ANDREA TIRELLI. = M is the manifold M with the opposite orientation and B 1

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1 AN INTRODUCTION TO (, n)-categories, FULLY EXTENDED TOPOLOGICAL QUANTUM FIELD THEORIES AND THEIR APPLICATIONS ANDREA TIRELLI Contents 1. Introduction Topological Quantum Field Theories Extended TQFTs and higher categories 2 2. Complete Segal spaces as models for (, n)-categories Simplicial sets, simplicial spaces and Kan complexes Models for (, 1)-categories Models for (, n)-categories Completions, truncations and symmetric monoidal structures Duals in (, n)-categories 9 3. The (, n)-category of cobordisms The complete Segal space Int The n-fold Segal space PBord n The (, n)-category Bord n and its symmetric monoidal structure Cobordisms with additional structure Fully extended topological quantum field theories Examples and applications Fully dualizable objects in Bord fr 2 : an informal proof Fully dualizable objects in Bord fr 2 : a rigorous proof An -model for the Morita bicategory of a monoidal category Full dualizability in Alg 1 (S) Applications to fully extended TQFTs Introduction In this section we will briefly introduce the notion of Topological Quantum Field Theory (TQFT), following Atiyah s axioms [Ati88], and explain the issues that arise in this classical setting. This will motivate the need for extending the definition of TQFT, which is done by means of higher category theory Topological Quantum Field Theories. In theoretical physics, a particle may be modelled as a physical field, which can be regarded as a smooth section of a vector bundle over space-time. A quantum field theory is a model for studying the interactions of particles through the underlying physical fields. We focus our attention on topological quantum field theories, which are those invariant under diffeomorphisms of the underlying space-time. TQFTs turn out to have many interesting applications in mathematics, for example in knot theory, the classification of 4-manifolds and in the study of moduli spaces in algebraic geometry. The first mathematically rigorous definition of topological quantum field theory was given by Atiyah in [Ati88], which we will explain in the modern categorical language. Definition 1.1. Given n 1, the oriented bordism category Cob or n objects are compact oriented (n 1)-dimensional manifolds; is defined as follows: for any pair of objects in M, N Cob or n a bordism from M to N is a compact oriented n-manifold B with an oriented boundary B = B 0 B 1, where B 0 = M is the manifold M with the opposite orientation and B 1 = N; let Bord n (M, N) be the set of bordisms from M to N; we define Hom Cob or n (M, N) := Bord n(m, N)/, where is the equivalence relation that identifies two bordisms B and B if there is an orientation preserving diffeomorphism B B that restricts to diffeomorphisms B 0 B 0 and B 1 B 1. 1

2 2 ANDREA TIRELLI Disjoint union of manifolds endows Cob or n with a symmetric monoidal structure. Definition 1.2 (Atiyah). Let k be a field. A n-dimensional Topological Quantum Field Theory F is a symmetric monoidal functor F : Cob or n Vect(k). More explicitly, a n-dimensional TQFT is given by the following set of data: (1) for each (n 1)-manifold M, a vector space F (M), such that F ( ) = k and disjoint unions of manifolds are mapped to the tensor products of the corresponding vector spaces; (2) for each bordism B : M N, a k-linear map F (M) F (N) satisfying the usual coherence axioms for categories. Remark 1.3. In general, if we replace Vect(k) by an arbitrary symmetric monoidal category C, we get a C-valued TQFT. It is worth noting that the idea behind Atiyah s definition of TQFT is that such a theory should be, in some sense, local: one can to compute what is attached to a n-manifold by cutting it along codimension 1 submanifolds. This principle works really well in low dimensions; in particular, for n = 1, 2, TQFTs are completely classified. Example 1.4. For n = 1, the objects of Cob or n are isomorphic to finite unions of copies of + := pt + and := pt (a fixed point with either a positive or negative orientation). It is possible to prove that the left and the right arcs and establish a perfect duality between F (+) and F ( ), which forces them to be finite dimensional and dual to each other. Thus, F is completely described, up to isomorphism, by dim F (+), which can be seen to be equal to F (S 1 ). Note that an isomorphism between two TQFTs is simply a natural isomorphism between the corresponding functors. It is worth noting that this easy example contains the germ of the so called Cobordism Hypothesis, largely discussed later in the paper, that classifies fully extended TQFTs in terms of the datum F (+). Example 1.5. It is not difficult to prove an analogous result for the 2-dimensional case, where a TQFT is completely determined by the image of S 1 together with its algebraic structure. Indeed, it is possible to see that a 2-dimensional TQFT is equivalent to the datum of a commutative Frobenius algebra structure on a finite-dimensional vector space A, which is precisely F (S 1 ) - see [Koc04] for a proof of this result. Remark 1.6. It is important to notice that the classification results in dimension 1 and 2 are made possible by the fact that 1- and 2-dimensional (compact and oriented) manifolds with boundary can be easily described from a topological point of view. Indeed, there are well known decomposition theorems that describe how such manifolds can be decomposed into simple pieces. For example, it is known that any compact oriented surface with boundary can be decomposed into discs and pairs of pants, which are precisely the surfaces that give rise to the commutative Frobenius algebra structure of F (S 1 ) Extended TQFTs and higher categories. In higher dimensions, classifying TQFTs is much harder: indeed, the topology of a n-manifold, for n 3, can be really complicated and there are no easy-to-use classification theorems as in dimension 1 and 2 that make cutting along codimension one submanifolds an effective computational tool. Indeed, we would like to be able to attach invariants not only to n- and (n 1)-manifolds, but to any k-manifold, for k = 0,..., n. In this way it would be possible, at least in principle, to recover the invariants attached to higher dimensional manifolds by cutting them into simple manifolds with corners of all codimensions. Moreover, TQFTs, as defined above, are too restrictive in many circumstances. For example, in Cob or 2, the only objects are disjoint unions of circles and bordisms are oriented 2-manifolds with boundaries being disjoint unions of circles. We would also like to include objects as closed intervals and bordisms between them. Thus, we need an extended definition of TQFT that takes into account the aforementioned issues and, to perform this extension, we also need to modify the construction of the bordism category Cob or 2, in order it to have a richer algebraic structure. A way to encode such an algebraic structure is by means of higher category theory. Roughly speaking, a higher category is a collection of objects, (1-)morphisms between objects, (2-)morphisms between morphisms, (3-)morphisms between 2-morphisms and so on and so forth. A higher category with n levels of morphisms is called an n-category. These higher-order morphisms are supposed to satisfy compatibility properties analogous to those of ordinary morphisms, except that we will not require them to be valid on the nose but only in a weak sense, due to a certain notion of homotopy. We will be particularly interested in higher categories with k-morphisms for every positive integer k, such that all k-morphisms of order k > n are isomorphisms (up to homotopy), these are called (, n)-categories. Example 1.7 (Sketch). A first example is the bordism (, n)-category Cob or n, whose objects are (oriented) 0-manifolds, 1-morphisms are (oriented) 1-manifolds with boundary, 2-morphisms are (oriented) manifolds

3 (, n)-categories, FULLY EXTENDED TQFTS AND APPLICATIONS 3 with corners and so on and so forth up to order n; (n + 1)-morphisms between n-morphisms are orientation preserving diffeomorphisms, (n + 2)-morphisms are isotopies between diffeomorphisms, and so on and so forth. It is possible to see that what one gets is a higher category where all k-morphisms, for k > n, are invertible up to coherent homotopy. Example 1.8 (fundamental -groupoid of a topological space). Let X be any topological space. The fundamental -groupoid of X, denoted by π (X), is given as follows: objects: points of X; 1-morphisms: paths between points; 2-morphisms: homotopies between 1-morphisms; n-morphisms: homotopies between (n 1)-morphisms, for n > 2. Composition is given by concatenation of paths (homotopies) and the inverse of a path (homotopy) is given by the inverse path (homotopy), which is the path (homotopy) that goes in the opposite time direction. Thus, bearing in mind the above sketchy definition of (, n)-category, we have constructed an example of (, 0)-category. The so called Homotopy Hypothesis states that any (, 0)-category is of this form, i.e. it is equivalent to the fundamental -groupoid of a topological space. Although it is clear from the previous example how the language of higher categories arises intuitively, it is important to say that, in order to make computations with (, n)-categories, it is necessary to work in a rigorous and axiomatised setting. A model for (, n)-categories is given by complete n-fold Segal spaces, a concise and example-oriented treatment of which will occupy part of this paper. This language will enable us to define rigorously fully extended Topological Quantum Field Theories and state the Cobordism Hypothesis, first conjectured by Baez and Dolan, [BD95, Extended TQFT Hypothesis Part I and II], a proof of which was first proposed by Lurie in [Lur09, 3]. In this framework, we will discuss some applications and examples, e.g. the relations between TQFTs and the Hochschild (co)homology of certain dg-algebras. Acknowledgements. I would like to thank Dr. Travis Schedler for the guidance and patience in supervising this project and Claudia Scheimbauer and Christopher Schommer-Pries for useful discussions. 2. Complete Segal spaces as models for (, n)-categories There are several models for (, n)-categories; in this section we will introduce the reader to the theory of complete (n-fold) Segal spaces, which constitute one of the possible models for higher categories. Note that other possible models, like Segal categories or Θ n -spaces, can be proven to be equivalent, in an appropriate sense, to complete n-fold Segal spaces. We will mainly follow the approach taken in [Lur09] and [CS15] Simplicial sets, simplicial spaces and Kan complexes. In this subsection we recall the basic definitions we need to develop the theory of Segal spaces and we fix some notation. Definition 2.1. Given a category C, a simplicial object X in C is a functor X : op C, where is the simplex category, whose objects are ordered sets of the form [m] = (0 < 1 < < m), for m 0, and morphisms are monotonically increasing maps. When C = Set a simplicial object in C is called simplicial set. A map f : X Y between two simplicial sets is a natural transformation between functors; we will denote the category of simplicial sets with sset. Remark 2.2. More explicitly, a simplicial set is the given by a sequence of sets X = (X n ) n 1 and a map X n X m for any monotonically increasing map [m] [n]. The image of the map f i : [m] [m + 1] sending j to j for j = 1,..., i 1 and j to j + 1 for j = i,..., m is called the i-th face map f i : X m+1 X m, for i = 1,..., m. If instead we consider the image of the map d i : [m+1] [m] sending j to j for j = 1,..., i and j to j 1 for j = i + 1,..., m + 1 we get the i-th degeneracy map d i : X m X m+1, i = 1,..., m + 1. It is easy to prove that any morphism in is given by the composition of face and degeneracy maps. Thus, by functoriality, all the maps of the simplicial set X are generated by d i and f i for i = 1,..., n. Moreover, d i and f i satisfy the so called simplicial identities. Conversely, a sequence of sets (X n ) n 1 equipped with maps d i and f i that satisfy the simplicial relations can be given the structure of a simplicial set. Remark 2.3. There is an operation, called geometric realisation, that builds from a simplicial set X a topological space X, / X := X n n n 1,

4 4 ANDREA TIRELLI obtained by interpreting each element in X n as one copy of the standard topological n-simplex n and then gluing together all these along their boundaries to a big topological space, using the information encoded in the face and degeneracy maps of X on how these simplices are supposed to be stuck together. Example 2.4 (Nerve of a category). A fundamental example of simplicial set is the nerve N(C) of a category C. Given any two objects x, y Ob(C), we will denote by C(x, y) the set of morphisms between them. N(C) is defined as follows: N(C) n = C(x 0, x 1 ) C(x n 1, x n ). x 0,...,x n Ob(C) The i-th face and degeneracy maps - which determine completely N(C) as a simplicial set - are given by composition of morphisms in C(x i 1, x i ) with morphisms in C(x i, x i+1 ) and insertion of the identity map Id xi respectively. Example 2.5. A way of extracting a simplicial set from a topological space, which will play a fundamental role in the construction of the (, n)-category Bord n as a complete n-fold Segal space, is given by the Sing functor Sing : Top sset, that sends a topological space X to its singular chain complex Sing(X), whose n-th level is Sing(X) n = C 0 ( n, X), by which we mean the space of continuous maps from the geometric realisation of the n-th standard simplex to X. Face and degeneracy maps are defined exaclty as one might expect. It is also possible to prove that the geometric realisation functor is : sset Top is the left adjoint of the Sing functor. Example 2.6. An easy but nonetheless useful example of simplicial set is given by the n-simplex n = Hom (, [n]) : op Set. It follows immediately from Yoneda s Lemma that, given a simplicial set X = X, there is a natural isomorphism of sets X n = sset( n, X). Moreover, for any i = 1,..., n, we define the i-th horn Λ i n to be the simplicial set given by (Λ n i ) k = {f ( n ) k i / f([k])}, for k 0, where the face and degeneracy maps are those induced by n - this defines a simplicial set thanks to the Remark 2.2. It is possible to prove that applying the geometric realisation functor to n and Λ i n gives exactly what one might expect: n is the standard topological n-simplex and Λ i n is the topological space obtained by taking n and deleting the face opposite to the i-th vertex. The simplicial sets n and Λ i n play a fundamental role in the definition of Kan complex. Definition 2.7. Let f : X Y be a map of simplicial sets. We say that f is a Kan fibration if, for any n 0 and any 0 i n, it satisfies the horn-filling condition, i.e. given a commutative diagram Λ i n X there exists a filling map g : n X that lifts the map n X, n Y f Λ i n X g f. n Moreover, a simplicial set X is a Kan complex if the map X { } to the simplicial set given by a point is a Kan fibration. We will refer to Kan complexes as spaces. Remark 2.8. Our convention to refer to Kan complexes as spaces is partly motivated by the fact that it is possible to prove that the category of Kan complexes KComp and the category of CW-complexes CW are equivalent as model categories. For the purpose of the present work we will not need to spell the details of such equivalence - for which we refer to [Zha13, 2.1]. We only need to know that Kan complexes can be treated as topological spaces. Definition 2.9. We define a simplicial space X to be a simplicial object in KComp, Y X : op KComp.

5 (, n)-categories, FULLY EXTENDED TQFTS AND APPLICATIONS Models for (, 1)-categories. Looking at the ambiguous definition of (, n)-categories given in the previous section, we realise that one possible way to construct rigorously such higher categories would be by induction on n. Indeed, the idea is that an (, n)-category should be a category enriched in (, n 1)- categories. Thus, the first case we need to consider is that of (, 0)-categories, which, in analogy with ordinary category theory, are also called -groupoids. In this context there is a basic hypothesis upon which we will build the theory of higher categories, see Example 1.8. Hypothesis 2.10 (Homotopy Hypothesis). Spaces are models for -groupoids. Remark The reader should refer to Example 1.8 and Remark 2.8 for a justification of the above hypothesis. Definition An (, 0)-category is a space. Remark Note that, given a topological space X, the simplicial set Sing(X) is actually a Kan complex, so we can identify π (X) with the Kan complex Sing(X). This is justified by Remark 2.8, since homotopies between homotopies more naturally yield CW-complexes. Given that (, 0)-categories are spaces, one might want to define (, 1) as topological categories, that is to say categories enriched in (topological) spaces. This definition turns out to be too strict for our purposes: we will need a space structure not only on the set of morphisms but also on the collection of objects, which explains why we will define (, 1)-categories as particular class of simplicial spaces. Definition A Segal space X = X is a simplicial space satisfying the following Segal condition: let m, n 0 and consider the following commutative diagram X n+m X n X m X 0 induced by the maps [m] [m + n], (0 < < m) (0 < < m) and [n] [m + n], (0 < < n) (m < < m + n). Then the induced map X n+m X m h X 0 X m is a weak equivalence, where h X 0 stands for the homotopy fiber product, which is a homotopical version of the usual pullback of spaces (see, e.g., [Lur09, Remark ]) for a definition) and weak equivalence means a map of simplicial sets whose geometric realisation is a usual weak homotopy equivalence of topological spaces. Morphisms between Segal spaces are just maps between the underlying simplicial spaces. We will denote by SeSp the category of Segal spaces. Example A straightforward example of Segal space is given by the nerve N(C), when C is a small category in spaces. Remark The previous example is useful to justify why we will use Segal spaces as models for (, 1)- categories. Indeed, we see that, given a category internal to spaces C, the 0-th level of its nerve N(C) 0 is the space of objects if C, while the first level is the space of all (1-)morphisms between objects. Analogously, given any Segal space X = X, we can view the set of 0-simplices (points) of X 0 as the set of objects. Moreover, given x, y X 0, we can interpret as the -groupoid of (1-)morphisms from x to y. X(x, y) := {x} h h X 1 {y} X 0 X 0 Lemma Let X = X be a Segal space. Then the Segal condition is equivalent to the following condition: consider any m 1 and define, for k = 1,..., m, the map f k : [1] [l] by (0 < 1) (k 1 < k): then the map induced by f 1,..., f l, h h X l X 1 X 1 X 0 X 0 is a weak equivalence. Proof. That the Segal condition implies the second one is an easily seen by induction on m. To prove the converse, fix m, n 0 and note that we can put together the weak equivalences given by the second condition, for l = n, m and n + m, to get a weak equivalence X n+m X m h X 0 X m, which, thanks to the definition of the maps f k condition. is the weak equivalence in the statement of the Segal

6 6 ANDREA TIRELLI Thus, the previous lemma and remark explain how Segal spaces might be interpreted as (, 1)-categories. However, in order to have a good model for such higher categories, Segal spaces are not enough. Indeed, we need a further condition, called completeness. Remark To explain why, in general, Segal spaces are not a completely satisfactory model for (, 1)- categories we give an example, also mentioned in [Lur09, Example ], that shows how non isomorphic Segal spaces can be interpreted as the same (, 1)-categories. Let C be an ordinary category. We can regard its nerve N(C) as a simplicial space, endowing each level with the discrete topology and taking the Sing functor. Applying the construction of Remark 2.16 we get the original category C. However, the Segal space X functorially associated to C, when C is viewed as an (, 1)-category, does not coincide with N(C) : indeed the space X 0 is usually not discrete, not even up to homotopy. Thus, N(C) and X are non isomorphic Segal spaces, but both define the same (, 1)-category C. To define completeness, we introduce the notion of homotopy category of a Segal space. Definition The homotopy category h 1 (X) of a Segal space X is the ordinary category whose objects are points of X 0 and whose morphism set between two objects x, y X 0 is given by h 1 (X) = π 0 (X(x, y)). Composition of morphisms is defined by the following sequence of maps: ( {x} h X 0 X 1 ) ( h {y} {y} h X 1 X 0 X 0 ) h {z} {x} h X 1 X 0 X 0 h X 0 X 1 {x} h h X 2 {z} X 0 X 0 {x} h h X 1 {z}. X 0 X 0 h X 0 {z} Roughly speaking, the completeness condition, formulated in the next definitions, says that the space of homotopy invertible morphisms contracts onto the space of identity morphisms. Definition Let X = X be a Segal space and let f be an element of X 1. Let x and y be the endpoints of f, i.e. the images of f under the maps X 1 X 0. Then, f is said to be invertible if its image under the composition of the maps {x} X 0 X 1 X 0 {y} {x} h X 0 X 1 ( h {y} π 0 {x} h X 1 X 0 X 0 ) h {y} = h 1 (X)(x, y) X 0 is an invertible morphism in the homotopy category of X. We will denote by X inv 1 the subspace of X 1 given invertible morphisms. Remark Note that the map φ : X 0 X 1 factors through the subspace X inv 1. To see this, consider an element x in X 0 and let y = φ(x), which can be mapped to the space {x} X0 X 1 X0 {x}. Thus, by composing with the two obvious maps we get a map X 0 Hom h1(x)(x, x) and, by spelling out the definition of homotopy fibre product, it is immediate to see that the image of x is the identity Id x. Definition A Segal space is complete if the map X 0 X inv 1 is a weak equivalence. Complete Segal spaces form a category CSeSp, which is a full subcategory of SeSp. It is worth noting that not every Segal space is complete. examples of Segal spaces which are not complete. Indeed the following construction produces Example Let G be a non-trivial discrete group. Consider the ordinary groupoid BG with only one object and G as set of morphisms. Let X be the Segal space obtained by taking the nerve of this category X = N(BG). Then we have that X 0 is a point, whereas X 1 = X1 inv = G is not homotopically equivalent to X 0 unless G is trivial. Now we have all we need to define our chosen model of (, 1)-categories. Definition An (, 1)-category is a complete Segal space. In Example 2.15 we have given a way of constructing a Segal space starting from a small category. In the following example we define the relative version of the nerve construction, which turns out to give rise to complete Segal space in many cases.

7 (, n)-categories, FULLY EXTENDED TQFTS AND APPLICATIONS 7 Example Let (C, W) be a relative category and consider the simplicial object C in Cat given by C n = Fun(n, C), where n, as a category, is defined as follows: Ob(n) = {0, 1,..., n} and there is a unique morphism for i to j if and only if i j. C has a subobject C W, where C W n is the category whose objects are the same as Fun(n, C) but whose morphisms are those natural transformations constructed using only morphisms in W. Taking the nerve levelwise we obtain the simplicial space N(C, W) n = N(C W n ). It is possible to prove ([Rez01]) that, under certain hypotheses on the relative category (C, W) this construction gives rise to a complete Segal space. For instance, when C is a small category and W is the set Iso(C) of isomorphisms of C it is easy to check directly that we obtain a complete Segal space. The following lemma proves this statement. Lemma Let C be a small category. Then, N(C, Iso(C)) is a complete Segal space. Proof. Without loss of generality, we can assume that C is skeletal, i.e. any isomorphism class of object has only one element. We can make this assumption because any category is equivalent to a skeletal one and equivalent categories have homotopically equivalent nerves. Then, when C is skeletal X1 inv := N(Fun(1, C, C inv )) inv = N(Fun(1, C)), where C inv denotes the groupoid obtained form C by discarding all the non invertible morphisms. Now it is clear that the map F un(1, C inv ) C inv evaluating at 0 in {0, 1} is an equivalence of categories; thus, it induces a homotopy equivalence between the nerves N(Fun(1, C inv )) and N(C inv ). But these are the spaces X1 inv and X 0 respectively. This proves our claim Models for (, n)-categories. Once the language and the definitions for (, 1)-categories are established it is relatively easy to extend them to the context of higher categories for n > 1. Definition Let n 1. An n-fold simplicial space X = X,..., is a functor X : op op KComp. Remark Analogously to Remark 2.2, we can spell out the above definition explicitly: an n-fold simplicial space is a collection X,..., = (X k1,...,k n ) k1,...,k n 0 of Kan complexes equipped with a map X k1,...,k n X l1,...,l n for any n-uple of maps [l i ] [k i ], where [l i ] [k i ] is a morphism in Hom ([l i ], [k i ]) for i = 1,..., n. Moreover, a map of n-fold simplicial spaces f : X,..., Y,..., is simply a collection of maps of simplicial sets X k1,...,k n Y k1,...,k n satisfying the obvious naturality conditions. Definition An n-uple Segal space is an n-fold simplicial space such that for any 1 i n and for any k 1,..., k i 1, k i+1,..., k n 0, the simplicial space is a Segal space. X k1,...,k i 1,,k i+1,...,k n To define n-fold Segal space we need an additional property. Definition Let X,..., be an n-fold simplicial space. We will say that X is essentially constant if the map of n-fold Segal spaces X 0,...,0 X,...,, where X 0,...,0 is viewed as a constant n-fold Segal space, given by the degeneracy maps, is a weak equivalence. Definition An n-fold Segal space is a n-uple Segal space such that, for any 1 i n and any k 1,..., k i 1 0, the n i-uple Segal space X k1,...,k i 1,0,,..., is essentially constant. A map of n-fold Segal spaces f : X Y is a map of the underlying n-fold simplicial spaces. We will denote the category of n-fold Segal spaces by SeSp n. Remark There is an alternative way to formulate the above definition: we could define a Segal n-space to be a simplicial object X in the category of (n 1)-fold Segal spaces which satisfies the Segal condition. Then, an n-fold Segal space is a Segal n-space such that X 0 is essentially constant as an (n 1)-fold Segal space. Exactly as in the case of n = 1, to get a satisfactory model for (, n)-categories we need to consider a particular class of n-fold Segal spaces, namely those that are complete. The completeness condition, defined below, is a straightforward generalisation of Definition Definition An n-fold Segal space X,..., is complete if, for any 1 i n, the following condition is satisfied: for any k 1,..., k i 1 0, the Segal space is complete. X k1,...,k i 1,,0,...,0

8 8 ANDREA TIRELLI In order to explain why we can interpret n-fold Segal spaces as higher categories we briefly introduce the notion of internal n-uple category. This is defined recursively as a category internal to the category of (n 1)- uple categories internal to spaces. See [Hor15, 3] for the general definition of internal category. In the case n = 2, an internal n-uple category is given by a space of objects, a space of horizontal 1-morphisms, a space of vertical 1-morphisms and a space of 2-morphisms, together with unit and composition maps, see [Shu11, 4]. For larger n, there is a space of objects and suitable spaces of higher morphisms in all directions. Note that that all composition laws are defined on the nose (this is not true for the model given by n-fold Segal spaces). The fact that n-fold Segal spaces reflect the sketchy definition of (, n)-category given in Example 1.7 is made clear by passing through the model given by internal n-uple categories, see [Hor15]. Indeed, the first condition in the definition of complete n-fold Segal space ensures that there are several ways (or directions) to compose morphisms: an element of X k1,...,k n should be interpreted as a composition of k i composed morphisms in the i-th direction, for i = 1,..., n, which says that the an n-fold Segal space is an n-fold category. Moreover, the constancy condition guarantees that n-fold Segal spaces have an higher categorical structure : an n-morphism has as source and target two (n 1)-morphisms which themselves have the same (in the sense that they are homotopic) source and target. In the following example we analyse the case of a 2-fold Segal space and compare the strict 2-categorical model to the weakened one. Example It it very useful to describe more explicitly the above interpretation in the easiest case, i.e. when n = 2. First, consider a 2-morphism in an ordinary 2-category. It can be depicted as a double arrow in the following diagram where the top and the bottom arrows are the source and the target, which are 1-morphisms between the same two objects. Now, consider a 2-fold Segal space X,. Then, an element of the space X 1,1 can be considered as a higher categorical analogue of the previous diagram where: the bullets are elements of X 0,0 ; the horizontal arrows are 1-morphisms, which are the images under the source and the target maps in the first direction X 1,1 X 1,0 ; the vertical arrows, which are essentially the identity maps up to homotopy since X 0,1 X 0,0, are the images under the source and the target maps in the second direction X 1,1 X 0,1. Thus we see that the above pictures differ only by the fact that in the second one the source and the target of the two horizontal 1-morphisms are not the same, but they are homotopic. The same ideas can be applied to get an analogous picture in the case n = 3, see [CS15, 2.2.1] for a detailed discussion Completions, truncations and symmetric monoidal structures. Completion. In Example 2.23 we have seen that not every Segal space is complete and it is not difficult to extend the construction to any n > 1 to prove that the same result holds for n-fold Segal spaces. Despite this fact, it is possible, given an n-fold Segal space X, to construct from it a complete n-fold Segal space X, called the completion of X. Indeed, for the case n = 1, in [Rez01] Rezk gave an explicit construction of the completion functor: to every Segal space X it associates a complete Segal space X and a map i X : X X, which is a Dwyer-Kan equivalence (roughly speaking, an equivalence of Segal spaces, see [CS15, Definition 1.1]). Now, taking into consideration the iterative definition of n-fold Segal space described in Remark 2.32 we can extend the completion functor to the category of n-fold Segal spaces: to do so, consider an n-fold Segal space X as a simplicial (n 1)-fold Segal space object, X : op SeSp n 1. Suppose, by induction, that the completion functor is defined for (n 1)-fold Segal spaces and apply it to each of the X n. Then, since the definition of completeness is given iteratively, the resulting n-fold Segal space is complete. Therefore, given an n-fold Segal space X,...,, one can apply the completion functor iteratively to obtain a complete n-fold Segal space ˆX,...,. Truncation. Another important construction, that gives an (, k)-category C from a (, n)-category C, for any k n, is the k-truncation. Informally, C is obtained by C by discarding all the non-invertible m-morphisms, for any k < m n. This process is formulated in the context of complete Segal spaces as follows: fix n 1, k n and consider the functor τ k : SeSp n SeSp k

9 (, n)-categories, FULLY EXTENDED TQFTS AND APPLICATIONS 9 that sends a given n-fold Segal space X,..., to τ k (X) = X,...,,0,..., 0. }{{}}{{} k times n k times It follows from Definition 2.33 that, if X is complete, then so is its k-truncation τ k (X) for all k n. Symmetric monoidal structures. We now want to define the -analogue of the concept of symmertic monoidal structure of an ordinary category, which, as we saw in Definition 1.2, is crucial when dealing with TQFTs. In order to do this, we need to introduce the category Γ of finite pointed sets. Definition Segal s category Γ is the category whose objects are given by the sets m = {0,..., m} for any m 0 and whose morphisms between two objects m and l are functions f : m l such that f(0) = 0. Remark For any m 0 and any 1 k n, consider the morphisms γ k Hom Γ ( m, 1 ) given by These morphisms are called the Segal morphisms. γ k : m 1, γ k (l) = δ kl. Definition A symmetric monoidal structure on a (complete) n-fold Segal space is a functor from the category Γ to the category of (complete) n-fold Segal spaces such that, for any m 0, the induced map A 1 k m A : Γ (C)SeSp n γ k : A m (A 1 )) m is an equivalence of (complete) n-fold Segal spaces. Given a (complete) n-fold Segal space X, a symmetric monoidal structure on X is a symmetric monoidal structure A such that A 1 is X. Remark It is possible to prove that symmetric monoidal structures, i.e. functors Γ (C)SeSp n that satisfy the above property, form an (, 1)-category ([JS15]). A 1-morphism in this category is called a symmetric monoidal functor of (, n)-categories. We now state a useful result, whose validity is justified by the fact that the completion map X ˆX is a weak equivalence and preserves finite products of Segal spaces up to weak equivalence, that says that if X is a symmetric monoidal n-fold Segal space, we can extend the symmetric monoidal structure to its completion. This lemma will be useful in the construction of Bord n as a complete n-fold Segal space. E, we obtain the following Lemma If A : Γ SeSp is a symmetric monoidal n-fold Segal space, then  : Γ CSeSp n, m  m is a symmetric monoidal complete n-fold Segal space Duals in (, n)-categories. We now want to define the concept of fully dualizable object in an (, n)-category. To do so, we have to make the discussion about the interpretation of an n-fold Segal space as a higher category more rigorous, i.e., we need to give a formal definition of r-morphisms for any r 0. To this end, we will exploit Remark 2.32 regarding the iterative definition of n-fold Segal space. Let X be an n-fold Segal space, X : op SeSp n 1. Let ObX be the set of points of X 0,...,0. For any x 0,..., x k ObX, we define map 1 X (x 0,..., x k ) to be the homotopy fibre of the map (n 1)-fold Segal spaces X k (X 0 ) k+1 at (x 0,..., x k ). It is possible to check that map 1 X (x 0,..., x k ) is an (n 1)-fold Segal space, and thanks to the Segal condition, we have the weak equivalence map 1 X(x 0,..., x k ) map 1 X(x 0, x 1 ) h h map 1 X(x k 1, x k ). Inductively, for 1 r n and f 0,..., f l map r 1 X (x 0,..., x k ), let map r X (f 0,..., f l ) be the homotopy fibre of (f 0,..., f l ) of the map of (n r)-fold Segal spaces map r 1 X (x 0,..., x k ) l (map r 1 X (x 0,..., x k ) 0 ) l+1. Then, a 1-morphism is an object f : x y in map 1 X (x, y) for objects x, y ObX. An r-morphism f : x y is an object of map r X (x, y), where x and y are (r 1)-morphisms. Two r-morphisms f, g : x y

10 10 ANDREA TIRELLI are homotopic if they lie in the same component of map r X (x, y) 0,...,0. In Definition 2.19, we defined the homotopy category h 1 (X) of a Segal space. We now want to extend this construction to an n-fold Segal space, for arbitrary n 1. Definition Let X be an n-fold Segal space. The homotopy category h 1 (X) of X is the (ordinary) category with objects ObX and, for each pair x, y ObX, h 1 (X)(x, y) = π 0 map 1 1(x, y) 0,...,0 is the set of path components of the space map 1 X (x, y) 0,...,0. It is also possible to define a higher categorical version of the homotopy categories and the following definition spells out the construction in the case of the homotopy 2-category. We refer to [Zha13, Proposition 2.5.8], for a proof that the following definition gives indeed a bicategory. Definition Fix n 2. Given a n-fold Segal space X, its homotopy 2-category h 2 (X) is a bicategory defined as follows: Ob(h 2 (X)) = ObX, for each pair of objects x, y Ob(h 2 (X)), h 2 (x, y) = h 1 map 1 X (x, y). In order to define the concepts of dual of an object and adjoint of an r-morphism in the context of n- fold Segal spaces, we will assume that the reader is familiar to the corresponding notions in the ordinary (bi)categorical context, for a treatment of which we refer to [ML98, XII.3]. Definition Fix n 2. Let X be an n-fold Segal space. We say that X has adjoints for 1-morphisms if the homotopy 2-category h 2 (X) has adjoints for for 1-morphisms. For 1 < r < n, we say that X has adjoints for r-morphisms if for all x, y ObX, map 1 X (x, y) has adjoints for (r 1)-morphisms. X has adjoints for morphisms if X has adjoints for r-morphisms for any 1 r < n. Definition Let X be a symmetric monoidal n-fold Segal space. We say that X has duals for objects if h 1 (X) has duals for objects. We say that X has duals if it has duals for objects and for morphisms. The next result, whose proof is omitted for the sake of brevity, states that to every symmetric monoidal (, n)-category we can associate a symmetric monoidal (, n)-subcategory with duals. Proposition Let X be a symmetric monoidal (complete) n-fold Segal space. There exists a symmetric monoidal (complete) n-fold Segal space X fd with duals and a symmetric monoidal functor X fd X that is final in the category of symmetric monoidal functors Y X, where Y is a symmetric monoidal (complete) n-fold Segal space with duals. Definition An object x ObX of a n-fold Segal space X is fully dualizable if it is contained in the essential image of the of X fd X, i.e. if x is homotopic to some element y in the image of the map. Note that the full dualizability condition is, in general, very difficult to check and one often seeks criteria that can be used in actual computations. In the case of (, 2)-categories we have such a criterion, which is stated in the following proposition, whose proof is given in [Zha13, Proposition 4.2.2]. Proposition Let C be a symmetric monoidal (, 2)-category. Then an object X Ob(C) is fully dualizable if and only if X has a dual X and the evaluation map ev : X X 1 has both right and left adjoints. 3. The (, n)-category of cobordisms In this section we will construct a complete n-fold Segal space Bord n, which is our higher categorical model for the category of cobordisms in Definition 1.1. We will define Bord n in several steps and we will explain how we can interpret it in the light of the sketchy definition given in Example 1.7. Moreover, we will extend the definition of Bord n to the case of structured cobordisms and, finally, we will give a rigorous definition of a fully extended topological quantum field theory and state the Cobordism Hypothesis. Since going into all the details of the construction is beyond the purposes of this paper and our main goal is to convey only the main ideas of the construction, we refer to our main reference [CS15] for a complete treatment of the topic The complete Segal space Int. In order to define the complete n-fold Segal space Bord n we need to give an auxiliary construction, namely the complete Segal space of closed intervals in R. Let Int the simplicial space defined as follows: for any k 0, Int k = {(a, b) = (a 0,..., a k, b 0,..., b k ) : a j < b j, for 0 j k, and a j 1 a j and b j 1 b j for 1 j k} R 2(k+1).

11 (, n)-categories, FULLY EXTENDED TQFTS AND APPLICATIONS 11 Note that the (2k + 2)-uple (a, b) subject to the conditions in the above definition can be interpreted as an ordered (k + 1)-uple of closed intervals in (a 0, b k ) with non-empty interior I = I 0 I k, where I k = [a k, b k ] (a 0, b k ), whose left endpoints and right endpoints appear in left-to-right order. Note that Int k is a contractible topological space, Int k R 2(k+1). Therefore, by applying the smooth Sing functor, it is a contractible Kan complex. Explicitly, (Int k ) l is the set of smooth maps l R, s a j (s), b j (s) for j = 0,..., k such that, for any s l, the following inequalities hold for any i = 0,..., k: a i (s) < b i (s), a i 1 (s) a i (s), b i 1 (s) b i (s). We will denote such an l-simplex with (I 0 (s) I k (s)) s l. The simplicial structure on Int is then given in the following way: given a morphism f : [m] [n] ([m], [n]) let f : m n the corresponding map between standard simplices. Then, by precomposing with f we get a map f from n-simplices to m-simplices, f ((I 0 (s) I k (s)) s n ) = (I 0 ( f (s)) I k ( f (s))) s n. The face and degeneracy maps d j : Int k+1 Int k and s j : Int k Int k+1, respectively, for k 0 and j = 0,..., k are given explicitly as follows: d j ((I 0 I k+1 )) = (I 0 Îj I k+1 ), s j ((I 0 I k )) = (I 0 I j I j I k ). In fact, Int is a complete Segal space, as shown in the following Proposition 3.1. Int is a complete Segal space and the inclusion Int is an equivalence of complete Segal spaces. Proof. The fact that each level Int k of Int is contractible implies that Int k h h Int1... Int 1 Int 0 Int 0 is a weak equivalence, which by Lemma 2.17 is equivalent to the Segal condition. It also ensures completeness and the fact that the given inclusion is a level-wise equivalence. Definition 3.2. We define the n-fold simplicial space Int n,..., as Int n,..., = (Int ) n. An element of the space Int n k 1,...,k n will be denoted by I = (a, b) = (I i 0 I i k i ) i=1,...,n. Lemma 3.3. Int n,..., is a contractible complete n-fold Segal space. Proof. This is an immediate consequence of Proposition 3.1. We now introduce the following maps, which we will use in the next step of the construction of Bord n as a complete n-fold Segal space. For k 0, define the boxing map by the following formula: B : Int k Int 0, I(s) = (I 0 (s) I k (s)) s l B(I) = (a 0, b k ). The notation B(I(s)) s l l will be used to denote the total space of B(I) l. The definition of the boxing map can be straightforwardly extended to the n-fold Segal space Int n,..., as follows: B : Int n k 1,...,k n Int n, I = (I i 0 I i k i ) i=1,...,n B(I) = (a 1 0, b 1 k 1 ) (a n 0, b n k n ). Another map that we will use is the box rescaling map ρ, which, for any I Int k1,...,k n, is the map ρ(i) : B(I) (0, 1) n defined by the restriction of the product of the affine maps R R sending a i 0 and b i k i to 0 and 1 respectively, for i = 1,..., n.

12 12 ANDREA TIRELLI 3.2. The n-fold Segal space PBord n. In this subsection we will focus on the core of the construction of the (, n)-category of cobordisms. Indeed, we will define an n-fold Segal space PBord n as a limit of simplicial spaces PBord V n, where the parameter V is a finite dimensional vector space, and in order to get Bord n as a complete n-fold Segal space we will only have to apply the completion functor defined in the previous section. We will define the PBord V n in several steps: first, we will present it simply as an n-fold simplicial set; then, we will endow each level with the structure of a Kan complex and finally we will prove that the two defined structures are compatible, giving rise to an n-fold simplicial space. Let us fix some notation: for any subset S {1..., n}, we denote with π S the projection π S : R n R S on the coordinates parametrised by S. Definition 3.4. Let V be a finite dimensional real vector space, which we identify with R r for some r, and fix n 1 and k 1,..., k n 0. Then, we define (PBord V n ) k1,...,k n to be the collection of tuples (M, I = (I0 i Ik i i ) n i=1 ), such that: (1) I = (I0 i Ik i i ) n i=1 Intn k 1,...,k n ; (2) M is a closed and bounded n-dimensional submanifold of V B(I) and the composition π : M V B(I) B(I) is a proper map; (3) for every S {1,..., n} let p S be the composition of π with the projection π S to the S-coordinates. Then, for every 1 i n and every 0 j i k i, at every x p 1 {i} (Ii j i ) the map p {i,...,n} is submersive. Remark 3.5. One way on interpreting (PBord V n ) k1,...,k n is the following: its elements are collections of k 1 k n composed bordisms, with k i composed bordisms with collars in the i-th direction. For more details on this interpretation, see [CS15, Remark 5.2] and [CS15, Proposition 5.4]. In order to define the space structure on each level set we proceed in a similar way as in the construction of the Segal space Int, i.e. by taking smooth singular chains. To do so we need to endow each set (PBord V n ) k1,...,k n with a topology. This is pursued as follows: (1) identify the set of closed n-dimensional submanifolds of V (0, 1) n with the quotient Sub(V (0, 1) n ) [M] Emb(M, V (0, 1) n )/Diff(M), where Emb(M, V (0, 1) n ) stands for the set of embeddings form M to V (0, 1) n and the disjoint union is over all diffeomorphism classes on n-dimensional manifolds; (2) endow the set Emb(M, V (0, 1) n ) with the Whitney C -topology, see [OR98, 9.3] for a definition; (3) if M is an n-dimensional submanifold of V (0, 1) n, define the neighbourhood basis at M to be the collection U M = {U K,W } K CV,W N (M,ι) given by U K,W = {N Sub(V (0, 1) n ) : N K = j(m) K, j W }, where C V is the collection of compact submanifolds of V (0, 1) n and N (M,ι) is the collection of neighbourhoods of the inclusion ι: M V (0, 1) n, obtaining a topology on Sub(V (0, 1) n ) Int n k 1,...,k n where we view Int n n 1,...,n n as a topological subspace of R 2(k+1) ; (4) identify an element (M, I) (PBord V n ) k1,...,k n, whose underlying manifold is the image of an embedding ι : M V B(I), with the element ([ϕ ι], I) Sub(V (0, 1) n ) Int n k 1,...,k n, where ϕ : V B(I) V (0, 1) n is the diffeomorphism ϕ := (Id V, ρ(i)); (5) topologise (PBord V n ) k1,...,k n using the inclusion constructed above (PBord V n ) k1,...,k n Sub(V (0, 1) n ) Int n k 1,...,k n. With the topological space structure on (PBord V n ) k1,...,k n in hand, we could define the space structure by taking continuous singular chains, but this is not the most suitable definition for our purposes and we redirect the reader to [CS15] and [GRW10] for a more detailed discussion, which also justifies the following definition. Definition 3.6. An l-simplex in (PBord V n ) k1,...,k n is given by tuples (M, I(s) = (I i 0(s) I i k i (s)) i=1,...,n ) s l such that: (1) I(s) = (I i 0(s),..., I i k i (s)) s l is an l-simplex of Int n k 1,...,k n ; (2) M is a closed and bounded (n + l)-dimensional submanifold of V B(I(s)) s l l such that

13 (, n)-categories, FULLY EXTENDED TQFTS AND APPLICATIONS 13 (a) the composition π : M V B(I(s)) s l B(I(s)) s l of the inclusion with the projection is a proper map, (b) its composition with the projection on l is a submersion M l ; (3) denoting with p S the composition of π with the projection on the S-coordinates, for any S {1,..., n}, for every 1 i n and 0 j i k i, at every point x p 1 i (Ij i i ) the map p {i,...,n} is submersive. Remark 3.7. If we set l = 0 in the previous definition, we get exactly Definition 3.4. Moreover, we see that, for every s l, the fibre M s of M l determines an element of (PBord V n ) k1,...,k n (M s ) = (M s V B(I(s)), I(s)), and we will use the notation π s : M s B(I(s)) for the composition of the embedding with the projection. Let us now construct (PBord V n ) k1,...,k n as a space. To this purpose, consider a map f : [m] [l] ([m], [l]) and let f : m l the associated map between the geometric realisations m and l. We define the map f sending an l-simplex in (PBord V n ) k1,...,k n to an m-simplex in the following way: the m-simplex in Int n k 1,...,k n is obtained by precomposing with f f ((I i 0(s) I i k i (s)) s l ) = (I i 0( f (s)) I i k i ( f (s))) s m given a (n + l)-dimensional submanifold of V B(I(s)) s l l, its image under the map f is obtained by the pullback of M l under the map f : f M = M f (s). s m The proof of proposition below follows directly from the way (PBord V n ) k1,...,k n is constructed as a simplicial space, so it is left to the reader. Proposition 3.8. (PBord V n ) k1,...,k n is a Kan complex. Since we want to consider arbitrary closed and bounded manifolds M, we need to take the limit over all finite dimensional vector spaces V. In order to do this, we give the following definition. Definition 3.9. Fix the countably infinite dimensional vector space R. Then, PBord n = lim PBord V n. V R Our purpose now is to make the collection of spaces (PBord n ),..., into an n-fold simplicial space. Firstly, we need to put a structure of n-fold simplicial set, i.e. define a functor op op sset: in order to do so, we need to extend the assignment [k i ] (PBord n ) k1,...,k n to a functor from op, for 1 i n, in the following way: for every 1 i n, consider a morphism g : [m i ] [k i ] ([m i ], [k i ]) and let g = Πg i ; then we define the map g : (PBord n ) k1,...,k n (PBord n ) m1,...,m n that sends an element (M V B(I), I = (I i 0 I i k i ) i=1,...,n ) to ( ) g M = π 1 (B(g (I))) V B(g (I)), g (I) = (I g(0) Ig(m i ) i) i=1,...,n. In this way, we have obtained a simplicial set structure on PBord n. We can also define the map g on l-simplices: the image of an element is (M V B(I(s)) s l l, I(s) = (I i 0 I i k i ) i=1,...,n (s)) (g M = π 1 (B(g (I(s)) s l l )) V B(g (I(s)) s l l ), g (I)(s) = (I g(0) I i g(m i) ) i=1,...,n(s)). The following result, whose proof can be found in [CS15, Proposition 5.18] and is thus omitted, says that the spatial and simplicial structure on (PBord n ),..., are compatible. Proposition Given any f : [l] [p] and g i : [m i ] [k i ] for 1 i n, the induced maps f and g commute, which means that we have an n-fold simplicial space structure on (PBord n ),...,. Proposition (PBord n ),..., is an n-fold Segal space.

14 14 ANDREA TIRELLI Sketch of the proof. We only spell out which are the conditions that need to be verified to have a proof of the above statement. Firstly, one proves that the Segal condition is satisfied, and to do so one just needs to show that (PBord n ) k1,...,k i+k i (PBordn ),...,kn k1,...,k i,...,k n (PBord n ) k1,...,k,...,kn. i (PBord n) k1,...,0,...,kn Moreover, the second step of the proof is to show that, for every i and every k 1,..., k i 1, the (n i)-fold Segal space (PBord n ) k1,...,k i 1,0,,..., is essentially constant: to do so, one shows that the degeneracy inclusion map (Pbord n ) k1,...,k i 1,0,0,...,0 (PBord n ) k1,...,k i 1,0,k i+1,...,k n admits a deformation retraction and thus is a weak equivalence. For the construction of such a retraction, see [CS15, Proposition 5.20] The (, n)-category Bord n and its symmetric monoidal structure. In the previous subsection we constructed (PBord n ),..., as an n-fold Segal space. Since our model for (, n)-categories are complete n-fold Segal space we give the following definition. Definition The (, n)-category of cobordisms Bord n is the completion PBord n of PBord n, which is a complete n-fold Segal space. Remark For n = 1, 2, PBord n is already complete, thanks to the classification theorems for 1- and 2-manifolds. On the other hand, for n 6, PBord n is not complete, see [Lur09, Warning 2.2.8] for more details. In Definition 1.1 the ordinary category Bord n was endowed with a symmetric monoidal structure by taking disjoint unions of manifolds. This can be carried out also in the context on n-fold Segal space in the following way: we construct a sequence of n-fold Segal spaces (PBord V n [m]) which form a Γ-object in SsSp n and thus, by Lemma 2.39, endow Bord n with a symmetric monoidal structure. Definition Consider a finite dimensional vector space V and fix n 1. For every k 1,..., k n 0 and m 1, let (PBord V n [m]) k1,...,k n be the collection of tuples (M 1,..., M m, (I i 0 I i k i ) i=1,...,n ), where, for each j = 1,..., m, (M j, (I0 i Ik i i ) i=1,...,n ) is an element of (PBord V n ) k1,...,k n and M 1,..., M m are disjoint. As in Definition 3.9 we take the limit over all V R and apply the completion functor to this n-fold Segal space, obtaining the complete n-fold Segal space Bord n [m]. It is clear that Bord n [1] = Bord n. Proposition There is a functor Γ CSeSp n, m Bord n [m] endowing Bord n with a symmetric monoidal structure. Proof. By Lemma 2.39, to prove the above result it suffices to define a functor Γ SeSp n, sending m to PBord n [m]. Firstly, given a morphism f : m l Γ( m, l ) we define the morphism PBord n [m] PBord n [l], (M 1,..., M m, I) ( M α,..., α f 1 (1) α f 1 (l) M α, I). Thus, we get a functor Γ SeSp n and we are left to show that the property in Definition 2.37 holds, i. e. we need to show that the map γ β : PBord n [m] (PBord n [1]) m 1 β m is an equivalence of n-fold Segal spaces. The map i β m is a level-wise inclusion and it remains to show that level-wise it is a weak equivalence. To do so, it suffices to construct a path from a generic element (M 1,..., M m, I) (PBord n [1]) m to an element in the image of the map 1 β m γ β, which induces a strong homotopy equivalence between the two spaces. First, there is a path to an element for which all (M α ) have the same specified intervals by composing all except one with a suitable smooth rescaling. Secondly, there is a path with parameter s [0, 1] given by composing the embedding M α V B(I) with the embedding into R V B(I) given by the map V R V, v (sα, v).

15 (, n)-categories, FULLY EXTENDED TQFTS AND APPLICATIONS Cobordisms with additional structure. In this subsection, we construct an enriched version of the n-fold Segal space Bord n : we equip bordisms with additional structures, such as an orientation or a framing, which turn out to be fundamental in the study of fully extended topological quantum field theories. Let us start with the definition of structured manifold. Definition Let M be a smooth n-dimensional manifold, X a topological space and E X a topological n-dimensional vector bundle. An (X, E)-structure on M is the datum of: a map f : M X an isomorphism of vector bundles φ : T M f (E). We will denote with Man (X,E) n the set of (X, E)-structured manifolds of dimension n. Example Consider a topological group G with a continuous homomorphism e : G O(n). By abuse of notation, we will denote with e : BG BGL(R n ) the induced map between the classifying spaces. Let us consider the vector bundle (R n EG)/G, on BG, where EG denotes the total space of the universal bundle on BG. A (BG, EG)-structure on an n-dimensional manifold is called a G-structure on M. The set of G-structured manifolds will be denoted by Man G n. The three examples of G-structures on manifolds which we will be mostly interested in are the following: G is the trivial group, which implies that X = BG = and E is trivial; in this case a G-structure on M is a trivialization of its tangent bundle, i.e. a framing; G = O(n) and e = Id O(n) ; in this case, given that the group Diff(R n ) retracts onto O(n), and O(n)-structured manifold is simply a smooth manifold; G = SO(n) and the map e is the inclusion; an SO(n)-structured manifold is a oriented manifold. Remark Note that, given two (X, E)-structured manifolds M and N, it is possible to define morphisms between them, so that we can turn the set Man (X,E) n into a category Man (X,E) n. For the sake of brevity, we will not define the morphisms in general, for a treatment of which we refer to [CS15, Definition 9.6]. We only point out that, in the case of G = SO(n) a morphism between oriented manifold is an orientation preserving map. Moreover, in the case of G = {1}, a morphism between two framed manifolds M and N is a pair (f, h) such that f : M N is an embedding and h is a homotopy between the trivialization of T M induced by the framing of M and that induced by the pullback along f of the framing on T N. We will now fix a (X, E)-structure and define the (complete) n-fold Segal space of (X, E)-structured cobordisms Bord (X,E) n. The steps of the construction are almost the same as for Bord n : we define the simplicial and spacial structures, which turn out to be compatible and then we apply the completion functor. Definition Fix a finite dimensional vector space V, n 1 and k 1,..., k n 0. We then let (PBord (X,E),V n ) k1,...,k n be the collection of tuples (M, f, φ, (I i 0 I i k i ) i=1,...,n ), where: (1) (M, (I i 0 I i k i ) i=1,...,n ) is an element of (PBord V n ) k1,...,k n, (2) (f, φ) is an (X, E)-structure on on the manifold M. The level sets can be endowed with a spatial structure in the following way. Definition An l-simplex in (PBord (X,E),V n ) k1,...,k n is given by a tuple (M, f, φ, I(s) = (I i 0(s) I i k i (s)) s l ) such that I = (I0 i Ik i i ) i=1,...,n l is an l-simplex of Int n k 1,...,k n, M is a closed and bounded (n + l)-dimensional submanifold of V B(I(s)) s l l such that (1) the composition π : M V B(I(s)) s l l B(I(s)) s l l of the inclusion with the projection is proper, (2) its composition with the projection onto l is a submersion. denoted with p S {1,..., n}, for every 1 i n and 0 j i k i, at every point x p 1 i submersive. the composition of π with the projection on the S-coordinates, for any S (Ij i i ) the map p {i,...,n} is Remark From the previous definitions, we see that the only difference between PBord V n and PBord (X,E),V n is the additional requirement that the manifold M is equipped with an (X, E)-structure. Thus, all the results and proofs given for PBord V n hold for PBord (X,E),V n. Following the principle of the previous remark, we construct the complete n-fold Segal space of structured cobordisms as follows: first, we take the limit over all finite dimensional vector spaces V R and define PBord (X,E) n = lim PBord (X,E),V n. V R

16 16 ANDREA TIRELLI Then, we take the completion and get the following definition. Definition The (, n)-category of (X, E)-structured cobordisms is Bord (X,E) n = PBord (X,E) n, which can be endowed with a symmetric monoidal structure by using a (X, E)-structured version of Definition In particular, when (X, E) = (, R n ), we will use the notation Bord fr n Fully extended topological quantum field theories. We now have all we need to give a rigorous definition of a fully extended topological quantum field theory. Definition Let n 1 and C be a symmetric monoidal (, n)-category. A C-valued fully extended n-dimensional topological quantum field theory is a symmetric monoidal functor of (, n)-categories with source Bord n and target C, F : Bord n C. We will also be interested in fully extended TQFTs that have as source the (, n)-category of (X, E)- structured cobordisms. Definition A fully extended n-dimensional (X, E)-topological quantum field theory is a symmetric monoidal functor of (, n)-categories with source the complete n-fold Segal space Bord (X,E) n, F : Bord (X,E) n C. Remark The most important cases of the previous definition are when (X, E) is equal to either (, R n ) - such theories are called framed - or (BSO(n), ESO(n)) - such theories are called oriented. The main result on fully extended TQFTs is the aforementioned Cobordism Hypothesis: loosely speaking, it says that a fully extended n-dimensional framed TQFT Z : Bord fr n C is completely determined by the evaluation at a point and that, for every object X in a symmetric monoidal (, n)-category with duals C, there is a field theory Z such that X = Z( ). More rigorously, we have the following result. Theorem 3.26 (Cobordism Hypothesis). Let C be a symmetric monoidal (, n)-category. The evaluation functor Fun(Bord fr n, C) C : Z Z( ) factors through the fully dualizable sub- -groupoid (C fd ) 0 of C and the induced functor is a Dwyer-Kan equivalence. Fun(Bord n, C) (C fd ) 0 Remark Notice that, in the previous statement, we have used the following notation: given an (, n)- category C, we have denoted with (C) 0 the sub- -gropuoid of C which is obtained from C discarding all non-invertible morphisms. In fact, (C) 0 = τ 0 (C), where τ 0 is the 0-th truncation functor. 4. Examples and applications In the last part of the present work we want to give some examples and applications of the concepts introduced in the previous sections. In particular, we would like to see how heuristic arguments concerning higher categories can be rephrased in the rigorous language of complete n-fold Segal spaces. Note that this is not always an easy task and, for this reason, we will present the heuristic arguments anyway and, when possible, make them rigorous. We will focus our attention on using the Cobordism Hypothesis (Theorem 3.26) to construct fully extended TQFTs, which boils down to finding fully dualizable objects in certain (, n)-categories. We will consider two different cases Fully dualizable objects in Bord fr 2 : an informal proof. In this subsection we give an example of an explicit calculation concerning fully dualizable objects in an (, 2)-category. In particular, we want to show a possible application of Proposition 2.46 and use it to prove that any object in the (, 2)-category Bord fr 2 of framed 2-bordisms is fully dualizable. We first give the proof of this result by using the heuristic, following the convention of [DSS13]. We will later explain how the proof needs to be adjusted to fit in the framework of Segal spaces. First, let us give some definitions and fix some notation. Given two non-negative integers k and n such that 0 k < n, we define an n-framed k-manifold to be a k-dimensional manifold M together with a trivialisation of the stabilised-up-to-dimension-n tangent bundle T M R n k. Heuristically, we could define description of the Bord fr 2 the (, n)-category HBord fr n as follows: objects: n-framed 0-manifolds; 1-morphisms: n-framed bordisms between objects;

17 (, n)-categories, FULLY EXTENDED TQFTS AND APPLICATIONS 17 2-morphisms: n-framed 2-manifold bordisms between 1-morphisms;... n-morphisms: n-framed n-manifold bordisms between (n 1)-morphisms; (n + 1)-morphisms: diffeomorphisms of framed n-bordisms; (n + 2)-morphisms: isotopies beteween n-morphisms;... In this setting, we also want to give a convenient way to present an n-framing of a k-manifold. This is done by using normally framed immersions: let us consider a k-manifold M for k n; then, an n-framing of k can be given by an immersion ι : M R n together with a normal framing, i.e. a trivialisation φ of the normal bundle ν(ι) of the immersion. The normally framed immersion (ι, φ) that we obtain in this way gives an n-framing of M as follows: T M R n k = T M ν(ι) = R n. In order to prove full dualizability in this context we will always use the above normally framed immersion notation: we leave completely implicit the induced n-framing and, in the case the immersion has codimension 1, we will specify the normal framing by a grey corona on the immersed manifold. Remark 4.1. A remark is in order to explain how we can extend the description of n-framings of k-manifolds to the case of manifolds with boundary and corners, which turns out to be fundamental, given our definition of Bord fr 2 in terms of bordisms between bordisms. Let us start with the case of a bordism M with boundary but without corners: each boundary component needs to be labelled in or out according to whether it is part of the source or the target of the bordism. We now explain how, given a normally framed immersion (ι, φ), we can induce a normally framed immersion on each component of the boundary: let us consider the case of an incoming component N ( M) in ; then, the immersion is the restriction of the immersion of M and the framing is given by the pair (l, s) ν(n, R n ), where l is a section of the normal bundle of N pointing into the bulk of the manifold M and s ν(m, R n ) is a given normal framing on M. In a similar way, an outgoing component of the boundary inherits the normal framing ( l, s), which means that the first normal frame point out of the bulk of the bulk of the manifold. Note that, in the case of manifold s with boundary and corners, when we draw a normal framing, we also need to specify which parts of the boundary are incoming and which are outgoing. The first will be undecorated (as if there are arrows pointing into the bulk), the latter will be indicated with arrows pointing out of the bulk of the manifold. Example 4.2. It is possible to prove, [DSS13], that, up to homotopy, all normally framed immersed circles in R 2 are the ones drawn in the following picture. Indeed, if we pick a fixed framed circle, then all framings on the circle are given by maps from S 1 to SO(2) (rotating the framing everywhere by the target). Thus, isomorphism classes of framings correspond to homotopy classes of such maps, which indeed form the fundamental group π 1 (SO(2)), which is isomorphic to Z. Remark 4.3. It is important to note that not every n-framed n-manifolds can be realised with normally framed immersion. An easy example, already when n = 1, is given by S 1, which has a unique up to diffeomorphism 1-framing, but can not be immersed in R. On the other hand, the Hirsch-Smale immersion theorem [Hir59] ensures that very n-framing of an (n k)-manifold can be realised by a normally framed immersion, if either k > 0 or each component of the manifold is not closed. We now explicitly describe the duals and the adjunctions in Bord fr 2. First, let us start with understanding the objects of such a higher category. Up to 2-framed diffeomorphism, there are two 2-framed points, pt + and pt, whose classes are represented by the following pictures. We prove that these two points are dual to each other as follows: we need to exhibit an evaluation bordism ev : pt + pt and a coevaluation bordism coev : pt pt + satisfying the zig-zag equation, which is how the duality condition is often called. We see that such 2-framed 1-bordisms are represented by the following pictures.

18 18 ANDREA TIRELLI Remark 4.4. Note that, in the pictures, we use the conventions introduced in Remark 4.1: the first bordism has no outgoing arrows at its boundary points, which means that both of them are the source of the morphism ev, whose target is thus the empty set. Moreover, when representing morphisms from or to disjoint unions of two points, the first one is drawn on top of the picture and the second one at the bottom. Since Bord fr 2 is symmetric monoidal, any right dual is also a left dual. Thus, we are left to show that the evaluation bordism ev has both right and left adjoints. Let us start with the left adjoint: ev L is a 2-framed 1-bordism from to pt + pt, ev L : pt + pt, and we say that it is given by the following bordism. To prove that this is in fact the felt adjoint of the evaluation we need to provide the unit and counit 2- morphisms of the adjuction ev L ev. The unit u 1 is a 2-morphism Id ev ev L and it is given by the following 2-framed 2-bordism. : Id Note that in the picture of the 2-manifold u 1, the corona indicates that the boundary is outgoing. That boundary is the circle with the outward trivialization of its normal bundle, and so the two uses of the corona are consistent, as mentioned previously. For what concerns the counit, it is a 2-morphism v 1 : ev L ev Id pt+ pt and it is represented by the following 2-framed surfaces with cuspidal corners. Remark 4.5. Comparing the two 2-morphisms v 1 and u 1, we can understand how the conventions in Remark 4.1 work in practice: the fact that the grey corona is drawn on the boundary of the 2-framed 2-bordism u 1 means that the source and the target of u 1 are the empty set and the whole boundary of the disk respectively. On the other hand, only two out of the four components of the boundary of the 2-bordism v 1 have a grey corona: these constitute the target of the 2-morphism and the remaining ones are the source. A similar discussion can be done for the right adjoint ev R of the evaluation bordism. A possible choice for it is the one shown in the following figure. Given that we allow immersed manifolds and not only embedded ones, we can deform ev R and allowing it to self-intersect as follows. In this case, the unit on the adjunction ev ev R are the 2-bordism u 2 : ev ev R shown in the following picture and the 2-bordism v 2 : ev ev R drawn as follows Fully dualizable objects in Bord fr 2 : a rigorous proof. The above proof is direct and rigorous except for the fact that we have used a heuristic model for the (, 2)-category Bord fr 2. We will now repeat the proof using the 2-fold Segal space model for such a higher category and we will later explain how the two proofs are related. The pictures below are taken from [CS15].

19 (, n)-categories, FULLY EXTENDED TQFTS AND APPLICATIONS 19 We first note that PBord fr 2 = Bord fr 2. Thus, an object in Bordfr 2, in the sense explained in Subsection 2.5 of Section 2, is an element of the form (M V (a, b) (c, d), F, (a, b), (c, d)), where F is a framing of M. Moreover, condition 3 in Definition 3.4 implies that M is a disjoint union of manifolds that are diffeomorphic to the open square (0, 1) 2. We thus are left to prove full dualizability of the object Q + = ((0, 1) 2 (0, 1) 2, F, (0, 1), (0, 1)), where F is a framing of (0, 1) 2. We can draw this element as follows, and it should be thought as the 2-framed point pt + introduced above. We then want to prove that the dual of such a framed manifold, which we call Q, is in fact the element corresponding to the 2-framed point pt, which is drawn in the next figure. In this case, the elements that witness this duality belong to the space (Bord fr 2 ) 1,0, which, by Example 2.34 and Subsection 2.5, is in fact the space of 1-morphisms. In particular, the evaluation 1-morphism ev pt+ is given by a strip, which is the open square (0, 1) 2 with the framing given by rotating the framing by π and is embedded in R (0, 1) 2. In fact, this is a 1-morphism in h 2 (Bord fr n ): it is depicted as follows. The coevaluation 1-morphism coev pt+ is given in a similar way: it is a strip, i.e. the open square (0, 1) 2 but the framing rotates in the opposite direction of the framing of the evaluation bordism ev pt+. It is then immediate to show that the duality condition is satisfied: indeed, the composition is connected by a path to a strip with the framing given by pulling back the ends of the strip to flatten it, as shown in the following figure. The above composition is homotopic to the same strip decorated with the trivial framing. Thus, thic composition is the identity in the homotopy category. We now exhibit the counit and the unit of the adjunction, which is the last step needed to prove the full dualizability of the element Q +. The counit is given by a cap with the framing coming from the trivial framing on the disk.