The space of traces in symmetric monoidal infinity categories

We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,C}: hom_C(x,x) \to hom_C(1,1)$ for all dualisable objects $x$ in any symmetric monoidal infinity-category $C$. This generalises the trace from linear algebra that assigns a scalar $Tr(f) \in k$ to any endomorphism $f:V \to V$ of a finite-dimensional $k$-vector space. Our main theorem computes the moduli space of tracelike transformations using the one-dimensional cobordism hypothesis with singularities. As a consequence we show that the trace $Tr$ can be uniquely extended to a tracelike transformation up to a contractible space of choices. This allows us to give several model-independent characterisations of the infinity-categorical trace. Restricting our notion of tracelike transformations from endomorphisms to automorphisms we in particular recover a theorem of To\"en and Vezzosi. Other examples of tracelike transformations are for instance given by $f \mapsto Tr(f^n)$. Unlikefor $Tr$ the relevant connected component of the moduli space is not contractible, but ratherequivalent to $B\mathbb{Z}/n\mathbb{Z}$ or $BS^1$ for $n=0$. As a result we obtain a $\mathbb{Z}/n\mathbb{Z}$-action on $Tr(f^n)$ as well as a circle action on $Tr(id_x)$.


Introduction
The trace, originally defined in the context of linear algebra, admits a well-known generalisation to arbitrary symmetric monoidal categories. Often the trace can be thought of as measuring 'fixed-points' of an endomorphism, most prominently in the case of the stable homotopy category, where it computes the Lefschetz fixed-point number, as we explain below. Recently this has been vastly generalised to indexed, relative, and equivariant settings using bicategories. We refer the reader to Ponto and Shulman's paper [PS14] for an introduction to traces from this perspective and further references.
For bordism categories the trace behaves like the 'closing up' operation that sends a braid to its associated link. In the context of topological quantum field theories this corresponds to taking the so-called 'state-sum', see for example Stolz and Teichner's work [ST12]. The example of bordism categories plays an important role later since, by the cobordism hypothesis, bordism categories are the universal setting for taking traces. This generalises the strategy of Toën and Vezzosi [TV15].
Another example, which highlights the importance of treating the trace ∞-categorically, is the derived Morita category of ring spectra and bimodules. Here the trace is given by topological Hochschild homology (THH). We will return to this example in the final section where we show that in this case Theorem A induces a circle action on THH(R) and a Z/nZ-action on THH(R; M ⊗ R . . . ⊗ R M ).

Traces in symmetric monoidal 1-categories
Before we consider ∞-categories, let us recall the trace in 1-categories. The trace of an endomorphism f : V → V on a finite-dimensional k-vector space V may be defined as where (b i ) i∈I is a basis of V and (β i ) i∈I is the dual basis of V * := hom Vect k (V, k) uniquely characterised by β i (b j ) = δ ij . Let us define the evaluation and coevaluation of V by and rewrite the trace as the composition This definition generalises to all symmetric monoidal categories (C, ⊗, 1): an object x ∈ C is called dualisable if there is a dual y ∈ C with an evaluation e : y ⊗ x → 1 and a coevaluation c : 1 → x ⊗ y satisfying: id x = (id x ⊗ e) • (c ⊗ id x ) and id y = (e ⊗ id y ) • (id y ⊗ c). (1) The data (y, e, c) is essentially unique, if it exists. For such x the trace of f : x → x is defined as: Tr x (f ) := e • swap x,y • (f ⊗ id y ) • c ∈ hom C (1, 1).
Definition 1.1. In analogy to the category of k-vector spaces we will refer to endomorphisms of the unit object as scalars. They form a commutative monoid and taking scalars defines a functor: Sc : Cat ⊗ → CMon, C → Sc(C) := hom C (1, 1).
To a symmetric monoidal functor F : C → D with unitor λ F : F (1 C ) ∼ = 1 D this assigns the map Sc(C) → Sc(D) sending f to λ F • F (f ) • λ −1 F . This generalisation of the trace retains the cyclicity of the trace that is well-known in the context of linear algebra [PS14,Proposition 2.4]: when f : x → z and g : z → x are morphisms between dualisable objects, then Tr x (g • f ) = Tr y (f • g).
The main advantage of having such a general definition is that we are now able to compare traces in different symmetric monoidal categories [PS14, Proposition 6.2]. Let F : C → D be a symmetric monoidal functor, x ∈ C dualisable, and f : x → x an endomorphism. Then F (x) is dualisable and the trace of F (f ) is Tr D,F (x) (F (f )) = F (Tr C,x (f )). (2) An interesting example is the stable homotopy category C = h 1 Sp with the smash-product as symmetric monoidal structure. Its scalars are homotopy classes of maps S → S and these are classified by integers. For a finite CW-complex X the suspension spectrum Σ ∞ + X is a dualisable object of h 1 Sp with dual the Spanier-Whitehead dual of X. A continuous map f : X → X induces an endomorphism Σ ∞ + f ; its trace Tr(Σ ∞ + f ) ∈ Hom h 1 Sp (S, S) ∼ = Z is the so-called Lefschetz-number Λ(f ) of f , which can be understood in terms of fixed-points of f . The functor that takes a spectrum to its cohomology with coefficients in a field is symmetric monoidal by the Künneth-theorem. The naturality (2) implies that we can understand Λ(f ) in terms of the action f has on the homology of X. This is the Lefschetz fixed-point theorem.

Axiomatic description: tracelike transformations
Our axiomatisation of the properties of the trace is loosely based on Kelly and Laplaza's notion of a 'trace function' from [KL80]. We discuss how they are related below. The main difference is that we will study compatible choices of traces for all symmetric monoidal categories at the same time: Definition 1.2. A tracelike transformation T is a family of maps T C,x : End C (x) → Sc(C) = End C (1) for all symmetric monoidal categories C and dualisable objects x ∈ C satisfying the following axioms: • Conjugation invariance: T y (ϕ • e • ϕ −1 ) = T x (e) for all endomorphisms e : x → x and isomorphisms ϕ : x ∼ = y.
A restricted tracelike transformation is a family of maps T C,x : Aut C (x) → Sc(C) satisfying the same axioms. We denote the set of tracelike transformations by T and the set of restricted tracelike transformations by T r .
Since Sc(C) is a commutative monoid for all symmetric monoidal categories C the sets T and T r inherit commutative monoid structures defined by multiplying tracelike transformations pointwise. Moreover, the multiplicative monoid (N, ·) acts on these monoids by maps P n : T → T defined as P n (T )(e) := T (e n ). Starting from the classical trace Tr ∈ T that we discussed above, we can therefore construct, for every finite sequence of natural numbers k 1 , . . . , k n ∈ N, a tracelike transformation: Θ k 1 ,...,kn x (e) = Tr x (e k 1 ) • · · · • Tr x (e kn ).
In fact, if we only want Θ to be a restricted tracelike transformation, we can allow the k i to be integers.
We now describe an equivalent definition of tracelike transformations: Let Cat ⊗  denote the category with objects symmetric monoidal categories and morphisms natural isomorphism classes of symmetric monoidal functors. We define a functor E fd : Cat ⊗  → Set by as the subset of conjugacy classes represented by automorphisms. By construction, a tracelike transformation is simply a natural transformation E fd ⇒ Sc of functors Cat ⊗  → Set. Similarly, a restricted tracelike transformation is a natural transformation A fd ⇒ Sc.
Note that our E fd (C) is similar to Kelly and Lapaza's set of cycles [D], which they define in [KL80] for any category D. If we let C fd ⊂ C denote the full subcategory on the dualisable objects and (C fd ) ⊂ C fd the maximal subgroupoid, then there is a canonical surjection E fd (C) → [C fd ], and a canonical bijection A fd (C) → [(C fd ) ]. The crucial difference is that they require a cyclicity condition T (f • g) = T (g • f ) for any two morphisms f : x → y and g : y → x, whereas this only follows from our conjugation invariance if one of f and g is an isomorphism.

1-categorical classification of tracelike transformations
We will show that the Θ k 1 ,...,kn are indeed the only examples of tracelike transformations: a variant of the cobordism hypothesis in dimension 1 implies that certain cobordism categories Cob N 1 ⊂ Cob Z 1 are freely generated by a dualisable object * + and α : * + → * + an endomorphism or automorphism, respectively. Hence there is a bijection between the set of tracelike transformations and the scalars of Cob N 1 . These can be described as Mfd N 1 /Diff + : diffeomorphism classes of closed 1-dimensional manifolds labelled by natural numbers. This in turn is in bijection with N[x], the free commutative monoid on the set {1, x, x 2 , . . . }. In summary this leads to: Theorem (see 4.6). There are compatible isomorphisms of commutative monoids

Traces in ∞-categories
If C is a symmetric monoidal ∞-category then its scalars Sc(C) = hom C (1, 1) are no longer just a set, but a space. In fact, they naturally carry the structure of an E ∞ -space. Consider for instance the ∞-category of spectra C = Sp: the endomorphisms of the unit object are: For a finite CW-complex X the 1-categorical trace on the homotopy category h 1 Sp can be promoted to a map of spaces: In definition 2.16 and 2.17 we will construct ∞-functors Sc, E fd , and A fd from the ∞-category of symmetric monoidal ∞-categories Cat ⊗ ∞ to the ∞-category of spaces Spc. For every symmetric monoidal ∞-category C and dualisable object x ∈ C there are compatible maps hom C (x, x) → E fd (C) and hAut C (x) → A fd (C). Definition 1.3. An ∞-categorical tracelike transformation is a natural transformation T : Similarly, a restricted tracelike transformation is a natural transformation T : A fd ⇒ Sc. We denote the space of (restricted) ∞-categorical tracelike transformations by T The ∞-functors Sc, E fd , and A fd recover their 1-categorical analogues on the homotopy category h 1 C of C in the sense that π 0 Sc(C) ∼ = Sc(h 1 C), π 0 E fd (C) ∼ = E fd (h 1 C), and π 0 A fd (C) ∼ = A fd (h 1 C), see 2.21. For any tracelike transformation T , symmetric monoidal ∞-category C and dualisable object x ∈ C fd naturality with respect to the functor C → h 1 C yields the following commutative diagram: Here we implicitly used the map T ∞ → T defined by sending an ∞-categorical tracelike transformation to its restriction to 1-categories. This map encodes how an ∞-categorical tracelike transformation behaves on the level of connected components.
Our main theorem describes the homotopy-type of T ∞ and the map to the discrete set T . Generalising the commutative monoid structure on T (r) there is an E ∞ -algebra structure T (r) ∞ . For simplicity, we will only identify the underlying space. To state the theorem, let Free E∞ (X) denote underlying space of the free E ∞ -algebra on X.
Theorem A (5.11). There is a commutative diagram of spaces: where the horizontal maps are equivalences. Moreover, the vertical maps identify the sets in the bottom layer as the set of connected components of the top layer: T ∼ = π 0 T ∞ and T r ∼ = π 0 T r ∞ . Warning 1.4. The reader should be warned that in identifying T ∞ we use the cobordism hypothesis with singularities, which was sketched in [Lur09c], but a full proof of which has not yet appeared in the literature. We will therefore be treating it as a conjecture and all our statements about T ∞ are dependent on this conjecture. Note, however, that the analogous statements about the space of restricted tracelike transformations T r ∞ only use the standard cobordism hypothesis in dimension 1, which has been proved in detail. (See [Lur09c] and [Har12].) Of particular interest is the homotopy-type of the moduli space of those ∞-categorical tracelike transformations that recover the classical trace on homotopy categories. In their work on the derived Chern-character [TV15] Toën and Vezzosi show that for restricted tracelike transformations the space is contractible and that therefore there is an essentially unique ∞categorical generalisation of the trace when applied to automorphisms.
In theorem A we use methods similar to theirs to compute the full homotopy-type of T (r) ∞ . Their result can now be read off by considering the fibre of T r ∞ → T r over the classical trace. Since we study T ∞ as well we can now generalise their result, removing the artificial restriction to automorphisms.
Using our knowledge of the structure of π 0 T ∞ it is in fact not hard to see that to uniquely characterise the ∞-categorical trace we only need to specify its behaviour on the category of vector spaces.
Corollary C (5.13). Any ∞-categorical tracelike transformation whose value on the category of complex vector spaces agrees with the trace from linear algebra is canonically equivalent to the ∞-categorical trace.
More informally, there is a unique extension of the trace from linear algebra to a family of maps Tr (x,C) : End C (x) → hom C (1, 1) for any symmetric monoidal ∞-category C and any dualisable object x ∈ C while preserving the conjugation invariance of the trace and its naturality with respect to symmetric monoidal functors. We can also give an alternative characterisation of Tr as the unique generating tracelike transformation. This characterisation is purely categorical and does not require one to first define a trace for vector spaces.
Definition 1.5. The monoid (N, ·) acts on T and T ∞ by taking powers of any morphism before applying the tracelike transformation P n (T )(a) := T (a n ). We call a tracelike transformation T ∈ T ∞ generating if the monoid π 0 T ∞ is generated by the orbit of T under the N-action.
In other words, T is generating if every other tracelike transformation S ∈ T ∞ is equivalent to one of the form P k 1 (T ) • · · · • P kn (T ).
Corollary D (5.19). The space of generating tracelike transformations in T ∞ is contractible and its image in T is the usual trace.

Notation
We will assume that the reader has a convenient model of (∞, 1)-categories at hand. In this paper we will be working in the context of Joyal's quasicategories, but really any equivalent Cartesian closed ∞-cosmos in the sense of Riehl and Verity will do. We will refer to these (∞, 1)-categories as ∞-categories and to morphisms between them as ∞-functors, or sometimes just as functors.
We write Cat ∞ for the ∞-category of ∞-categories and Spc ⊂ Cat ∞ for the full subcategory of ∞-groupoids, which we will refer to synonymously as 'spaces'. The 1-category of topological spaces will be denoted by Top. There is a functor Top → Spc that 'forgets the point-set information', it sends a topological space to its ∞-groupoid of paths.
For ∞-categories C, D, E we denote the ∞-category of functors from C to D by Fun ∞ (C, D) and the maximal subgroupoid of E by E ∼ ∈ Spc. For objects a, b ∈ E the space of morphisms from The ∞-category of (simplicial) presheaves on C is P(C) := Fun ∞ (C op , Spc). The Yoneda embedding will be denoted by Y : C → P(C).

Structure of the paper
We begin by recalling complete Segal spaces and other ∞-categorical tools in section 2, where we also define the functors E fd and A fd . Then, in section 3, we define concrete models of the one-dimensional bordism category with and without marked points and show that they define symmetric monoidal ∞-categories. Using these we formulate the cobordism hypothesis, as well as a variant with singularities. In section 4 we complete the proof of the classification result for 1categorical tracelike transformations. Section 5 contains the homotopy-theoretic computations and the proofs of the main theorems. In the final section we discuss how non-contractible connected components of the moduli space T ∞ induce group actions on the trace. complete Segal spaces. In this section we recall how to relate this approach to the more standard theory of quasicategories, by giving a model independent description of complete Segal spaces in the language of ∞-categories. Once this is established we make precise the functors L fd , A fd , and Sc from the introduction.

Complete Segal spaces
2.1. We let Cat  denote the 1-category of 1-categories. It admits a functor to the ∞-category of ∞-categories Cat  → Cat ∞ ; in the quasicategory model this sends a category to its nerve. Write Cat 1 ⊂ Cat ∞ for the essential image of the inclusion. We will think of this as the ∞category of 1-categories. This in fact is also a (2, 1)-category: the hom-spaces of Cat 1 are the 1-groupoids of 1-functors and natural isomorphisms: The inclusion Cat 1 → Cat ∞ has a left-adjoint, the homotopy-category functor h 1 : Cat ∞ → Cat 1 and in this sense Cat 1 is a localisation of Cat ∞ . 2.2. We define the simplex category as the full subcategory ∆ ⊂ Cat 1 generated by the partially ordered sets [n] = {0 ≤ · · · ≤ n} thought of as categories, for n ≥ 0. Using the embedding Theorem 2.7 states that this functor is fully faithful. This is a way of saying that the objects [n] generate Cat ∞ strongly under colimits. The essential image of N are the complete Segal spaces: Definition 2.3. A simplicial space X ∈ P(∆) satisfies the Segal condition if for all n ≥ 2 the map (λ * 0 , . . . , λ * n−1 ) : X n −→ X 1 × X 0 · · · × X 0 X 1 induced by λ i : [1] → [n] with λ i (k) = i + k is an equivalence.
Definition 2.4. Let I ∈ Cat  be the contractible groupoid with two objects and * ∈ Cat  the discrete category with one object. Write N (I) and N ( * ) for the simplicial sets that are the nerves of these categories and interpret them as simplicial spaces that are discrete in every layer. 1 A simplicial space X ∈ P(∆) is called complete if the natural map coming from the 1-functor I → * is an equivalence.
Definition 2.5. The ∞-category of complete Segal spaces is defined as the full subcategory CSS ⊂ P(∆) spanned by the objects that are complete and satisfy the Segal condition.
Remark 2.6. Given a Segal space X there also is a simpler characterisation of the completeness condition due to Rezk. By [Rez01, Theorem 6.2] the space hom P(∆) (N (I), X) is equivalent to the subspace X eq 1 ⊂ X 1 on those 1-simplices that represent an isomorphism in the homotopy category h 1 X, which we describe in 2.9. Moreover, X eq 1 ⊂ X 1 is always a union of connected components. Therefore a Segal space is complete if and only if s 0 : X 0 → X eq 1 is an equivalence. Proof by citation. It is not difficult to see that for an ∞-category C the nerve N C indeed satisfies the Segal and the completeness condition. To prove that N induces an equivalence is more difficult.
The first result of this type was [Ber07], but there the model used for Cat ∞ was simplicial categories. Joyal and Tierney show in [JT07] that the model categories of quasicategories and complete Segal spaces are Quillen-equivalent. The ∞-categorical statement is an immediate consequence of their result, see [Lur09b, Corollary 4.3.16].
2.8. It is important to understand how standard constructions in Cat ∞ change under the equivalence to CSS. Let X = N (C) be the complete Segal space of some ∞-category C. By definition, the levels of X are of the form In particular X 0 is the maximal subgroupoid C ∼ of C and X 1 is the maximal subgroupoid of the arrow category C [1] .
We will say that an object of C is a functor * → C, where * is the terminal category. For two such objects a, b : * → C one can reconstruct the hom-space hom C (a, b) from the complete Segal space X as: We can recover the composition, up to inverting the weak equivalence X 2 → X 1 × X 0 X 1 : 2.9. The above suffices to reconstruct the homotopy category h 1 C of C up to categorical equivalence: we define the set of objects as O := π 0 X 0 and pick a section o : π 0 X 0 → X 0 to interpret them in the above sense. For two a, b ∈ O we define the morphism set as The composition in C is constructed by taking π 0 of what we did earlier. Note that this erases the ambiguity coming from inverting the equivalence.
Remark 2.10. The construction given above has the disadvantage that it is not functorial in X: we need to make the unnatural choice of a section o : π 0 X 0 → X 0 . In fact, we should not expect there to be a 1-categorical description since the ∞-functor h 1 : Cat ∞ → Cat 1 does not factor through the 1-category of 1-categories Cat  .

Dualisability
We recall the necessary definitions to talk about dualisable objects: Seg74]). Segal's category Γ is defined as a skeleton of the opposite category of the category of finite pointed sets. The objects are k := { * , 1, . . . , k} for k ≥ 0 and morphisms in Γ op are basepoint preserving maps. A functor X : Γ op → C is called a special Γ-object in C if X( 0 ) is a terminal object of C and for all finite pointed sets A, B the canonical maps Let CSS ⊗ denote the ∞-category of special Γ-objects in CSS. This can be thought of as a full subcategory of P(∆ × Γ). We write X k n for the value of X : Theorem 2.12. The ∞-category of special Γ-objects in C is a model for the commutative monoid objects in C. In particular, the nerve functor N : Cat ∞ CSS lifts to an equivalence Cat ⊗ ∞ CSS ⊗ between the ∞-category of symmetric monoidal ∞-categories and the ∞-category of special Γ-objects in CSS.
Proof by citation. Commutative monoids in C are by definition E ∞ -algebras in C with respect to the symmetric monoidal structure coming from the Cartesian product. That these are the same as functors Γ op → C satisfying the 'specialness condition' is for instance shown in [Lur18, Proposition 2.4.2.5].
Corollary 2.13. The localisation-adjunction h 1 I between Cat ∞ and Cat 1 lifts to a localisation- There is an adjunction on the functor categories and since both functors preserve the 'specialness' of Γ-objects this adjunction restricts to the full subcategories Cat ⊗ ∞ and Cat ⊗ 1 . The functor I * is fully faithful because I was and hence h ⊗ 1 is a localisation.
2.14. Note that here Cat ⊗ 1 is by definition the ∞-category of special Γ-objects in Cat 1 . It is a folklore theorem that this is equivalent to the (2, 1)-category of symmetric monoidal categories. For the readers convenience we will use this theorem and from now on think of h 1 X as a symmetric monoidal category. However, it is worth remarking that we could equally well work with (special) Γ-categories.
Definition 2.15. An object x in a symmetric monoidal ∞-category C is called dualisable if x is dualisable as an object of the symmetric monoidal 1-category h 1 C. The 1-categorical definition was given in the introduction. If all objects of C have duals we say that C has duals. Define C fd to be the maximal (full) subcategory of C that has duals.

Some functors of interest
Definition 2.16. For a complete Segal space X ∈ CSS we define its space of objects as obj(X) : Here X eq 1 ⊂ X 1 is the subspace of those connected components that represent invertible morphisms in the homotopy category.
Definition 2.17. For a special Γ-object in complete Segal spaces X ∈ CSS ⊗ we let obj fd (X) ⊂ obj(X) = X 1 0 denote the union of those connected components that correspond to dualisable objects in the homotopy category. Accordingly, we let E fd (X) ⊂ E(X) and A fd (X) ⊂ A(X) be the subspaces supported at the dualisable objects. Finally, we set Note that this agrees with the functor End (1) defined above [TV15, Proposition 2.7].
Remark 2.18. By construction there is a natural transformation A → E such that for each X the map A(X) → E(X) is an equivalence onto the connected components it hits.
Lemma 2.19. For any complete Segal space represented by a simplicial topological space X • we can compute E(X) as the space of tuples (γ, f ) where f ∈ X 1 and γ is a path from Proof. In order to compute the homotopy pullback A(X) = X 0 × h X 0 ×X 0 X eq 1 we replace the diagonal X 0 → X 0 × X 0 by the path fibration P (X 0 ) → X 0 × X 0 and obtain: Note that the right-hand space is indeed the space of tuples (γ, f ) as described in the lemma. For the second claim, recall from remark 2.6 that for a complete Segal space the degeneracy map s 0 : X 0 → X 1 induces an equivalence X 0 X eq 1 . We therefore have an further equivalence The latter space is the space of paths in X 0 whose start and end-point agree, i.e. the space of free loops Λ(X 0 ) = Map(S 1 , X 0 ).
Lemma 2.20. Write X ∈ CSS ⊗ as X N C for some symmetric monoidal ∞-category C. Then the above functors can be described as follows: • Sc(X) is the endomorphism space hom C (1, 1) of the unit object 1 ∈ C.
Proof. For obj(X) this is a direct consequence of the definition on N C, as discussed in 2.8. Since the notion of dualisability is defined via the homotopy-category the description of obj fd follows immediately. The definition of Sc(X) is somewhat cryptic, but in fact not much is happening: the space X 0 0 is contractible since the Γ-object is special. We may hence replace it by the terminal space * . The functor * → X 0 0 → X 1 0 = C ∼ picks out the unit object 1 of C and so the definition is equivalent to Finally, the claim that A fd (X) Map(S 1 , obj fd (X)) follows by the same argument as in lemma 2.19, now restricted to the full subcategory on dualisable objects.
For the second part let us assume that every object in C is dualisable so that E fd (C) = E(C).
Since duals are computed on the level of homotopy categories this will not cause a problem. The homotopy fibre of the map E(C) → N 0 C at an object x ∈ C is equivalent to hom C (x, x), so the connected components of the fibre are π 0 hom C (x, x) = End h 1 C (x). The fundamental group of N 0 C at x is Aut h 1 C (x). Therefore, choosing representatives x i for the isomorphism classes in C, we can write π 0 E(C) as the disjoint union of quotients: Here the action is by conjugation and the resulting set is canonically isomorphic to E(h 1 C). This isomorphism restricts to a bijection between A(C) and A(h 1 C).

The bordism category and the cobordism hypothesis
In order to apply the cobordism hypothesis, which is a key step in the proof of the main theorem, we need to first establish a concrete model for the one dimensional bordism category. In this section we take the model for Bord or 1 (X) from [SP17] and show that it is a special Γ-object in complete Segal spaces. Moreover, we define a "marked" version of the bordism category and show that it too is a special Γ-object in complete Segal spaces. Using this model we can then formulate a special case of the conjectural cobordism hypothesis with singularities.

The bordism category as a complete Segal space
We are going to define a symmetric monoidal ∞-category Bord or d (X) by constructing a 1-functor F : ∆ op × Γ op → Top based on [SP17] and then showing in 3.4 that after composing with Top → Spc it satisfies the required conditions to be in CSS ⊗ as in theorem 2.12.
In what follows we let R ∞ denote the countably dimensional vector space R ∞ = colim n→∞ R n and we let R 1+∞ denote the product R × R ∞ .
be the topological space of monotone maps [n] → R. We say that a submanifold W ⊂ R 1+∞ is admissible with respect to t ∈ R [n] if W is closed as a subset, the projection π : W → R to the first coordinate is proper, and there is an ε > 0 such that W is cylindrical over each of the intervals (t i − ε, t i + ε) for i = 0, . . . , n.
To topologise the space of bordisms we recall the plot-topologies from [SP17].
For any k-dimensional manifold U we say that a map f : is a smooth submanifold of U ×R ∞ and the map ϕ f : Γ(f ) → X is continuous. The plot topology on Ψ d (X) is the finest topology such that all smooth maps U → Ψ d (X) are continuous.
We are now ready to define the simplicial space that will give rise to the bordism category. Our model is almost identical to the one defined in [SP17, Definition 5.8] with the only difference being that we chose to encode the symmetric monoidal structure by using Γ-spaces whereas Schommer-Pries uses E ∞ -algebras. Definition 3.3. For any topological space X, we define Bord or d (X) by the functor ∆ op × Γ op → Top that sends ([n], k ) to the topological space of tuples (t, (W 1 , ϕ 1 ), . . . , (W k , ϕ k )) where t ∈ R [n] and the W i are pairwise disjoint d-dimensional oriented submanifolds of R × (−1, 1) ∞ admissible with respect to t. The ϕ i are continuous maps ϕ i : W i → X. This is topologised as a subspace of R [n] × (Ψ d (X)) k . Functoriality in the Γ-direction is defined as follows. For λ : k → l we set λ * (t, (W 1 , ϕ 1 In the ∆-direction a morphism ρ : [n] → [m] acts by reindexing the t i : (ρ * t) j = t ρ(j) .
We now proceed to show that the above satisfies the Segal and completeness condition, so that it lies in CSS ⊗ ⊂ Fun ∞ (∆ op × Γ op , Spc) and gives rise to a symmetric monoidal infinity category via theorem 2.12. Proof. We begin with the Segal condition for Bord or , (W, ϕ)) | . . . } be the simplicial topological space from definition 3.3 that represents Bord or d (X) 1 . For any n we let B n ⊂ B n be the subspace of those (t, (W, ϕ)) satisfying that t i = i, that W is cylindrical over (−∞, 0] and [n, ∞), and that the two restrictions ϕ : W (−∞,0] → X and ϕ : W [n,∞) → X factor through the projection to W {0} and W {n} respectively. These conditions ensure that (t, (W, ϕ)) is uniquely determined by the restriction (W [0,n] , ϕ |W [0,n] ). This encodes exactly the data of n composable bordisms of length 1. Put into formulas we have a homeomorphism: . To be precise we should actually translate W [i,i+1] by i to the left and then extend it cylindrically over (−∞, 0] and [1, ∞), then it is a well-defined point in B 1 ⊂ B 1 .
The above homeomorphism already indicates that B n satisfies some type of Segal condition. However, one should note that B n ⊂ B n is not closed under face or degeneracy operators, so the B n do not actually define a simplicial space. Still, this will be very useful as we have the following homotopy commutative diagram: Our goal is to show that the map S is an equivalence -this is the Segal condition. The map i n is the canonical inclusion map i n : B n → B n . It has a homotopy inverse B n → B n defined by piece-wise linearly rescaling the first coordinate direction of R×(−1, 1) ∞ to that t i = i and then pushing off everything outside of [0, n] to infinity. This is a standard type of argument and we refer the reader to [GRW10, Proof of 3.9] for details. This also implies that j is an equivalence as it is a homotopy pullback of i 1 and i 0 . It remains to check that the map q that compares the strict pullback with the homotopy pullback is an equivalence. For this it will suffice to show that the maps B 1 ⇒ B 0 involved in the pullback are Serre fibrations. But now observe that B 1 is in fact homeomorphic to the space [W ] M X (W ) discussed in the start of [ERW19, section 3.1]. This is true because the bordism category from [ERW19] agrees with the one from [GRW10] after removing units and [SP17,Theorem A.2] shows that their topology in turn is homeomorphic to the plot topology we used. We can therefore cite [ERW19, Proposition 3.2.4(ii)], which tells us that the source and target maps s, t : B 1 → B 0 are Serre fibrations. This shows that q is an equivalence and hence the map S also has to be, and B • is a Segal space for any d and X as claimed.
The next claim is that B • is complete for d ≤ 2. As in remark 2.6 we let B eq 1 ⊂ B 1 denote the subspace of those 1-simplices that represent invertible morphisms in the homotopy category h(B • ). This is always a union of connected components. By Finally, it remains to show that Bord or d (X) satisfies the specialness condition as a Γ-object in simplicial spaces. This can be checked in each level separately, and relies on the idea that (−1, 1) ∞ is "large enough" for us to make any two submanifolds disjoint, canonically up to a contractible space of choices. We refer the reader to [CS19, Proposition 7.2] for details.
Remark 3.5. The first model for a bordism category satisfying the Segal condition was constructed in [Sch14]. There the completeness condition is enforced by replacing Bord or d with its completion. This is not necessary for d ≤ 2 as the more naive construction is already complete -see [CS19, Remark 5.25]. Since we have not been able to find a proof for this claim in the literature, we give the one above. We now show that Bord or d (X) indeed recovers the desired 1-category Cob d (X) as its homotopy category. The lemma is based on [CS19,proposition 8.20], which shows the same in a different model for Bord or d (X). Definition 3.7. The 1-category Cob d (X) has as objects tuples (M, ϕ) where M is a closed oriented (d − 1)-manifold and ϕ : M → X is a continuous map. A morphism (M, ϕ) → (N, ψ) is represented by a triple (W, i, χ) where W is a compact oriented d-manifold, i : M − N ∼ = ∂W is an orientation-preserving diffeomorphism, and χ : W → X is a continuous map such that χ • i = ϕ ψ. Two such triples (W, i, χ) and (W , i , χ ) represent the same morphism if there is an orientation preserving diffeomorphism f : W ∼ = W such that f |∂W • i = i and that χ • f is homotopic to χ relative to ∂W . Composition is defined by gluing cobordisms. This category has a symmetric monoidal structure induced by disjoint union.
Lemma 3.8. For all X the homotopy category h 1 Bord or d (X) is canonically equivalent to the symmetric monoidal category Cob d (X) described in definition 3.7.
Proof. As discussed in 2.9 we are free to choose representatives for π 0 Bord or d (X) 0 . In fact choosing multiple representatives per connected component yields an equivalent category, so we can simply set the objects of h 1 Bord or d (X) to be tuples (R × M, ϕ • pr M ) where M ⊂ (−1, 1) ∞ is a closed oriented (d − 1)-manifold and ϕ : M → X is a continuous map. This maps to the objects of Cob d (X) by forgetting the embedding.
The set of morphisms (M, ϕ) → (N, ψ) can be computed as π 0 of the homotopy pullback Up to equivalence we may replace the middle space with the space B 1 of bordism of length 1 from the proof of theorem 3.4. So we are interested in the homotopy fibre of the map B 1 → (B 0 ) 2 sending W to (W 0 , W 1 ). By [ERW19, Proposition 3.2.4(ii)] this map is a fibration, so we can consider the strict fibre instead. From the description in [ERW19, section 3.1] it follows that this is given by: Here W only runs over equivalence classes of abstract bordisms that go from M to N and the maps W → X are required to restrict to ϕ and ψ, respectively. There is a canonical functor h 1 Bord or d (X) → Cob d (X) defined by forgetting the embedding on objects and sending bordisms to their equivalence class. It follows from the above that this functor is fully faithful. Moreover, it is essentially surjective because every closed oriented (d − 1)-manifold can be embedded into (−1, 1) ∞ .
Moreover, if we define a Γ-structure on Cob d (X) the same way we did on Bord or d (X), then this is an equivalence of Γ-categories, and the Γ-structure on Cob d (X) corresponds to the standard symmetric monoidal structure defined by disjoint union. We refer the reader to the proof of [CS19, proposition 8.20] for more details on how this is compatible with the Γ-structure. Proof. By definition 2.15 we need to show that every object in the homotopy category h 1 Bord or d (X) is dualisable. Because of lemma 3.8 we can equivalently construct duals for the category Cob d (X). This is well-known to those familiar with the cobordism hypothesis, but we include a brief description in the interest of completeness.

The cobordism hypothesis
Let (M, ϕ) be some object of Cob d (M ). Then we will show that (M − , ϕ) is the dual. Remark 3.10. One can also equip bordism categories with more general tangential structures. In the context of the cobordism hypothesis one usually uses the framed bordism category Bord fr d where each W comes with a framing. There always is a functor Bord fr d (X) → Bord or d (X) that forgets the framing and only remembers the induced orientation. For d = 1 the space of framings on a fixed W that are compatible with a preferred orientation is contractible. Therefore Bord fr d (X) → Bord or d (X) is a level-wise equivalence of Segal spaces and the associated symmetric monoidal ∞-categories are equivalent. As we will only need the one-dimensional cobordism hypothesis we can therefore state it using the oriented bordism category Bord or 1 (X) rather than Bord fr 1 (X). 3.11. To formulate the cobordism hypothesis in dimension 1 we need to construct a natural map X → obj fd (Bord or 1 (X)) for any space X. As we saw above all objects in Bord or 1 (X) are dualisable, and so the space of dualisable objects is simply Bord or 1 (X) 0 . We can define the map f : X → Bord or 1 (X) 0 by sending x ∈ X to the (positively oriented) manifold W := R × {0} ⊂ R × (−1, 1) ∞ equipped with the constant map ϕ x : W → {x} → X. This is natural in X and induces a map: Theorem 3.12 (Cobordism hypothesis in dimension 1, [Lur09c], for more details see [Har12]). The map constructed above is an equivalence for all X ∈ Top and C ∈ Cat ⊗ ∞ : hom Cat ⊗ ∞ (Bord or 1 (X), C) hom Spc (X, obj fd (C)).
Since α is represents an invertible morphism in the homotopy category this defines a point ( * + , α) ∈ A(Bord or 1 ) in the space of automorphisms.
Corollary 3.14. For every Y ∈ CSS ⊗ evaluating on ( * + , α) ∈ A fd (Bord or 1 (S 1 )) yields an equivalence Proof. Consider the full subcategory Y fd ⊂ Y on the dualisable objects. By the cobordism hypothesis 3.12 and lemma 2.20 we have equivalences This equivalence corresponds to the evaluation at the point of A(Bord or 1 (S 1 )), which is represented by the free loop γ : S 1 → obj(Bord or 1 (S 1 )) sending x to the manifold * + labeled by l : * → {x} ⊂ S 1 . Following through the equivalence in the proof of lemma 2.19 we see that this is in the same connected component as the preferred endomorphism (α : * + → * + ) ∈ A(Bord or 1 (S 1 )). Therefore the natural transformation described in the claim is an equivalence as well.

The cobordism category with marked points
We now introduce a variant of the bordism category where morphisms W : M → N come equipped with a finite subset A ⊂ W \ ∂W of marked points. Then we show that it is a complete Segal space and use it to formulate a special case of Lurie's cobordism hypothesis with singularities in our setting.
is a d-dimensional oriented submanifold and A ⊂ W is a 0-dimensional submanifold. We topologise this with a plot topology as in definition 3.2 except that now both the graph obtained from W and the graph obtained from A have to be smooth.
Note that Ψ m can also be thought as the subspace of Ψ d × Ψ 0 containing precisely those tuples (W, A) where A ⊂ W . where t ∈ R [n] and the W i are pairwise disjoint d-dimensional oriented submanifolds of R × (−1, 1) ∞ , each of which is admissible with respect to t. The A i are finite subsets of W i such that π(A i ) ⊂ n j=1 (t j−1 , t j ). This is topologised as a subspace of R [n] × (Ψ m ) k . Functoriality in ([n], k ) is as in definition 3.3, except that we also need to forget all points of A i that lie outside of (t 0 , t n ) after applying a face operator.
We have an analogue of 3.4: Our proof of the Segal property relied on the fact that the maps s, t : B 1 → B 0 that send a bordism of length one to its source or target are fibrations. In the next paragraph we will argue that the map p : B m 1 → B 1 that forgets the markings is a locally trivial fiber bundle whose fiber at a point W ∈ B 1 is the unordered configuration space Conf * (W (0,1) ). In particular p is a Serre fibration. Since B 0 = B m 0 it follows that the source and target maps of B m • are composites of Serre fibrations B m 1 → B 1 → B 0 = B m 0 . Therefore the same proof as in theorem 3.4 applies and B m • is a Segal space. To see that p : B m 1 → B 1 is a fiber bundle let W ∈ B 1 any point in the base. We choose a tubular neighbourhood N ⊂ R × (−1, 1) ∞ of W and an identification of N with the normal bundle ν W . For any smooth section of the normal bundle f ∈ Γ(ν W ) the image f (W ) ⊂ N ⊂ R×(−1, 1) ∞ is a smooth submanifold and for certain admissible f it is an element f (W ) ∈ B 1 : let U W ⊂ Γ(ν W ) denote the subset of such f . This is in bijection with the set U W ⊂ B 1 of those V ∈ B 1 such that there is an f ∈ Γ(ν W ) for which f (W ) = V . It follows from [CS19, Lemma A.5] that U W and U W are homeomorphic and that U W is a neighbourhood of W ∈ B 1 . (For this it is important to note that any V ∈ B 1 is uniquely determined by its intersection with the compact subspace [0, 1] 1+∞ ⊂ R 1+∞ .) All that is left to do is to trivialise p : B m 1 → B 1 over U W . Indeed we have the following homeomorphism: Next, we check the completeness condition for B m • . For this it is useful to think of B • as the subspace of B m • given by the manifolds with empty configurations. From the definition of the topology we can see that we cannot change the number of points in a configuration by a continuous path and so B n ⊂ B m n is a union of connected components for each layer n. Moreover, note that for a morphism to be homotopy invertible in B m • both it and its inverse have to be labeled by an empty configuration: this is true because composition adds the number of points in the configuration and the identity morphisms have empty configurations. Therefore, the spaces of equivalences agree (B m 1 ) eq = B eq 1 . Since we also have B 0 = B m 0 this means that B m • is complete if and only if B • is. We have argued why this is true for d ≤ 2 in 3.4. Finally, the specialness condition for the Γ-direction is again standard.
We pick the positively oriented point * + as a preferred object in Bord or,m 1 and construct an endomorphism β : * + → * + . Construction 3.19. To define the functor L we first need to enhance the ∞-category Bord or,m 1 to contain the (contractible) data of disjoint small ε-balls around the marked points. For any oriented 1-manifold W ⊂ R 1+∞ we can define a signed distance function d W : W ×W → R {∞} by setting d(x, y) = ±l where l is the length of the shortest path from x to y and the sign depends on whether that path agrees with the orientation of W . Let B be the simplicial Γ-space defined just like Bord or,m 1 except that each (W i , A i ) comes with a function ε i : A i → (0, ∞) satisfying, for all a ∈ A i : Because the space of possible choices for ε i is convex, the forgetful map B → Bord or,m 1 is a level-wise equivalence. As a result B is a complete Segal space.
We will now construct a functor L : B → Bord or 1 (S 1 ). Let (t, (M 1 , A 1 , ε 1 ), . . . , (W k , A k , ε k )) be a point in B k n . Then we define l i : W i → S 1 = R/Z as follows: otherwise.
One checks that this is continuous and loops around S 1 once in every ε i (a)-ball.
We can now state our interpretation of Lurie's cobordism hypothesis with singularities in the one-dimensional case. As explained in warning 1.4 some of our theorems are conditional on this conjecture. Remark 3.21 (Comparison to Lurie's conjecture). We will now informally derive our formulation from Lurie's more general cobordism hypothesis with one type of singularity as stated in [Lur09c, Proposition 4.3.1].
In the general setting one fixes a d − 1-dimensional manifold Y and then allows bordisms to have singularities of the form of a cone on Y . We will only need this in dimension 1 and for the case Y = S 0 . Of course a cone on S 0 is diffeomorphic to D 1 and so instead of introducing actual singularities in a bordism W it suffices to keep track of the cone points. This is the set of marked points A ⊂ W . A small ball around each marked point a ∈ A is then is a cone on S 0 . The cobordism hypothesis with singularities says that there is an equivalence between functors Bord or,m 1 → C and functors Z : Bord or 1 → C together with a choice of morphism α : 1 → Z(S 0 ). By the cobordism hypothesis without singularities Fun ⊗ ∞ (Bord or 1 , C) is equivalent to (C fd ) via the evaluation on the positively oriented point. Let x ∈ C denote Z( * + ). Then the value In summary, the data of a symmetric monoidal functor Bord or,m 1 → C is equivalent to that of a dualisable object x ∈ C fd together with an endomorphism α # : x → x. In other words, to a point (x, α # ) ∈ E(C fd ).

3.22.
By construction the functor L : Bord or,m 1 → Bord or 1 (S 1 ) sends the preferred endomorphism β : * + → * + in Bord or,m 1 to the preferred automorphism α : * + → * + in Bord or 1 (S 1 ). Therefore the following diagram commutes: Moreover by the two variants of the cobordism hypothesis the horizontal morphisms in this diagram are equivalences for all C. This means that via the Yoneda lemma the functor L corresponds to the natural transformation A fd ⇒ E fd .

1-categorical classification
In this section we complete the proof of the classification of 1-categorical tracelike transformations sketched in the introduction.
Definition 4.1. The category Cob Z 1 has as objects finite sets M equipped with an orientation M → {+, −}. A morphism X : M → N is a diffeomorphism class of 1-dimensional oriented bordisms X equipped with ∂X ∼ = M − N and a relative integral cohomology class ξ ∈ H 1 (X, M N ). The composition of two morphisms (X, ξ) : M → N and (Y, ζ) : N → L is the morphism (X N Y, χ) where χ is defined as the image of (ξ, ζ) under The symmetric monoidal structure is defined by taking disjoint unions, the unit is the empty set.

4.2.
Since the bordisms X : M → N are oriented 1-manifolds we can canonically identify H 1 (X, M N ) with Z π 0 X . Therefore choosing ξ is equivalent to labelling every connected component of X by an integer. The composition adds integers of connected components that are joint in the process of glueing bordisms. Graphically this can be described as: We write * + for the object defined by one positively oriented point and α : * + → * + for its automorphism defined by the trivial bordism labelled by the integer 1.
Definition 4.3. Let Cob N 1 ⊂ Cob Z 1 denote the subcategory that contains all objects, but only those morphisms that are labelled by non-negative integers under the identification in 4.2.

4.4.
We will now show that Cob Z 1 is equivalent to the homotopy category of Bord or 1 (S 1 ) and that Cob N 1 is equivalent to the homotopy category of Bord or,m 1 . Recall that in the process of defining h 1 X for a complete Segal space X : ∆ op → Spc we had to choose a section o : π 0 X 0 → X 0 . For Bord or 1 (S 1 ) we may choose o to take values (t, (M, ϕ)) such that ϕ : M → S 1 only hits the base-point of S 1 .
Lemma 4.5. There is a commutative diagram of symmetric monoidal functors where the vertical functors are equivalences.
Proof. The rightmost vertical functor is the symmetric monoidal equivalence from 3.8. To prove the the lemma we have to provide compatible lifts G and G .
For G this means that we have to give, for each [X, ϕ] : (M, ϕ |M ) → (N, ϕ |N ) a class α ∈ H 1 (X, M N ). Because of the choice we made in 4.4 we may assume that both ϕ |M and ϕ |N are the constant maps to the basepoint * ∈ S 1 . Therefore, the pullback of the canonical generator [S 1 ] ∈ H 1 (S 1 , * ) gives a well-defined class α := ϕ * (S 1 ) ∈ H 1 (X, M N ). This construction is compatible with gluing and disjoint union and therefore yields a symmetric monoidal functor G. Moreover, the assignment [ϕ] → ϕ * [S 1 ] defines a bijection between π 0 Map((X, M N ), (S 1 , * )) and H 1 (X, M N ) for any bordism X and hence G is an equivalence of categories.
To obtain the functor G , first consider the composite G • h 1 L. Concretely, let (W, A) : M → N be some marked bordism. Then L(W, A) = (W, ϕ) is equipped with a labelling such that ϕ : W → S 1 winds around the circle once for each marking. This is done compatibly with the orientation and hence every connected component W 0 ⊂ W will be labeled in (G • h 1 L)(W, A) by the number of elements of W 0 ∩ A. Since this is non-negative it lies in the subcategory Cob N 1 and the functor G • h 1 L can be factored through a unique symmetric monoidal functor G : h 1 Bord or,m 1 → Cob N 1 . This functor is an equivalence of categories because up to diffeomorphism the marking of a bordism is uniquely determined by the number of marked points in each connected components.
Note that warning 1.4 applies to the first line of the following proposition as we use the conjectural cobordism hypothesis with singularities.
Proposition 4.6. There are compatible isomorphisms of commutative monoids under which the tracelike transformation Θ k 1 ,...,kn is sent to n i=1 x k i .
Proof. We begin by constructing the isomorphisms in the second line, the first line is then obtained similarly. As a consequence of lemma 4.5, the h 1 N adjunction 2.13, the cobordism hypothesis 3.14 and lemma 2.21 we have isomorphisms natural in C ∈ Cat ⊗  : The object * + and automorphism α from 4.2 define an element of A fd (Cob Z 1 ); this is precisely the element that corresponds to id Cob Z 1 when we apply the above bijection in the case C = Cob Z 1 . We now apply the 1-categorical coYoneda lemma for the 1-category h 1 Cat ⊗  . (See theorem 5.2 where we recall the ∞-categorical version.) Recall that in this category objects are symmetric monodial 1-categories and morphisms are isomorphism classes of symmetric monoidal functors. So the above A fd (C) can be described as Hom h 1 Cat ⊗  (Cob Z 1 , C) in this category. Therefore the coYoneda lemma implies that sending T to T Cob Z 1 ( * + , α) defines a bijection: This is an isomorphism of monoids since the monoid structure on T r is defined by pointwise multiplication.
The scalars of Cob Z 1 are diffeomorphism classes of closed oriented 1-manifolds labelled by integers, this is precisely how the set Mfd Z 1 /Diff + was defined in the introduction. Recall that N[x ±1 ] is the free commutative monoid on the set {. . . , x −1 , x 0 , x 1 , x 2 , . . . } ∼ = Z. We define a homomorphism N[x ±1 ] → Mfd Z 1 /Diff + by sending x k to the circle labelled by k. This is a bijection since all closed 1-manifolds can be written as the disjoint union of circles.
This establishes the isomorphisms in the second line of the proposition. The first line is obtained similarly, except that we use the cobordism hypothesis with singularities 3.20 instead of 3.14.
To complete the proof we need to compute the value of the tracelike transformation Θ k 1 ,...,kn on ( * + , α). In other words, we have to compute Tr(α k 1 )•· · ·•Tr(α kn ) using the classical definition of the trace. Using the evaluation and coevaluation provided in 3.9 one sees that the trace of a bordism (X, ∂X ∼ = M M − ) is given by gluing the two boundaries of X. See also [ST12] for a different perspective on this. In the case at hand we compute: This verifies that Θ k 1 ,...,kn ( * + , α) is a disjoint union of n circles labelled by the integers k 1 , . . . , k n . Hence Θ k 1 ,...,kn is sent to the polynomial x k 1 + · · · + x kn as claimed.

∞-categorical classification
In this section we combine the cobordism hypothesis with the Yoneda lemma and make the necessary computations to prove the main theorems.
5.1 Applying the Yoneda lemma and the cobordism hypothesis Evaluating at a certain object x ∈ C gives a functor ev x : Fun ∞ (C, D) → D and consequently a map hom Fun∞(C,D) (F, G) → hom D (F (x), G(x)).
Theorem 5.2 (coYoneda lemma, [Lur09a, Lemma 5.5.2.1]). For any functor F : D → Spc and object x ∈ C the following composition of evaluations is a natural equivalence We can now use the coYoneda lemma in conjunction with the one-dimensional cobordism hypothesis to express the moduli spaces T ∞ and T r ∞ as the space of scalars of Bord or 1 (S 1 ) and Bord or,m 1 , respectively. Note, however, that to compute T ∞ we use the conjectural cobordism hypothesis with singularities, so warning 1.4 applies. Proof. By corollary 3.14 of the cobordism hypothesis 3.12, the functor A fd is naturally equivalent to the corepresented functor hom Cat ⊗ ∞ (Bord or 1 (S 1 ), ). Then first equivalence now follows from the coYoneda lemma 5.2 for D = Cat ⊗ ∞ , x = Bord or 1 (S 1 ), and F = Sc. Similarly, the cobordism hypothesis with singularities 3.20 in dimension 1 says that E fd is naturally equivalent to the corepresented functor hom Cat ⊗ ∞ (Bord or,m 1 , ). Hence the second equivalence also follows from the coYoneda lemma.

Identification of the homotopy-type of the scalars of the bordism category
We give a more geometric description of Sc(Bord or 1 (S 1 )): Definition 5.4. For a topological space X, let Mfd X d ⊂ Ψ d (X) be the space of of closed dmanifolds submanifolds M ⊂ (−1, 1) ∞ equipped with an X-structure ϕ : M → X.
Lemma 5.5. There is an equivalence Sc(Bord or 1 (X)) Mfd X 1 . Proof. The monodial unit of Bord or 1 (X) is given by the empty manifold, so by lemma 2.20 the space of scalars is equivalent to hom Bord or 1 (X) (∅, ∅), which is indeed equivalent to Mfd X d .
Definition 5.6. We write T for S 1 with the usual group structure, i.e. T ∼ = U 1 ∼ = SO 2 . For a space X ∈ Spc we denote its free loop space by ΛX := Map(T, X).
The group T acts on this by precomposition and we denote the homotopy-orbits by (ΛX) hT := (ET × ΛX)/T.
The space of closed manifolds is best understood as a free E ∞ -algebra on the space of connected closed manifolds. Recall the following: Theorem 5.7. For a topological space X the underlying space of the free E ∞ -algebra on X is given by: Proof. This is a classical theorem by Segal: his proof of the Barratt-Priddy-Quillen theorem in [Seg74, Proposition 3.5 and 3.6] in fact shows that the unordered configuration space Conf * (R ∞ ; X) is a model for the free Γ-space on X. A modern reference is [Lur18, Proposition 3.1.3.13].
The following is fairly standard, see for instance [Gia19, Lemma 7.1], but we give a proof for completeness sake.
Lemma 5.8. For any topological space X the space of closed 1-manifolds with map to X is: Here, as in theorem 5.7, Free E∞ (Y ) denotes the free E ∞ -algebra on a space Y .
Proof. The space Mfd 1 has a natural N-grading induced by the number of connected components. All closed 1-manifolds are disjoint unions of circles, so the nth level of the grading is space of submanifolds of (−1, 1) ∞ that are abstractly diffeomorphic to (S 1 ) n . In the case X = * this space is equivalent to the classifying space BDiff + ((S 1 ) n ) by [SP17,Corollary B.5]. A similar argument shows that for general X it is the homotopy-orbits of Map((S 1 ) n , X) with respect to the action of Diff + ((S 1 ) n ): The group Diff + ((S 1 ) n ) can be decomposed as a wreath product Σ n Diff + (S 1 ) acting componentwise Map(S 1 , X) n . Since we are working with homotopy actions, we may replace the group Diff + (S 1 ) by the equivalent group T. Homotopy-orbits with respect to the action of a wreath product can be computed as Putting all the parts of the N-grading back together we get Map(S 1 , X) hT n hΣn Free E∞ Map(S 1 , X) hT as claimed. Here the last equivalence is as in theorem 5.7.
Lemma 5.9. In the case X = S 1 that is most relevant to us, we compute: Proof. The space Map(S 1 , S 1 ) has as connected components the spaces Map k (S 1 , S 1 ) of maps with winding number k for k ∈ Z. We need to compute Map k (S 1 , S 1 ) hT for all k ∈ Z. Let X k be the space S 1 ⊂ R 2 with the action of T defined by λ.ζ := λ k · ζ. There is an Tequivariant embedding ι : X k → Map k (S 1 , S 1 ) that identifies X k with the space of degree k maps S 1 → S 1 of constant speed. Non-equivariantly ι is a homotopy equivalence with the inverse Map k (S 1 , S 1 ) → S 1 given by evaluation on the base-point. Therefore ι is a Borel weak equivalence and induces an equivalence on the homotopy-orbits: To compute (X k ) hT for k = 0 observe that X k can be thought of as the quotient S 1 /(Z/kZ) with T-action induced from the standard action of T on S 1 . Therefore (X k ) hT = (X k × ET)/T ∼ = ((S 1 /(Z/kZ)) × ET)/T ∼ = (S 1 × ET)/T /(Z/kZ) The space (S 1 × ET)/T is homeomorphic to ET and therefore contractible. The action of Z/kZ is free and hence (X k ) hT is a model for B(Z/kZ). The remaining case k = 0 is easy: the space X 0 is S 1 with the trivial T-action, therefore the homotopy fixed-points decompose as (X 0 ) hT S 1 × BT.

5.3
The space of marked 1-manifolds Lemma 5.10. There is an equivalence: Proof. Just like in lemma 5.5 we see that the space of scalars Sc(Bord or,m 1 ) is equivalent to the space of closed marked oriented 1-dimensional submanifolds of (−1, 1) ∞ . By an argument as in proposition 3.17 this space is homeomorphic to a Borel construction: Here the diffeomorphism group Diff + (M ) acts diagonally on Conf * (M ) ⊂ k≥0 M k /Σ k . The configuration space has the property that there is a canonical homeomorphism Conf * (M M ) ∼ = Conf * (M ) × Conf * (M ) for any two manifolds M and M . In particular we have Conf * (S 1 ). We can therefore argue just like in lemma 5.8 to see that Mfd or ,m 1 is equivalent to the underlying space of a free E ∞ -algebra: We are now left to compute the homotopy T-orbits of Conf * (S 1 ). To do so, we can decompose the configuration space by the cardinality of the configuration. In the k = 0 case we have only the empty configuration, so Conf 0 (S 1 ) = {∅} is a point and the homotopy orbits are BT. For k > 0 let, as before, X k be the space S 1 ⊂ C with the action of T defined by λ.ζ := λ k · ζ. There is an equivariant map i : X k → Conf 0 (S 1 ) which sends ζ to the subset {ξ | ξ k = ζ} ⊂ S 1 . This map identifies X k with the subspace of 'equally spaced' configurations of k points on the circle and therefore is an equivalence. As in lemma 5.9 we have that (X k ) hT BZ/k and hence the result follows.

Proof of theorem A
Theorem 5.11 (Theorem A). There is a commutative diagram of spaces: where the horizontal maps are equivalences. Moreover, the vertical maps identify the sets in the bottom layer as the set of connected components of the top layer: T ∼ = π 0 T ∞ and T r ∼ = π 0 T r ∞ .
Proof. The bijections in the bottom row are discussed in proposition 4.6 about the classification of the 1-categorical tracelike transformations. For the top row corollary 5.3 identifies T r ∞ and T ∞ with Sc(Bord or 1 (S 1 )) and Sc(Bord or,m 1 ), respectively. These spaces of scalars are then computed in lemmas 5.5, 5.8, 5.9, and 5.10.
Since the ∞-categorical classification followed the same steps as the 1-categorical one, the left-hand square commutes. By 3.22 the top-right map can be understood as Sc(L) : Sc(Bord or 1 (S 1 )) → Sc(Bord or,m 1 ). Combining the definition of L with the equivalences in lemmas 5.8, 5.9, and 5.10 it follows that it is the inclusion BS 1 ∼ = {0} × BS 1 → BS 1 on the first component and the identity on BZ/k → BZ/k for k > 0. The right-most vertical map is Sc(Bord or 1 (S 1 )) → Sc(Cob Z 1 ), which is a bijection on connected components because Cob Z 1 h 1 Bord or 1 (S 1 ). This implies that π 0 T ∞ ∼ = T and by the same argument π 0 T r ∞ ∼ = T r .
Note that the vertical map sends S 1 × BS 1 to {x 0 } and BZ/kZ to {x k }. Corollary 5.13 (Corollary C). Any ∞-categorical tracelike transformation whose value on the category of complex vector spaces agrees with the trace from linear algebra is canonically equivalent to the ∞-categorical trace.
Proof. By corollary 5.12 the connected component of Tr in T ∞ is contractible. Since we have a full classification of tracelike transformations it is enough to show that for any N-polynomial n i=1 x k i the map Θ k 1 ,...,kn Vect k : E fd (Vect C ) → Sc(Vect C ) ∼ = C, (V, f : V → V ) → Tr(f k 1 ) · · · Tr(f kn ) is the trace from linear algebra only if n = 1 and k 1 = 1.
Comparing coefficients we see that this can only be equal to Tr(A λ ) = 1 + λ for all λ ∈ C if n = 1 and k 1 = 1.
Note that by the same argument corollary C also holds for restricted tracelike transformations.
Remark 5.14. The statement of corollary C remains true if we replace C by any other infinite field. It then however is not sufficient to look only at the matrices A λ . If, for instance, k is of characteristic 3, then Θ 1,0,0 Vect k (k 2 , A λ ) = (1 + λ) · 2 · 2 = 1 + λ. This problem can be solved by considering larger matrices. Note that corollary C is not true for finite fields: over F p the tracelike transformations Θ p and Θ 1 = Tr agree.

The action of N and the proof of corollary D
We would like to give a purely categorical characterisation of the trace among the other tracelike transformations that does not rely on a pre-defined notion of trace. To define the notion of generating tracelike transformation we first need to specify the N-action outlined in the introduction.
Definition 5.15. The multiplicative monoid (N, ·) acts on the sets T and T r by (P n T )(e) := T (e n ).
It is clear that P n • P m = P nm , but we need to briefly check that P n T is indeed still conjugation invariant: We will now lift the maps P n to maps P n : T (r) ∞ of the moduli space of ∞-categorical tracelike transformations. Once could assemble these maps into a coherent action of (N, ·), but that is not necessary for our purposes. Instead we will just check that the P n induce P n on π 0 T (r) ∞ ∼ = T (r) . Construction 5.16. Fix n ∈ N and some complete Segal space X • . There is a diagonal map ∆ n X 1 : X 0 × (X 0 ) 2 X 1 → X 0 × (X 0 ) 2 (X 1 × X 0 · · · × X 0 X 1 ), where as usual all pullbacks are computed in the infinity category of spaces. We can pick a homotopy inverse to the Segal map X n → X 1 × X 0 · · · × X 0 X 1 and construct the composite Here d : X n → X 1 is the face operator coming from the long edge {0, n} ⊂ [n]. This defines a natural transformation p n : E ⇒ E that commutes with the projection E(X) → X 0 . Given a symmetric monoidal structure on X the transformation p n therefore preserves the subspace E fd (X) ⊂ E(X). We can hence define a map P n : T ∞ → T ∞ , (T : E fd ⇒ Sc) → (T • p n : E fd ⇒ E fd ⇒ Sc).
Lemma 5.17. For T ∈ T ∞ and n ∈ N the tracelike transformation P n T acts as (P n T )(e) = T (e n ) on the homotopy category.
Proof. The inverse of the Segal map composed with the face map: is given by (f 1 , . . . , f n ) → f n • · · · • f 1 on the homotopy category. Therefore p n X : E(X) → E(X) yields E(h 1 X) → E(h 1 X), (x, e) → (x, e n ).
Precomposing with this induces P n : T → T as described in definition 5.15.
We can now make the definition from the introduction precise: Definition 5.18. A tracelike transformation T ∈ T ∞ is generating if the monoid π 0 T ∞ is generated by the set {[P n (T )] | n ∈ N}. Equivalently, T is generating if for every other S ∈ T ∞ there are non-negative integers k 1 , . . . , k n such that S is equivalent to P k 1 (T ) · · · P kn (T ).
Corollary 5.19 (Corollary D). The space of generating tracelike transformations is contractible and its image in T is the classical trace Tr.