang-Baxter equation so that (C5) is satisfied as well.

Proof of Theorem 1.2
The proof of Theorem 1.2 presented here is based on the proof of Theorem 1.1 given in Chapter 5 of [13]. It is divided into two parts as follows. The first part culminates in Corollary 4.3, which states that the given symmetric strict monoidal functor Y : cBr Ñ Vect is faithful if and only if it is faithful on loops (a notion that will be defined below). As it turns out, Corollary 4.3 is a consequence of Proposition 4.2, which takes place solely in the chromatic Brauer category. Secondly, we prove Theorem 4.4 which classifies all symmetric strict monoidal functors Y which are faithful on loops. We will also give a second proof of Theorem 4.4 in Section 4.1.
We introduce some operations ϕ op , o ϕ and ϕ o on morphisms ϕ : prms, cq Ñ prm 1 s, c 1 q in cBr. If the morphism ϕ is represented by the pair pW, Ωq (together with an embedding i : W ãÑ r0, 1sˆR 3 ), then there is a morphism ϕ op : prm 1 s, c 1 q Ñ prms, cq which can also be represented by pW, Ωq but where the embedding is given by op˝i, where op : pt, x, y, zq Þ Ñ p1´t, x, y, zq. Note that for isomorphisms α : prms, cq Ñ prms, c 1 q the identity α op " α´1 holds by means of Lemma 3.1. Indeed, for the underlying permutation σ α , σ α op of α, α op , respectively, we have σ´1 α " σ α op . Also note the validity of the equations i op pkq " e pkq and e op pkq " i pkq . Furthermore, we can express ϕ op : prm 1 s, c 1 q Ñ prms, cq in terms of ϕ : prms, cq Ñ prms, c 1 q via the formulas ϕ op "`e prm 1 s,c 1 q b 1 prms,cq˘˝`1prm 1 s,c 1 q b ϕ b 1 prms,cq˘˝`1prm 1 s,c 1 q b i prms,cq˘, or  , respectively. Then one can check (by comparing) that ϕ op is also represented by Figure 1.
A morphism ϕ : prms, cq Ñ prm 1 s, c 1 q can be written for some suitable pl k q kPN P À 8 k"0 N in the form where α : prms, cq Ñ prms, c 0 q and β : prm 1 s, c 1 0 q Ñ prm 1 s, c 1 q are isomorphisms in cBr, and ϕ 0 : prms, c 0 q Ñ prm 1 s, c 1 0 q is a loop-free morphism in cBr that can be written for some suitable pp k q kPN , pq k q kPN P À 8 k"0 N in the form (In particular, note that the maps c 0 : rms Ñ N and c 1 0 : rm 1 s Ñ N are monotonous, and satisfy |c´1 0 pkq|´2p k " |c 1´1 0 pkq|´2q k for all k P N.) Let us compute the expression ϕ op˝ϕ seperately by using the normal form of ϕ as described above, Now, for the isomorphism α we have the relations valid in cBr given by e prms,cq˝p 1 prms,cq b α´1q " e prms,c0q˝p α b 1 prms,c0q q, and p1 prms,cq b αq˝i prms,cq " pα´1 b 1 prms,c0q q˝i prms,c0q .
(Indeed, if we write α as a product of adjacent transpositions α " T 1˝¨¨¨˝TN , then we can shift the T i along e prms,‹q (resp. i prms,‹q ) from T i b 1 to 1 b T i , but in the reverse order. Note that during this procedure, the coloring ‹ is changing after each step, after the last shift of T N it has become c 0 .) This leads to wherec : r2 ř k p k s Ñ N is monotonous and satisfies |c´1pkq| " 2p k .
Conversely, if the morphisms ϕ, ψ : prms, cq Ñ prm 1 s, c 1 q satisfy (4.2), it suffices to show that ?pϕq " ?pψq in Br. After applying ? to (4.2) we obtain a ϕ 0˝ψ a 0 " λ b 1 2 pm`m 1 q , where ϕ 0 " ?pϕq, ψ 0 " ?pψq and a " ?˝o. Therefore, it suffices to show that for every component C of a representative W p a ϕ 0 q of a ϕ 0 there exists a component C of a representative W pψ a 0 q of ψ a 0 such that C and C have the same endpoints in rm`m 1 s. Let P be a point in rm`m 1 s and let P ϕ and P ψ denote the other endpoint of the connected component C and C, respectively, containing P . Note first, that the number 1 2 pm`m 1 q is the maximal number of loops λ which can be contained in a ϕ 0˝ψ a 0 , since W pψ a 0 q and W p a ϕ 0 q each consist of 1 2 pm`m 1 q distinguished connected components. This means that for every component C of W p a ϕ 0 q there is a component C of W pψ a 0 q such that C and C close up to S 1 . In other words, if P P C and P P C, then P ϕ " P ψ .
The symmetric strict monoidal functor Y : cBr Ñ Vect is called faithful on loops if for any two morphisms ϕ, ψ : prms, cq Ñ prm 1 s, c 1 q in cBr the condition Y pϕq " Y pψq implies that there are a sequence pl k q kPN P À 8 k"0 N and loop-free morphisms ϕ 0 and ψ 0 such that ϕ " Â k λ bl k pkq b ϕ 0 and ψ " Â k λ bl k pkq b ψ 0 . Recall from the discussion in Section 3.2 that Y ppr1s, kqq " V k is for all k P N a finite dimensional vector space, whose dimension will be denoted by d k . Furthermore, recall that Y pλ pkq q " d k .
As an immediate consequence of Proposition 4.2 we obtain the following corollary. Proof. What we need to show is that Y being faithful is implied by Y being faithful on loops. Let ϕ 0 , ψ 0 : prms, cq Ñ prms, c 1 q be loop-free such that Y pϕ 0 q " Y pψ 0 q, Now, under the assumption that Y is faithful on loops, Proposition 4.2 implies that ϕ 0 " ψ 0 . Hence, Y is also faithful on loop-free morphisms. Now let ϕ, ψ : prms, cq Ñ prm 1 s, c 1 q be morphisms (possibly containing loops) such that Y pϕq " Y pψq, and rewrite them as ϕ "`Â k λ bl k pkq˘b ϕ 0 and ψ "`Â k λ bl k pkq˘b ψ 0 , for a sequences pl k q kPN , P À k N and loop-free morphisms ϕ 0 , ψ 0 : prms, cq Ñ prm 1 s, c 1 q. Since Y is faithful on loops we have in particular Y pλ pkq q " d k ą 0 for all k P N. We obtain i.e. ϕ 0 " ψ 0 and therefore ϕ " ψ. Hence, Y is faithful.
If Y is faithful on loops, it is clear that the dimension d k of V k needs to satisfy for all pl k q kPN P À 8 k"0 Z the implication (1.1), namely Theorem 4.4. Suppose that d k ą 0 for all k P N, and that the implication (1.1) holds for all sequences pl k q kPN P À 8 k"0 Z . Then, the functor Y : cBr Ñ Vect is faithful on loops.
Proof. Let ϕ 0 : prms, cq Ñ prms, c 1 q be a loop-free morphisms presented in its normal form ϕ 0 " β˝φ 0˝α , withφ 0 " Â k 1 pr|c´1pkq|´2p k s,kq be bp k pkq bi bq k pkq . We will compute TrpY pϕ op 0˝ϕ 0 qq. Recall that the trace is invariant under cyclic permutation, and the trace of the tensor product of two matrices is the product of their traces. We will also use the identity Trpi pkq˝epkq q " d k . Indeed, this follows from the invariance of the trace under cyclic permutation and Y pλ pkq q " e pkq˝ipkq " d k . Thus, we obtain TrpY pϕ op 0˝ϕ 0 qq " Tr`Y pα´1q˝Y pφ op 0 q˝Y pβ´1q˝Y pβq˝Y pφ 0 q˝Y pαq" Tr`Y pφ op 0 q˝Y pφ 0 q" Note two things: The number TrpY pϕ op 0˝ϕ 0 qq does not vanish in any case and it only depends on the domain prms, cq and codomain prms, c 1 q of the morphism ϕ ∅ . Now, let ϕ, ψ : prms, cq Ñ prm 1 s, c 1 q such that Y pϕq " Y pψq. Then there are sequences pµ k q kPN , pν k q kPN P À 8 k"0 N and loop-free morphisms ϕ 0 , ψ 0 : prms, cq Ñ prm 1 s, c 1 q such that ϕ " Â 8 k"0 λ bµ k pkq b ϕ 0 and ψ " Firstly, we study the behavior of Y on isomorphisms.
(ii) If the claim holds for two isomorphisms α : prms, aq Ñ prms, a 1 q and β : prms, a 1 q Ñ prms, a 2 q in cBr, then it also holds for their composition β˝α : prms, aq Ñ prms, a 2 q. In fact, using σ β˝α " σ β˝σα , we obtain for all w i P V apiq , 1 ď i ď m, An immediate consequence of Lemma 4.5 is that Y is faithful on isomorphisms in cBr. Corollary 4.6. Suppose that d k ě 2 for all k P N. If ι 1 , ι 2 : prms, cq Ñ prms, c 1 q are two isomorphisms in cBr such that Y pι 1 q " Y pι 2 q, then ι 1 " ι 2 .
The next crucial step is to compute the preimage under Y of scalar square matrices.
Proposition 4.7. Suppose that d k ě 2 for all k P N. If ϕ P Hom cBr pprms, cq, prms, cqq satisfies Y pϕq " µ¨1 Â m i"1 V cpiq for some µ P R, then there exists pl k q kPN P À 8 k"0 N such that µ " ś 8 k"0 d l k k and ϕ "´Â Proof. The given morphism ϕ : prms, cq Ñ prms, cq can be written for some suitable pl k q kPN P À 8 k"0 N in the form where α : prms, cq Ñ prms, c 1 q and β : prms, c 2 q Ñ prms, cq are isomorphisms in cBr, and ϕ 0 : prms, c 1 q Ñ prms, c 2 q is a loop-free morphism in cBr that can be written for some suitable pp k q kPN , pq k q kPN P À 8 k"0 N in the form (In particular, note that the maps c 1 , c 2 : rms Ñ N are monotonous, and satisfy |c 1´1 pkq|´2p k " |c 2´1 pkq|´2q k for all k P N.) Applying Y and using that Y pλ pkq q " d k for all k P N, we obtain Using the assumption Y pϕq " µ¨1 Â m i"1 V cpiq and setting γ :" β´1˝α´1 : prms, c 1 q Ñ prms, c 2 q, we have µ For every k P N we fix an ordered basis tv k 1 , . . . , v k d k u of the real vector space V k " Y ppr1s, kqq. For any object prms, cq in cBr we assume that is equipped with the lexicographically ordered basis tv cp1q r1 b¨¨¨b v cpmq rm u r1,...,rm . Then, by Remark 2.2, the matrix representation of Y pϕ 0 q is the Kronecker product of the matrix representations of Y p1 pr|c 1´1 pkq|´2p k s,kq q " p1 V k q bp|c 1´1 pkq|´2p k q , Y pe bp k pkq q " Y pe pkq q bp k and Y pi bq k pkq q " Y pi pkq q bq k , k P N. Note that the matrix representation of 1 V k is the identity matrix of size d kˆdk . Moreover, the matrix representation of Y pe pkq q : V k b V k Ñ R is a p1ˆd 2 k q-matrix which contains at least two nonzero entries because d k ě 2, and the pd kˆdk q-matrix pY pe pkq qpv k i b v k j qq i,j is invertible according to the discussion in Section 3.3. Similarly, the matrix representation of Y pi pkq q : R Ñ V k b V k is a pd 2 kˆ1 q-matrix which contains at least two nonzero entries. Therefore, if there exists k P N such that p k ą 0 or q k ą 0, then the matrix representation of Y pϕ 0 q has a column or a row which contains at least two nonzero entries. On the other hand, it follows from Lemma 4.5 that Y pγq is represented by a permutation matrix because γ : prms, c 1 q Ñ prms, c 2 q is an isomorphism. In particular, every column and every row of Y pγq has exactly one nonzero entry, which yields a contradiction in Equation (4.3). Consequently, p k " 0 and q k " 0 for all k P N. Hence, we have c 1 " c 2 , and ϕ 0 " Â 8 k"0 1 pr|c 1´1 pkq|s,kq " 1 prms,c 1 q . Then it follows from Equation (4.3) that µ " ś 8 k"0 d l k k because Y pγq is a permutation matrix and Y pϕ 0 q is an identity matrix. Moreover, we can apply Corollary 4.6 to the isomorphisms γ, ϕ 0 : prms, c 1 q Ñ prms, c 1 q to obtain γ " ϕ 0 . Finally, we conclude that ϕ "´Â Proof of Theorem 4.4. Assume that Y pϕq " Y pψq for two given morphisms ϕ, ψ : prms, bq Ñ prns, cq in cBr. For some suitable pµ k q kPN , pν k q kPN P À 8 k"0 N we can write where α, α 1 : prms, cq Ñ prms, b 0 q and β, β 1 : prns, c 0 q Ñ prms, c 1 q are isomorphisms in cBr, and ϕ 0 , ψ 0 : prms, b 0 q Ñ prns, c 0 q are loop-free morphisms in cBr that can be written for some suitable pp k q kPN , pp 1 k q kPN , pq k q kPN , pq 1 k q kPN P À 8 k"0 N in the form (In particular, note that the maps b 0 : rms Ñ N and c 0 : rns Ñ N are monotonically increasing, and satisfy |b´1 0 pkq|´2p k " |c´1 0 pkq|´2q k and |b´1 0 pkq|´2p 1 k " |c´1 0 pkq|2 q 1 k for all k P N.) We have to show that ϕ and ψ have the same number of k-loops for each k P N, that is, pµ k q kPN " pν k q kPN . In the following, we will only show ν k ď µ k . Then, ν k " µ k follows by symmetry.
We will use Proposition 4.7 to reduce the assumption Y pϕq " Y pψq to equation (4.4) below, which is a statement in the chromatic Brauer category.
Applying the monoidal functor Y to the previous equation and using Y pϕq " Y pψq, we obtain Y pb˝ψ˝aq " Y pb˝ϕ˝aq " ote that the assumptions on the sequence d 0 , d 1 , . . . imply that d k ě 2 for all k P N. Hence, it follows from Proposition 4.7 that there exists pl k q kPN P À 8 k"0 N such that Note that the implication (1.1) implies that l k " µ k`pk`qk for all k P N. It suffices to show that for each k P N, ν k`pk`qk is an upper bound for the number of k-loops contained in the composition b˝ψ˝a. (Indeed, then it follows from equation (4.4) that ν k`pk`qk ď µ k`pk`qk . Thus, ν k ď µ k .) Setting ψ 1 0 :" β´1˝β 1˝ψ 0˝α 1˝α´1 , r a " Â 8 k"0 a k and r b " Fix k P N. It suffices to show that the number of loops of label k in r b˝ψ 1 0˝r a is ď p k`qk .
We choose 1-manifolds W 0 Ă r0, 1sˆR 3 , W 1 Ă r1, 2sˆR 3 and W 1 Ă r2, 3sˆR 3 which represent (up to translations along the first coordinate) the Brauer morphisms r a, ψ 1 0 and r b, respectively. Then, r b˝ψ 1 0˝r a is represented (after reparametrization of the first coordinate) by the union W : For a 1-manifold X Ă rs, tsˆR 3 which represents some morphism in cBr, let Xte pkq u (respectively, Xti pkq u) be the set of label k components of X whose endpoints are both contained in tsuˆR 3 (respectively, in ttuˆR 3 ). Moreover, denote by Xt1 pkq u the set of label k components of X which have one endpoint in tsuˆR 3 and the other one in ttuˆR 3 . Finally, let Xtλ pkq u be the set of closed label k components of X.
Note that the number of k-loops in r b˝ψ 1 0˝r a is given by the cardinality of W tλ pkq u, and we have to show that this number is ď p k`qk . By definition of r a and r b, |W 0 ti pkq u| " p k and |W 1 te pkq u| " q k . Hence, it suffices to construct an injective map W tλ pkq u Ñ W 0 ti pkq u Y W 1 te pkq u.
Let L P W tλ pkq u be a closed label k component of W . The intersections L X W 0 , LXW 1 and LXW 1 can be written as disjoint union of components of W 0 , W 1 and W 1 respectively. It follows from L X pt0uˆR 3 q " H, W 0 te pkq u " H and W 0 tλ pkq u " H that L X W 0 is a disjoint union of elements of W 0 ti pkq u. Analogously, it follows from L X pt3uˆR 3 q " H, W 1 ti pkq u " H and W 1 tλ pkq u " H that L X W 1 is a disjoint union of elements of W 1 te pkq u. Moreover, L has nonempty intersection with W 0 \ W 1 . (In fact, ψ 1 0 is a loop-free Brauer morphism, being the composition of the loop-free Brauer morphism ψ 0 and Brauer isomorphisms. Therefore, W 1 does not contain any closed components, so in particular W tλu " H. Hence, L cannot be entirely contained in W 1 . Thus, L X W 0 \ W 1 ‰ H.) Hence, we can pick an element of W 0 ti pkq uYW 1 te pkq u which is contained in L. This defines a map W tλu Ñ W 0 ti pkq u Y W 1 te pkq u. By construction, this map is injective. (Indeed, assume that L, L 1 P W tλ pkq u are mapped to the same element C P W 0 ti pkq u Y W 1 te pkq u. Then, H ‰ C Ă L X L 1 implies L " L 1 .) l

Positive TFTs, fold maps, and exotic Kervaire spheres
Using the Brauer category Br and singularity theory of fold maps, Banagl [3,4] has constructed a high-dimensional positive TFT which is defined on smooth cobordisms. He also showed that the state sum of the theory can distinguish exotic smooth structures on spheres from the standard smooth structure. The construction is sketched in Section 10 of [3] as an application of the general framework of positive TFTs, and has been worked out in full detail in [4]. In the present section, we construct a refinement of Banagl's theory in which we replace the Brauer category Br by its chromatic enrichment cBr. The power of our state sum invariant is illustrated by Theorem 5.7, where we show that the associated aggregate invariant can detect exotic Kervaire spheres in infinitely many dimensions.
The present section is structured as follows. In Section 5.1, we outline the general features of the framework of positive TFTs, and explain the process of quantization. Section 5.2 provides the concrete definitions of fold fields and the cBr-valued action functional while pointing out the changes that arise from using the chromatic Brauer category instead of Br. Quantization is discussed in Section 5.3, where we carefully indicate the necessary modifications in the algebraic process of profinite idempotent completion. Finally, we define the aggregate invariant of homotopy spheres, and sketch our application to exotic Kervaire spheres in Section 5.4. 5.1. General framework. In [3] Banagl presents an individual approach to the construction of certain TFTs in arbitrary dimension. The idea is to modify Atiyah's original axioms [1] by formulating them over semirings instead of rings. Compared to a ring, a semiring is not required to have additive inverses, i.e. "negative" elements. Banagl introduces the notion of positive TFTs, and shows that any system of so-called fields and action functionals gives rise to a positive TFTs by means of a process called quantization. This framework is inspired by quantization from theoretical physics. In order to avoid set theoretic difficulties that may arise in the definition of the Feynman path integral, Banagl employs the concept of complete semirings due to Eilenberg [7]. The reason is that a complete semiring has a summation law that allows to sum families of elements indexed by arbitrary index sets. Positive TFTs can motivate the construction of new invariants for smooth manifolds like the aggregate invariant of homotopy spheres (see Section 10 in [4]).
In the following, we outline Banagl's construction [3] of a n-dimensional positive TFT from given systems of fields and action functionals. Following Section 5 in [3], a system F of fields assigns to every closed pn´1q-manifold M and to every ncobordism W sets F pM q and F pW q of fields on M and W , respectively. Fields on a cobordism can be restricted to subcobordisms and to codimension 1 submanifolds. Apart from desirable behavior with respect to the action of homeomorphisms and disjoint union, fields are especially required to glue under the gluing of cobordisms. The axioms for a system T of action functionals (or action exponentials) with values in a strict monoidal category C are inspired by the exponential of the action that appears in the integrand of the Feynman path integral. To every n-cobordism W one associates a map T W : F pW q Ñ MorpCq in such a way that disjoint union of cobordisms is reflected by tensor product of morphisms in C, and gluing of cobordisms corresponds to composition of morphisms. More precisely, it is required that T W pf q " T U pf | U q b T V pf | V q for fields f on the disjoint union W " W 1 \ W 2 of cobordisms W 1 and W 2 , and T W pf q " T U pf | U q˝T V pf | V q for fields f on the gluing W " U Y N V along N of cobordisms U from M to N and V from N to P . Furthermore, the action functional is invariant under the action of homeomorphisms. In Section 5.2, we will specifically take C " cBr.
Next, we describe the process of quantization (see Section 6 in [3]). For this purpose, we fix a system F of fields, a C-valued system T of action functionals, and a complete semiring S. Following Section 4 in [3], one first constructs a complete additive monoid Q from the semiring S and the strict monoidal category C. The elements of Q are just maps MorpCq Ñ S. Then, one exploits the completeness of S to define two different multiplications on Q. As a result, one obtains a pair pQ c , Q m q of generally non-commutative complete semirings. Multiplication in Q c is based on the composition of morphisms in C, whereas multiplication in Q m exploits the monoidal structure of C. As explained in Section 6 of [3], one assigns to every n-cobordism W the composition T W : F pW q Ñ Q of T W : F pW q Ñ MorpCq with the map MorpCq Ñ Q that assigns to every morphism in C its characteristic function. Then, the state sum Z W : F pBW q Ñ Q is defined on a boundary condition where the sum ranges over all fields F on W that extend f , i.e., F | BW " f . Note that Z W is well-defined due to the completeness of Q. In analogy with the quantum Hilbert state from physics, the state module ZpM q of a closed n-manifold M consists of all maps ("states") F pM q Ñ Q that satisfy a certain constraint equation. It can be shown that Z W satisfies the constraint equation and is thus an element of the state module ZpBW q. Furthermore, the state modules and state sums thus defined can be shown to satisfy Banagl's axioms of a positive TFT, including the essential gluing axiom. For a topologically meaningful choice of fields and action functionals the state sum Z W is an invariant of n-cobordisms W that is interesting for further investigation.

Fold fields and cBr-valued actions.
Fix an integer n ě 2. In this section we specify the fields and actions that determine our modification of the ndimensional positive TFT constructed in [4]. All manifolds considered (with or without boundary) will be smooth, that is, differentiable of class C 8 .

5.2.1.
Cobordisms. We recall the terminology concerning manifolds and cobordisms from Section 7.1 of [4]. From now on, we always use the terminology M , N , P etc. for closed pn´1qdimensional manifolds. Fix an integer D ě 2n`1. We will always assume that any M is smoothly embedded in R D , and that every connected component of M is contained in a hyperplane of the form tkuˆR D´1 for some k P t0, 1, 2, . . . u.
Definition 5.1. A cobordism from M to N is a compact n-dimensional smoothly embedded manifold W Ă r0, 1sˆR D with the following properties: (1) the boundary of W is BW " M \ N , where M Ă R D " t0uˆR D is the ingoing boundary and N Ă R D " t1uˆR D is the outgoing boundary, (2) the interior of W satisfies W zBW Ă p0, 1qˆR D , (3) there exists 0 ă ε ă 1 2 such that W X r0, εsˆR D " r0, εsˆM and W X r1´ε, 1sˆR D " r1´ε, 1sˆN are product embeddings (any such ε is referred to as a cylinder scale), and (4) every connected component of W is contained in a set of the form r0, 1st kuˆR D´1 for some k P t0, 1, 2, . . . u.
The advantage of working with embedded cobordisms W Ă r0, 1sˆR D is that they are naturally equipped with time functions ω : W Ñ r0, 1s induced by projection to the first coordinate. For every regular value t P r0, 1s of the time function ω : W Ñ r0, 1s the preimage ω´1ptq is a smoothly embedded codimension 1 submanifold of W .

5.2.2.
System of fold fields. Our theory will use exactly the same definition of fold fields on n-cobordisms that is employed in the original construction. Thus, in this section we will outline the content of Section 7.2 of [4]. We also use the same sets of fields on closed pn´1q-manifolds although their definition relies on our modified action functional (see the end of Section 5.2.3).
The construction of fold fields on an n-dimensional cobordism W is based on the notion of fold maps from W into the plane R 2 -C. By definition, a fold map of an n-manifold X without boundary into the plane is a smooth map F : X Ñ R 2 such that for every point x P X there exist coordinate charts centered at x and F pxq in which F takes one of the following two normal forms: n´1˘( fold point of F ). Let SpF q denote the set of fold points of a fold map F : X Ñ R 2 . It can be shown that SpF q Ă X is a smoothly embedded 1-dimensional submanifold that is closed as a subset, and that F restricts to an immersion SpF q Ñ R 2 . In analogy with the Morse index of non-degenerate critical points, there is the following notion of an (absolute) index for fold points.
Proposition 5.2. To any fold map F : X Ñ R 2 one can associate a well-defined locally constant map where i P t0, . . . , n´1u is the number of minus signs that appear in the local normal form of fold points above.
Let W be an n-dimensional cobordism from M to N . A smooth map F : W Ñ R 2 is called fold map if F has for some ε ą 0 an extension to a fold map Given a fold map F : W Ñ R 2 , the intersection Sp r F q X W does not depend on the choice of the fold map extension r F , and will in the following be denoted by SpF q. (The independence of the choice of r F can be shown by using the characterization of fold maps by means of transversality in jet spaces as discussed in Section 3 of [4].) Furthermore, for an open subset U Ă BW we write SpF q&U if SpF q&U for some (and hence, any) fold map extensionF of F . If SpF q&BW , then SpF q Ă W is a 1-dimensional smoothly embedded compact submanifold with boundary BSpF q " SpF q X BW . In this case, we write ι F : SpF q Ñ N for the restriction of ι r F : Sp r F q Ñ N to SpF q for some (and hence, any) fold map extensionF of of F . Let ω : W Ñ r0, 1s denote the time function associated to W (see Section 5.2.1). Definition 5.3. Given a fold map F : W Ñ C, we set &pF q " t P r0, 1s ; t is a regular value of ω, and SpF q&ω´1ptq ( Ă r0, 1s .
Definition 5.4. A fold map F : W Ñ C has generic imaginary parts over t P r0, 1s if the restriction Im˝F | : SpF q X ω´1ptq Ñ R is injective. Let GenImpF q " tt P r0, 1s ; F has generic imaginary parts over tu Ă r0, 1s .
For k P t0, 1, 2, . . . u let F pkq denote the restriction of a fold map F : W Ñ C to the part of W that lies in r0, 1sˆtkuˆR D´1 (see Definition 5.1(4)): Fields on W are fold maps F : W Ñ C with certain properties concerning the subsets &pF pkqq and GenImpF pkqq of r0, 1s.
Definition 5.5. A fold field on W is a fold map F : W Ñ C so that for all k P t0, 1, 2, . . . u the following conditions hold: p1q 0, 1 P &pF pkqq X GenImpF pkqq, and p2q GenImpF pkqq is residual in r0, 1s.
Condition p1q is exploited in the construction of the Br-valued action functional S in Section 7.3 of [4] (as well as in our modified construction in Section 5.2.3). Condition p2q is crucial for the proof of the indispensable gluing theorem (see Section 7.7 in [4]).
Let F pW q denote the set of all fold fields on W . If W " H, then one puts F pW q " t˚u (set with a single element). Fields on closed pn´1q-dimensional manifolds will be introduced at the end of Section 5.2.3, which completes the definition of the system F of fields.

5.2.3.
System of cBr-valued action functionals. In Section 7.3 in [4], Banagl uses singularity theory of fold maps into the plane to construct a system S of Brvalued action functionals. Namely, for every n-cobordism W there is a function S : F pW q Ñ MorpBrq assigning to every fold field on W a morphism in Br that encodes the combinatorial information of the 1-dimensional singular set of the fold map. In the present section, we modify the original construction by replacing the Brauer category Br by its chromatic enrichment cBr. The idea is to capture not only the singular patterns provided by the singular sets of fold fields, but also to remember the indices of fold lines by using labels from the set N. Hence, we construct a system S of cBr-valued action functionals which is a lift of S under the forgetful map MorpcBrq Ñ MorpBrq. During the process of quantization in Section 5.3 we will linearize the system S of action functionals by means of a faithful linear representation Y : cBr Ñ Vect (see Theorem 1.2).
Let W be an n-cobordism from M to N . We construct the function S : F pW q Ñ MorpcBrq as follows. If W is empty, then we set Sp˚q " id pr0s,c H q . Next suppose that W is non-empty and entirely contained in a set of the form r0, 1sˆtkuˆR D´1 , where k P t0, 1, 2, . . . u. Let F p" F pkqq P F pW q be a field on W . By condition (1) for fold fields (see Definition 5.5), we have 0, 1 P &pF q, so that the intersections SpF q X M and SpF q X N are compact manifolds of dimension 0. Furthermore, since F has generic imaginary parts over 0 and 1 (see Definition 5.4), the composition Im˝F : W Ñ R restricts to injective maps on both SpF q X M and SpF q X N . Let m and m 1 denote the number of points in SpF qXM and SpF qXN , respectively. Then, we obtain orderings SpF q X M " tp 1 , . . . , p m u and SpF q X N " tq 1 , . . . , q m 1 u which are uniquely determined by requiring that pIm˝F qpp i q ă pIm˝F qpp j q if and only if i ă j, and pIm˝F qpq i q ă pIm˝F qpq j q if and only if i ă j. The resulting bijections SpF q X M -M rms, p i Þ Ñ i, and SpF q X N -M rm 1 s, q i Þ Ñ i, are exactly the same as those described in the original construction of S. We define maps c : rms Ñ N and c 1 : rm 1 s Ñ N by assigning to each point x P rms " M rms -SpF q X M and x 1 P rm 1 s " M rm 1 s -SpF qXN the index of the fold map F at x and x 1 , respectively (see Proposition 5.2). So far, we have constructed objects prms , cq and prm 1 s , c 1 q in cBr. The desired morphism SpF q : prms , cq Ñ prm 1 s , c 1 q in cBr is now represented by the pair pSpF q, ι F q, where the embedding SpF q Ă r0, 1sˆR 3 is defined in exactly the same manner as described in the construction of S. That is, every component of SpF q with non-empty boundary is embedded as a smooth arc that connects the corresponding points in pt0uˆM rmsq Y pt1uˆM rm 1 sq. (For components of SpF q without boundary one may choose an arbitrary embedding into p0, 1qˆR 3 .) Finally, for an arbitrary non-empty cobordism W , we define SpF q " Â 8 k"0 SpF pkqq. (Note that the tensor product is actually finite because W is compact.) This completes our construction of a system S of cBr-valued action functionals which lifts S under the forgetful map MorpcBrq Ñ MorpBrq. Note that Lemma 7.12 in [4] remains valid when replacing S with S in the formulation. That is, given a fold field F on W and some t P p0, 1q such that t P &pF pkqq X GenImpF pkqq for all k P t0, 1, . . . u, F restricts to fold fields F ďt on W X pr0, tsˆR D q and F ět on W X pr1´t, 1sˆR D q, and we have SpF q " SpF ďt q˝SpF ět q in cBr.
Finally, fields on a closed pn´1q-manifold M are defined to be certain fold fields on the cylinder r0, 1sˆM Ă r0, 1sˆR D , i.e., the trivial cobordism from M to M . Namely, when M is non-empty, we put F pM q " tf P F pr0, 1sˆM q; Spf q " 1 P MorpcBrqu, where 1 denotes some identity morphism in cBr. Note that, by Lemma 3.1, a fold field f P F pr0, 1sˆM q satisfies Spf q " 1 in cBr if and only if Spf q " 1 in Br. If M " H, then one puts F pM q " t˚u (set with a single element). Hence, the set of fields on closed pn´1q-manifold remains unchanged when replacing S by S. In particular, Lemma 7.13 and Lemma 7.14 (additivity axiom) in [4] remain valid when replacing S with our modified cBr-valued action functional S in the formulation.

5.3.
Quantization. As pointed out in Section 8 of [3], it can be advantageous to linearize the category-valued system of action functionals used for quantization. We construct a linearization T of our system S of cBr-valued action functionals from Section 5.2.3 as follows. Fix once and for all a faithful symmetric strict monoidal functor Y : cBr Ñ Vect by means of Theorem 1.2. Then, a Vect-valued action functional T is defined by assigning to every n-cobordism W the composition In the present section, we quantize the system F of fold fields of Section 5.2.2 and our system T of Vect-valued action functionals. For this purpose, we will first modify in Section 5.3.2 below the algebraic process of profinite idempotent completion (see Section 6 in [4]) to represent loops of different colors in cBr by a countable family of loop parameters. Then, we proceed to define our positive TFT Z. In Section 5.3.3 we will specify the state modules ZpM q of closed pn´1qmanifolds M , and define the state sums of n-cobordisms W as certain elements Z W P ZpBW q. First of all, Section 5.3.1 provides the necessary background on semirings.
5.3.1. Semirings and semimodules. We collect some basic material from the theory of semirings and semimodules that is needed for the process of quantization. A detailed background is provided in Section 2 of [3] and Section 4 of [4].
Recall that a (commutative) monoid is a triple M " pM,˚, eq, where M is a set equipped with a (commutative) associative binary operation˚and a two-sided identity element e P M , that is, e˚m " m˚e " m for all m P M . A semiring is a tuple S " pS,`,¨, 0, 1q, where S is a set together with two binary operations and¨and two elements 0, 1 P S such that pS,`, 0q is a commutative monoid, pS,¨, 1q is a monoid, the multiplication¨distributes over the addition from either side, and 0 is absorbing, i.e. 0¨s " 0 " s¨0 for every s P S. The semiring S is called commutative if the monoid pS,¨, 1q is commutative. A morphism of semirings sends 0 to 0, 1 to 1 and respects addition and multiplication. Fix a semiring S. A (left) S-semimodule is a commutative monoid M " pM,`, 0 M q together with a scalar multiplication SˆM Ñ M , ps, mq Ñ sm, such that for all r, s P S, m, n P M , we have prsqm " rpsmq, rpm`nq " rm`rn, pr`sqm " rm`sm, 1m " m, and r0 M " 0 M " 0m. Given a morphism ϕ : S Ñ T of semirings, it is clear that T becomes a S-semimodule via st " ϕpsqt.
A monoid pM,˚, eq is called idempotent if m˚m " m for all elements m P M . The semiring pS,`,¨, 0, 1q is idempotent if pS,`, 0q is an idempotent monoid. A semimodule is called idempotent if its underlying additive monoid is idempotent.
Next, we discuss the important notion of Eilenberg-completeness [7, p. 125] for semirings and semimodules. A complete monoid is a commutative monoid pM,`, 0q together with an assignment Σ, called a summation law, which assigns to every family pm i q iPI , indexed by an arbitrary set I, an element ř iPI m i of M (called the sum of the m i ), such that ÿ and for every partition I " A complete semiring is a semiring S for which pS,`, 0, Σq is a complete monoid, and infinite distributivity holds, that is, A semimodule M over a commutative semiring S is called complete if its underlying additive monoid is equipped with a summation law that makes it complete as a commutative monoid, and infinite distributivity ÿ iPI sm i " s˜ÿ iPI sm iḩ olds for every s P S and every family pm i q iPI in M . If ϕ : S Ñ T is a morphism of semirings and T is complete as a semiring, then T can be easily seen to be complete as an S-semimodule.
We will also need a notion of continuity for idempotent complete semirings. Here, we only state the definition, and refer to the discussion preceding Proposition 4.2 in [4] for more details. Observe that any idempotent monoid pM,˚, eq admits a natural partial order ď given by m ď m 1 if and only if m`m 1 " m 1 . An idempotent complete monoid pM,`, 0, Σq is continuous if for all families pm i q iPI , m i P M , and for all c P M , ř iPF m i ď c for all finite F Ă I implies ř iPI m i ď c. An idempotent complete semiring (semimodule) is called continuous if its underlying additive monoid is continuous.
It is useful to note that the product ś iPI M i of a family tM i u iPI of continuous idempotent complete monoids is a continuous idempotent complete monoid.
Example 5.6. The minimal example of a semiring that is not a ring is given by the Boolean semiring B. This is the set B " t0, 1u equipped with addition defined by 1`1 " 1 and multiplication given by 0¨0 " 0 (where 0 and 1 serve as identity elements for addition and multiplication, respectively). Distributivity holds, but in B there exists no additive inverse for 1. We leave it to the reader to check that the commutative semiring B is idempotent, complete, and continuous.

5.3.2.
Profinite idempotent completion. The algebraic process of profinite idempotent completion (see Section 6 in [4]) adapts the general construction of a pair pQ c , Q m q of complete semirings (see Section 4 in [3]) to reflect the specific inner structure of morphism sets of the Brauer category, as we recall next. Note that the sets Hom Br prms, rm 1 sq have the special property that they are naturally equipped with the action τ i ϕ " ϕ b λ bi of the (multiplicatively written) monoid N " tτ i ; i P Nu. Fix a linear representation U : Br Ñ Vect and write V " U pr1sq and p λ " U pλq P R. Then, the subset H m,n " U pHom Br prms, rm 1 sqq of the real vector space Hom Vect pV bm , V bm 1 q inherits an action of the monoid N via τ i f " p λ i f . Given a set A, let F M pAq denote the free commutative monoid generated by A. In particular, F M pH m,n q has the structure of a Nrτ s-semimodule by Lemma 4.1 in [4]. A Nrτ s-semimodule is given by the algebraic tensor product QpH m,n q " F M pH m,n q b Nrτ s Brrqss, where B denotes the Boolean semiring, and Brrqss is the associated semiring of formal power series. It can be shown (see Lemma 6.7 in [4]) that QpH m,n q is isomorphic as a Nrτ s-semimodule to a finite sum of copies of Brrqss, so that its elements consist of a number of power series in the loop parameter q. (This can be derived more abstractly by using minimal shells of projectively finite subsets of a real vector space, see Definition 6.1 in [4].) Finally, the profinite idempotent completion of the set U pMorpBrqq is the Nrτ s-semimodule Q " QpU q " ź m,m 1 PN QpH m,n q.
Provided that the underlying functor U is chosen to be faithful on loops, the additive monoid pQ,`, 0q can be promoted to idempotent complete semirings Q c (the composition semiring, see Proposition 6.12 in [4]) and Q m (the monoidal semiring, see Proposition 6.14 in [4]). It can be shown that the semirings Q c and Q m are both continuous (see Proposition 6.15 in [4]). Continuity is exploited in [4] to check several axioms of positive TFTs, namely the behavior of state sums under disjoint union (Proposition 7.22), and the gluing axiom (Theorem 7.26).
When using the chromatic Brauer category instead of Br, we need to replace the semimodule Brrqss over Nrτ s by the semimodule Brrqss " Brrq 0 , q 1 , . . . ss over Nrts " Nrτ 0 , τ 1 , . . . s. Here, the different parameters represent loops of different labels in cBr. Let us introduce the semirings Nrτ s and Brrqss (compare Section 4 in [4]). Set N " À 8 i"0 N, which is a commutative monoid with respect to component wise addition and identity element 0 " p0, 0, . . . q. In general, a (formal) power series in a countable number of indeterminates q " q 0 , q 1 , . . . and with coefficients in the Boolean semiring B is a function a : N Ñ B, written as a formal sum ř νPN apνqq ν , where q ν denotes the (finite) product ś 8 s"0 q νs s . The element apνq is referred to as the coefficient of q ν . Let Brrqss be the set of all power series over B having a countable number of indeterminates q " q 0 , q 1 , . . . . We write 0 for the power series a with apνq " 0 for all ν, and 1 for the power series a with ap0q " 1 and apνq " 0 for all ν ‰ 0. Define an addition on power series by a`b " c, where cpνq " apνq`bpνq for all ν. Define a multiplication on power series by the Cauchy product, that is, a¨b " c where cpνq " ř µ`κ"ν apµqbpκq. Then, pBrrqss,`,¨, 0, 1q is a commutative idempotent semiring, the semiring of power series over B in a countable number of indeterminates. In a similar way, using finite sums rather than power series, one defines the polynomial semiring Nrts in a countable number of indeterminates t " τ 0 , τ 1 , . . . . Note that Brrqss is a Nrts-semimodule via the semiring morphism Nrts Ñ Brrqss that extends the unique semiring morphism N Ñ B by τ k Þ Ñ q k , k P N. It can be shown that Brrqss is a complete semiring, and is hence complete as a Nrts-semimodule. Furthermore, the idempotent complete semiring Brrqss can be shown to be continuous.
Returning to the category cBr, we observe that the k-loops λ pkq , k P N, induce an action of the (multiplicatively written) commutative monoid N " tt nu ; ν P Nu on the morphism sets Hom cBr pprms, cq, prm 1 s, c 1 qq via pt ν , ϕq Þ Ñ t ν ϕ "´Â 8 k"0 λ bν k pkq¯b ϕ. Using the fixed linear representation Y : cBr Ñ Vect, we write V k " Y ppr1s, kqq and p λ pkq " Y pλ pkq q P R. Then, the subset H prms,cq,prm 1 s,c 1 q " Y pHom cBr pprms, cq, prm 1 s, c 1 qqq of the real vector space Hom Vect pY pprms, cqq, Y pprm 1 s, c 1 qqq " Hom Vect pV cp1q b¨¨¨bV cpmq , V c 1 p1q b¨¨¨bV c 1 pm 1 q q inherits an action of the monoid N via t ν f "´ś 8 k"0 p λ ν k pkq¯¨f . In analogy to Lemma 4.1 in [4], it follows that F M pH prms,cq,prm 1 s,c 1 q q, the free commutative monoid generated by H prms,cq,prm 1 s,c 1 q , has the structure of a Nrts-semimodule via ÿ νPN m ν t ν¨ÿ j α j f j " ÿ ν,j pm ν α j qpt ν f j q, m ν , α j P N, f j P H prms,cq,prm 1 s,c 1 q .
Using the algebraic tensor product of semimodules over the commutative semiring Nrts (compare Section 4 in [4]) we can now define a Nrts-semimodule by QpH prms,cq,prm 1 s,c 1 q q " F M pH prms,cq,prm 1 s,c 1 q q b Nrts Brrqss.
Let OP prms,cq,prm 1 s,c 1 q denote the (finite) set of loop-free morphisms prms, cq Ñ prm 1 s, c 1 q in cBr. Since the linear representation Y : cBr Ñ Vect has been chosen to be faithful, it can be shown that QpH prms,cq,prm 1 s,c 1 q q is isomorphic in the category of Nrts-semimodules to the product of copies of Brrqss indexed by the elements of OP prms,cq,prm 1 s,c 1 q . (In fact, in analogy to Lemma 6.6 in [4], one can show that every element in F M pH prms,cq,prm 1 s,c 1 q q can be uniquely written as r ÿ σ"1 p σ ptqY pϕ σ q for suitable polynomials p σ ptq P Nrts, where ϕ 1 , . . . , ϕ r is the list of elements of OP prms,cq,prm 1 s,c 1 q .) Finally, the profinite idempotent completion of the set Y pMorpcBrqq is the Nrτ s-semimodule Q " QpY q " ź prms,cq,prm 1 s,c 1 q QpH prms,cq,prm 1 s,c 1 q q.
Similarly to the construction in Section 6 in [4], the additive monoid pQ,`, 0q is complete, and can hence be advanced to idempotent complete semirings Q c and Q m . Since Brrqss is continuous, it follows that Q c and Q m are both continuous.

5.3.3.
State modules and state sums. Let Q denote the profinite idempotent completion of the set Y pMorpcBrqq associated to a fixed faithful linear representation Y : cBr Ñ Vect as constructed in of Section 5.3.2. We proceed to define our smooth positive TFT Z. The state module of a closed pn´1q-manifold is defined to be ZpM q " tz : F pM q Ñ Qu. By Proposition 3.1 in [3], ZpM q inherits the structure of a two-sided Q c -semialgebra and a two-sided Q c -semialgebra, and ZpM q is complete. Then, it follows from the corresponding properties of Q that ZpM q is idempotent and continuous. The construction of a contraction product x¨,¨y : pZpM q p bZpN qqˆpZpN q p bZpP qq Ñ ZpM q p bZpP q is analogous to the discussion in Section 7.4 of [4]. Here, p b denotes the complete tensor product of complete idempotent continuous semimodules (see Section 5 in [4]) rather than the algebraic tensor product b of function semimodules discussed in [2].
Let W n be a cobordism from M to N . The state sum will be defined as an element Z W P ZpM q p bZpN q. Fix a cylinder scale ε W for W . Given a boundary condition pf M , f N q P F pM qˆF pN q, we define F pW ; f M , f N q " tF P F pW q| Dεpkq, ε 1 pkq P p0, ε W q : F | r0,εpkqs«Mpkq « f M pkq, F | r1´ε 1 pkq,1sˆN pkq « f N pkq, @ku, where the equivalence relation « for fold fields on closed pn´1q-manifolds X from Definition 7.18 in [4] is used. Namely, two smooth maps f : ra, bsˆX Ñ C and f 1 : ra 1 , b 1 sˆX Ñ C are equivalent, f « f 1 , if there exists a diffeomorphism ξ : ra, bs Ñ ra 1 , b 1 s with ξpaq " a 1 such that f pt, xq " f 1 pξptq, xq for all pt, xq P ra, bsˆX. On pf M , f N q the state sum Z W is then defined as which is a well-defined element of the complete semiring Q. Note that, when SpF q : prms, cq Ñ prm 1 s, c 1 q in cBr, the element T W pF q is supposed to be identified with the element T W pF q b 1 P QpH prms,cq,prm 1 s,c 1 q q Ă Q.
In close analogy with the further steps in [4], one can prove that our assignment Z is in fact a positive topological field theory. Namely, following Section 7.6 in [4], one checks the correct behavior of our state sum under disjoint union. Moreover, following Section 7.7 in [4], one proves the essential gluing formula Z W " xZ W 1 , Z W 2 y (see Theorem 7.26 in [4]), where W is the result of gluing a cobordism W 1 from M to N with a cobordism W 2 from N to P along N . Note that the preparatory results Proposition 7.23, Lemma 7.24, and Proposition 7.25 in [4] need only be modified by replacing the Br-valued action functional S with the cBr-valued action functional S in the formulation. Diffeomorphism invariance ϕ˚pZ W q " Z W 1 (see Theorem 9.16 in [4]) of our state sum under diffeomorphisms ϕ : BW Ñ BW 1 that can be extended to so-called time consistent diffeomorphisms ϕ : W Ñ W 1 can be shown along the lines of Section 9 in [4]. In particular, Lemma 9.12 and Lemma 9.14 in [4] remain valid when replacing S with S in the formulation. The map ϕ˚: ZpBW q Ñ ZpBW 1 q can then be defined on a function z : F pBW q Ñ Q and a field g P F pBW 1 q by ϕ˚pzqpgq " zpg˝pid r0,1sˆϕ qq P Q.

5.4.
The aggregate invariant and exotic Kervaire spheres. Positive TFTs have been created with the intention to provide new topological invariants for highdimensional manifolds (see [3]). In this section, we explain how our positive TFT Z can be used to assign to any homotopy sphere M its aggregate invariant ApM q, an element of the complete semiring Q from Section 5.3.2. The construction of A is analogous to that of the aggregate invariant A studied Section 10 in [4]. While the invariant A is known to distinguish exotic spheres from the standard sphere (see Corollary 10.4 in [4]), we will indicate briefly that the invariant A can distinguish exotic Kervaire spheres from other exotic spheres in infinitely many dimensions.
Fix a closed pn´1q-manifold M which is homeomorphic (but not necessarily diffeomorphic) to the sphere S n´1 . Without loss of generality, we assume in the following that S n´1 and M are smoothly embedded in t0uˆR D´1 (compare Section 5.2.1). From now on, we suppose that n´1 ě 5. Then, by classical Morse theory, M admits Morse functions with exactly two non-degenerate critical points, namely one minimum and one maximum. Given any diffeomorphism ξ : r0, 1s Ñ ra, bs with ξp0q " a, and any Morse function f M : M Ñ R with exactly two non-degenerate critical points, we observe that the map is a fold field on M (see Definition 5.5). Let C 2 pM q Ă F pM q denote the (nonempty) subset of all such maps f M . Fix an element f S P C 2 pS n´1 q of the form f S " id r0,1sˆfS . Let us write CobpS n´1 , M q for the collection of all oriented cobordisms from S n´1 to M that are embedded in r0, 1sˆt0uˆR D´1 (compare property (4) of Definition 5.1). Since M is homeomorphic to S n´1 , it can be shown that CobpS n´1 , M q is non-empty (see Lemma 10.1 in [4]). Now, for any cobordism W P CobpS n´1 , M q and any fold field f M P C 2 pM q, the state sum Z W P ZpS n´1 q p bZpM q of Section 5.3.3 can be evaluated at pf S , f M q P F pS n´1 qˆF pM q to yield an element Z W pf S , f M q in the complete semiring Q from Section 5.3.2 that is associated to a faithful linear representation Y : cBr Ñ Vect. Hence, summation in the complete semiring Q yields a well-defined element In conclusion, we outline an application to Kervaire spheres, which are a concrete family of homotopy spheres that can be obtained from a plumbing construction as follows (see [10, p. 162]). The unique Kervaire sphere Σ n´1 K of dimension n´1 " 4r`1 can be defined as the boundary of the parallelizable p4r`2q-manifold given by plumbing together two copies of the tangent disc bundle of S 2r`1 . According to the classification theorem of homotopy spheres (see [12, Theorem 6.1, pp. 123f]), as well as recent work of Hill-Hopkins-Ravenel [8] on the Kervaire invariant one problem, it is known that Σ n´1 K is an exotic sphere, i.e., homeomorphic but not diffeomorphic to S n´1 , except when n´1 P t5, 13, 29, 61, 125u.
Note that, according to Remark 6.3 in [21], there are infinitely many dimensions of the form n´1 " 13 pmod 16q in which there exist exotic spheres that are not diffeomorphic to the Kervaire sphere Σ n´1 K . The following result shows that our aggregate invariant A can distinguish exotic Kervaire spheres from other exotic spheres in infinitely many dimensions. We give a sketch of the proof by referring to the results of [19]. A detailed proof is beyond the scope of this paper, and will appear elsewhere.
Theorem 5.7. Suppose that n´1 " 13 pmod 16q and n´1 ě 237. Then, an exotic pn´1q-sphere Σ n´1 is diffeomorphic to the Kervaire sphere if and only if ApΣ n´1 q " ApΣ n´1 K q. Sketch of proof. Recall from Section 5.3.2 that elements of Q are families of power series in Brrqss which are indexed by the loop-free morphisms of cBr. It follows from the construction of the state sum Z W (see Section 5.3.3) that non-trivial power series of the element ApM q P Q can only occur in the factor QpH pr2s,0q,pr2s,0q q " Brrqss ' Brrqss ' Brrqss, where the three copies of Brrqss correspond to the three possible loop-free morphisms pr2s, 0q Ñ pr2s, 0q in cBr, namely 1 pr2s,0q , b p0,0q , and i p0q˝ep0q . Let ζpΣ n´1 q P Brrqss denote the component of ApΣ n´1 q that corresponds to the loop-free morphism i p0q˝ep0q Then, for every ν P N the coefficient of q ν in ζpΣ n´1 q is nonzero if and only if there exists a fold field F P F pf S , f Σ q such that SpF q "´Â 8 k"0 λ ν k pkq¯b pi p0q˝ep0q q. We choose ν such that ν j " 1 for j " n{2 and ν j " 0 for j ‰ n{2. Then, it follows from Corollary 10.1.4 and Theorem 3.4.9 in [19] that the coefficient of q ν in ζpΣ n´1 q is 1 whenever Σ n´1 is diffeomorphic to Σ n´1 K . Conversely, if Σ n´1 is not diffeomorphic to Σ n´1 K , then Corollary 10.1.4 in [19] implies that the coefficient of q ν in ζpΣ n´1 q is 0.