Download AN INTRODUCTION TO (∞,n)-CATEGORIES, FULLY EXTENDED

AN INTRODUCTION TO (∞, n)-CATEGORIES, FULLY EXTENDED TOPOLOGICAL
QUANTUM FIELD THEORIES AND THEIR APPLICATIONS
ANDREA TIRELLI
Contents
1. Introduction
1.1. Topological Quantum Field Theories
1.2. Extended TQFTs and higher categories
2. Complete Segal spaces as models for (∞, n)-categories
2.1. Simplicial sets, simplicial spaces and Kan complexes
2.2. Models for (∞, 1)-categories
2.3. Models for (∞, n)-categories
2.4. Completions, truncations and symmetric monoidal structures
2.5. Duals in (∞, n)-categories
3. The (∞, n)-category of cobordisms
3.1. The complete Segal space Int•
3.2. The n-fold Segal space PBordn
3.3. The (∞, n)-category Bordn and its symmetric monoidal structure
3.4. Cobordisms with additional structure
3.5. Fully extended topological quantum field theories
4. Examples and applications
4.1. Fully dualizable objects in Bordf2 r : an informal proof
4.2. Fully dualizable objects in Bordf2 r : a rigorous proof
4.3. An ∞-model for the Morita bicategory of a monoidal category
4.4. Full dualizability in Alg∞
1 (S)
4.5. Applications to fully extended TQFTs
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1. 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.
1.1. 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 Cobor
n is defined as follows:
• objects are compact oriented (n − 1)-dimensional manifolds;
• for any pair of objects in M, N ∈ Cobor
n a bordism from M to N is a compact oriented n-manifold B
with an oriented boundary ∂B = ∂B0 t ∂B1 , where ∂B0 ∼
= M is the manifold M with the opposite
orientation and ∂B1 ∼
= N;
• let Bordn (M, N ) be the set of bordisms from M to N ; we define HomCobor
(M, N ) := Bordn (M, N )/ ∼,
n
where ∼ is the equivalence relation that identifies two bordisms B and B 0 if there is an orientation
preserving diffeomorphism B → B 0 that restricts to diffeomorphisms ∂B0 → ∂B00 and ∂B1 → ∂B10 .
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Disjoint union of manifolds endows Cobor
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 : Cobor
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 Cobor
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 ).
1.2. 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 Cobor
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 Cobor
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 Cobor
n , whose objects are (oriented)
0-manifolds, 1-morphisms are (oriented) 1-manifolds with boundary, 2-morphisms are (oriented) manifolds
(∞, n)-CATEGORIES, FULLY EXTENDED TQFTS AND APPLICATIONS
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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 Θn -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].
2.1. 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• = (Xn )n≥1 and a map
Xn → Xm for any monotonically increasing map [m] → [n]. The image of the map fi : [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 fi : Xm+1 → Xm ,
for i = 1, . . . , m. If instead we consider the image of the map di : [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 di : Xm → Xm+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 di and fi for i = 1, . . . , n. Moreover,
di and fi satisfy the so called simplicial identities. Conversely, a sequence of sets (Xn )n≥1 equipped with
maps di and fi 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|,

,
G
|X| := 
Xn × |∆n |
∼,
n≥1
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ANDREA TIRELLI
obtained by interpreting each element in Xn 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:
G
N (C)n =
C(x0 , x1 ) × · · · × C(xn−1 , xn ).
x0 ,...,xn ∈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(xi−1 , xi ) with morphisms in C(xi , xi+1 ) and insertion of the identity map
Idxi 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 Bordn 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 Xn ∼
= sSet(∆n , X). Moreover, for any i = 1, . . . , n, we define the i-th horn Λin to be
the simplicial set given by
(Λni )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 Λin
gives exactly what one might expect: |∆n | is the standard topological n-simplex and |Λin | is the topological
space obtained by taking ∂|∆n | and deleting the face opposite to the i-th vertex. The simplicial sets ∆n
and Λin 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
Λin
X
f
∆n
Y
there exists a filling map g : ∆n → X that lifts the map ∆n → X,
Λin
X
g
∆n
f
.
Y
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,
X : ∆op −→ KComp.
(∞, n)-CATEGORIES, FULLY EXTENDED TQFTS AND APPLICATIONS
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2.2. 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 2.11. The reader should refer to Example 1.8 and Remark 2.8 for a justification of the above
hypothesis.
Definition 2.12. An (∞, 0)-category is a space.
Remark 2.13. 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 2.14. A Segal space X = X• is a simplicial space satisfying the following Segal condition: let
m, n ≥ 0 and consider the following commutative diagram
Xn+m
Xn
Xm
X0
induced by the maps [m] → [m + n], (0 < · · · < m) 7→ (0 < · · · < m) and [n] → [m + n], (0 < · · · <
h
n) 7→ (m < · · · < m + n). Then the induced map Xn+m → Xm × Xm is a weak equivalence, where ×hX0
X0
stands for the homotopy fiber product, which is a homotopical version of the usual pullback of spaces (see,
e.g., [Lur09, Remark 2.1.12]) 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 2.15. A straightforward example of Segal space is given by the nerve N (C), when C is a small
category in spaces.
Remark 2.16. 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 X0 as the set of objects.
Moreover, given x, y ∈ X0 , we can interpret
h
h
X0
X0
X(x, y) := {x} × X1 × {y}
as the ∞-groupoid of (1-)morphisms from x to y.
Lemma 2.17. 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 fk : [1] → [l] by (0 < 1) 7→ (k − 1 < k):
then the map induced by f1 , . . . , fl ,
h
h
X0
X0
Xl → X1 × · · · × X1
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
h
Xn+m → Xm × Xm ,
X0
which, thanks to the definition of the maps fk is the weak equivalence in the statement of the Segal
condition.
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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 2.18. 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 2.1.18], 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 X0 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 2.19. The homotopy category h1 (X) of a Segal space X is the ordinary category whose objects
are points of X0 and whose morphism set between two objects x, y ∈ X0 is given by
h1 (X) = π0 (X(x, y)).
Composition of morphisms is defined by the following sequence of maps:
h
h
h
h
h
h
h
{x} × X1 × {y} × {y} × X1 × {z} −→ {x} × X1 × X1 × {z}
X0
X0
X0
X0
∼
X0
X0
h
h
X0
X0
h
h
X0
X0
X0
←− {x} × X2 × {z}
−→ {x} × X1 × {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 2.20. Let X = X• be a Segal space and let f be an element of X1 . Let x and y be the endpoints
of f , i.e. the images of f under the maps X1 ⇒ X0 . Then, f is said to be invertible if its image under the
composition of the maps
h
h
h
h
{x} × X1 × {y} −→ {x} × X1 × {y} −→ π0 {x} × X1 × {y} = h1 (X)(x, y)
X0
X0
X0
X0
X0
X0
is an invertible morphism in the homotopy category of X. We will denote by X1inv the subspace of X1 given
invertible morphisms.
Remark 2.21. Note that the map φ : X0 → X1 factors through the subspace X1inv . To see this, consider
an element x in X0 and let y = φ(x), which can be mapped to the space {x} ×X0 X1 ×X0 {x}. Thus,
by composing with the two obvious maps we get a map X0 → Homh1 (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 Idx .
Definition 2.22. A Segal space is complete if the map X0 → X1inv 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. Indeed the following construction produces
examples of Segal spaces which are not complete.
Example 2.23. 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 X0 is a point, whereas X1 = X1inv = G is not homotopically equivalent
to X0 unless G is trivial.
Now we have all we need to define our chosen model of (∞, 1)-categories.
Definition 2.24. 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.
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Example 2.25. Let (C, W) be a relative category and consider the simplicial object C• in Cat given by
Cn = 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 CnW 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 (CnW ).
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 2.26. 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 X1inv :=
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 X1inv and X0 respectively. This proves our claim.
2.3. 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 2.27. Let n ≥ 1. An n-fold simplicial space X = X•,...,• is a functor
X : ∆op × · · · × ∆op → KComp.
Remark 2.28. Analogously to Remark 2.2, we can spell out the above definition explicitly: an n-fold
simplicial space is a collection X•,...,• = (Xk1 ,...,kn )k1 ,...,kn ≥0 of Kan complexes equipped with a map
Xk1 ,...,kn → Xl1 ,...,ln for any n-uple of maps [li ] → [ki ], where [li ] → [ki ] is a morphism in Hom∆ ([li ], [ki ])
for i = 1, . . . , n. Moreover, a map of n-fold simplicial spaces f : X•,...,• → Y•,...,• is simply a collection of
maps of simplicial sets Xk1 ,...,kn → Yk1 ,...,kn satisfying the obvious naturality conditions.
Definition 2.29. An n-uple Segal space is an n-fold simplicial space such that for any 1 ≤ i ≤ n and for
any k1 , . . . , ki−1 , ki+1 , . . . , kn ≥ 0, the simplicial space
Xk1 ,...,ki−1 ,•,ki+1 ,...,kn
is a Segal space.
To define n-fold Segal space we need an additional property.
Definition 2.30. Let X•,...,• be an n-fold simplicial space. We will say that X is essentially constant if the
map of n-fold Segal spaces X0,...,0 → X•,...,• , where X0,...,0 is viewed as a constant n-fold Segal space, given
by the degeneracy maps, is a weak equivalence.
Definition 2.31. An n-fold Segal space is a n-uple Segal space such that, for any 1 ≤ i ≤ n and any
k1 , . . . , ki−1 ≥ 0, the n − i-uple Segal space
Xk1 ,...,ki−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 SeSpn .
Remark 2.32. 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 X0 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 2.22.
Definition 2.33. An n-fold Segal space X•,...,• is complete if, for any 1 ≤ i ≤ n, the following condition is
satisfied: for any k1 , . . . , ki−1 ≥ 0, the Segal space
Xk1 ,...,ki−1 ,•,0,...,0
is complete.
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 Xk1 ,...,kn should be interpreted as a composition of ki 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 2.34. 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 X1,1 can be
considered as a higher categorical analogue of the previous diagram
•
•
•
•
where: the bullets are elements of X0,0 ; the horizontal arrows are 1-morphisms, which are the images under
the source and the target maps in the first direction X1,1 ⇒ X1,0 ; the vertical arrows, which are essentially
the identity maps up to homotopy since X0,1 ∼ X0,0 , are the images under the source and the target maps
in the second direction X1,1 ⇒ X0,1 . 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.
2.4. 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
b called the completion of
an n-fold Segal space X, to construct from it a complete n-fold Segal space X,
X. Indeed, for the case n = 1, in [Rez01] Rezk gave an explicit construction of the completion functor: to
b and a map iX : X → X,
b which is a Dwyer-Kan
every Segal space X it associates a complete Segal space X
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 −→ SeSpn−1 .
Suppose, by induction, that the completion functor is defined for (n − 1)-fold Segal spaces and apply it to
each of the Xn . 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 X̂•,...,• .
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
τk : SeSpn → SeSpk
(∞, n)-CATEGORIES, FULLY EXTENDED TQFTS AND APPLICATIONS
9
that sends a given n-fold Segal space X•,...,• to
τk (X) = X•, . . . , •, 0, . . . , 0 .
| {z } | {z }
k times
n−k times
It follows from Definition 2.33 that, if X is complete, then so is its k-truncation τk (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 2.35. Segal’s category Γ is the category whose objects are given by the sets
hmi = {0, . . . , m}
for any m ≥ 0 and whose morphisms between two objects hmi and hli are functions f : hmi → hli such that
f (0) = 0.
Remark 2.36. For any m ≥ 0 and any 1 ≤ k ≤ n, consider the morphisms γk ∈ HomΓ (hmi, h1i) given by
γk : hmi → h1i, γk (l) = δkl .
These morphisms are called the Segal morphisms.
Definition 2.37. 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
A : Γ → (C)SeSpn
such that, for any m ≥ 0, the induced map


Y
A
γk  : Ahmi → (Ah1i))m
1≤k≤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 Ah1i is X.
Remark 2.38. It is possible to prove that symmetric monoidal structures, i.e. functors Γ → (C)SeSpn 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 → X̂ 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 Bordn as a complete n-fold Segal space. E,
we obtain the following
Lemma 2.39. If A : Γ → SeSp is a symmetric monoidal n-fold Segal space, then
 : Γ → CSeSpn ,
\
hmi 7→ Ahmi
is a symmetric monoidal complete n-fold Segal space.
2.5. 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 → SeSpn−1 . Let ObX be the set of points of X0,...,0 . For any
x0 , . . . , xk ∈ ObX, we define map1X (x0 , . . . , xk ) to be the homotopy fibre of the map (n−1)-fold Segal spaces
Xk → (X0 )k+1 at (x0 , . . . , xk ). It is possible to check that map1X (x0 , . . . , xk ) is an (n − 1)-fold Segal space,
and thanks to the Segal condition, we have the weak equivalence
∼
map1X (x0 , . . . , xk ) −→ map1X (x0 , x1 ) ×h · · · ×h map1X (xk−1 , xk ).
r
Inductively, for 1 ≤ r ≤ n and f0 , . . . , fl ∈ mapr−1
X (x0 , . . . , xk ), let mapX (f0 , . . . , fl ) be the homotopy fibre
of (f0 , . . . , fl ) of the map of (n − r)-fold Segal spaces
r−1
l+1
mapr−1
.
X (x0 , . . . , xk )l → (mapX (x0 , . . . , xk )0 )
Then, a 1-morphism is an object f : x → y in map1X (x, y) for objects x, y ∈ ObX. An r-morphism
f : x → y is an object of maprX (x, y), where x and y are (r − 1)-morphisms. Two r-morphisms f, g : x → y
10
ANDREA TIRELLI
are homotopic if they lie in the same component of maprX (x, y)0,...,0 .
In Definition 2.19, we defined the homotopy category h1 (X) of a Segal space. We now want to extend this
construction to an n-fold Segal space, for arbitrary n ≥ 1.
Definition 2.40. Let X be an n-fold Segal space. The homotopy category h1 (X) of X is the (ordinary)
category with objects ObX and, for each pair x, y ∈ ObX, h1 (X)(x, y) = π0 map11 (x, y)0,...,0 is the set of
path components of the space map1X (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 2.41. Fix n ≥ 2. Given a n-fold Segal space X, its homotopy 2-category h2 (X) is a bicategory
defined as follows:
• Ob(h2 (X)) = ObX,
• for each pair of objects x, y ∈ Ob(h2 (X)), h2 (x, y) = h1 map1X (x, y).
In order to define the concepts of dual of an object and adjoint of an r-morphism in the context of nfold 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 2.42. 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 h2 (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, map1X (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 2.43. Let X be a symmetric monoidal n-fold Segal space. We say that X has duals for objects
if h1 (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 2.44. 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 2.45. 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 2.46. 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 Bordn , which is our higher categorical
model for the category of cobordisms in Definition 1.1. We will define Bordn 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 Bordn 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.
3.1. The complete Segal space Int• . In order to define the complete n-fold Segal space Bordn 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,
Intk = {(a, b) = (a0 , . . . , ak , b0 , . . . , bk ) : aj < bj , for 0 ≤ j ≤ k, and
aj−1 ≤ aj and bj−1 ≤ bj for 1 ≤ j ≤ k} ⊂ R2(k+1) .
(∞, 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 (a0 , bk ) with non-empty interior I = I0 ≤ · · · ≤ Ik , where
Ik = [ak , bk ] ∩ (a0 , bk ), whose left endpoints and right endpoints appear in left-to-right order.
Note that Intk is a contractible topological space, Intk ⊆ R2(k+1) . Therefore, by applying the smooth Sing
functor, it is a contractible Kan complex. Explicitly, (Intk )l is the set of smooth maps
|∆l | → R, s 7→ aj (s), bj (s)
for j = 0, . . . , k such that, for any s ∈ |∆l |, the following inequalities hold for any i = 0, . . . , k:
ai (s) < bi (s), ai−1 (s) ≤ ai (s), bi−1 (s) ≤ bi (s).
We will denote such an l-simplex with (I0 (s) ≤ · · · ≤ Ik (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 ∆ ((I0 (s) ≤ · · · ≤ Ik (s))s∈|∆n | ) = (I0 (|f |(s)) ≤ · · · ≤ Ik (|f |(s)))s∈|∆n | .
The face and degeneracy maps dj : Intk+1 → Intk and sj : Intk → Intk+1 , respectively, for k ≥ 0 and
j = 0, . . . , k are given explicitly as follows:
dj ((I0 ≤ · · · ≤ Ik+1 )) = (I0 ≤ · · · ≤ Iˆj ≤ · · · ≤ Ik+1 ),
sj ((I0 ≤ · · · ≤ Ik )) = (I0 ≤ · · · ≤ Ij ≤ Ij ≤ · · · ≤ Ik ).
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 Intk of Int• is contractible implies that
h
h
Int0
Int0
∼
Intk −→ Int1 × . . . × Int1
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 Intn•,...,• as
Intn•,...,• = (Int• )×n .
An element of the space Intnk1 ,...,kn will be denoted by
I = (a, b) = (I0i ≤ · · · ≤ Iki i )i=1,...,n .
Lemma 3.3. Intn•,...,• 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 Bordn as
a complete n-fold Segal space. For k ≥ 0, define the boxing map by the following formula:
B : Intk −→ Int0 ,
I(s) = (I0 (s) ≤ · · · ≤ Ik (s))s∈|∆l | 7→ B(I) = (a0 , bk ).
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 Intn•,...,• as follows:
B : Intnk1 ,...,kn −→ Intn ,
I = (I0i ≤ · · · ≤ Iki i )i=1,...,n 7→ B(I) = (a10 , b1k1 ) × · · · × (an0 , bnkn ).
Another map that we will use is the box rescaling map ρ, which, for any I ∈ Intk1 ,...,kn , is the map
ρ(I) : B(I) → (0, 1)n defined by the restriction of the product of the affine maps R → R sending ai0 and biki
to 0 and 1 respectively, for i = 1, . . . , n.
12
ANDREA TIRELLI
3.2. The n-fold Segal space PBordn . 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 PBordn as a limit of
simplicial spaces PBordVn , where the parameter V is a finite dimensional vector space, and in order to get
Bordn 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 PBordVn 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 πS the projection πS : Rn → RS on
the coordinates parametrised by S.
Definition 3.4. Let V be a finite dimensional real vector space, which we identify with Rr for some
r, and fix n ≥ 1 and k1 , . . . , kn ≥ 0. Then, we define (PBordVn )k1 ,...,kn to be the collection of tuples
(M, I = (I0i ≤ · · · ≤ Iki i )ni=1 ), such that:
(1) I = (I0i ≤ · · · ≤ Iki i )ni=1 ∈ Intnk1 ,...,kn ;
(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 pS be the composition of π with the projection πS to the S-coordinates.
i
Then, for every 1 ≤ i ≤ n and every 0 ≤ ji ≤ ki , at every x ∈ p−1
{i} (Iji ) the map p{i,...,n} is
submersive.
Remark 3.5. One way on interpreting (PBordVn )k1 ,...,kn is the following: its elements are collections of
k1 · · · kn composed bordisms, with ki 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 (PBordVn )k1 ,...,kn
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
G
∼
Sub(V × (0, 1)n ) ←−
Emb(M, V × (0, 1)n )/Diff(M ),
[M ]
n
where Emb(M, V × (0, 1) ) 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 UM = {UK,W }K∈CV ,W ∈N(M,ι) given by
UK,W = {N ∈ Sub(V × (0, 1)n ) : N ∩ K = j(M ) ∩ K, ∀j ∈ W },
where CV 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 ) × Intnk1 ,...,kn
where we view Intnn1 ,...,nn as a topological subspace of R2(k+1) ;
(4) identify an element (M, I) ∈ (PBordVn )k1 ,...,kn , 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 ) × Intnk1 ,...,kn , where
ϕ : V × B(I) → V × (0, 1)n is the diffeomorphism ϕ := (IdV , ρ(I));
(5) topologise (PBordVn )k1 ,...,kn using the inclusion constructed above
(PBordVn )k1 ,...,kn ⊂ Sub(V × (0, 1)n ) × Intnk1 ,...,kn .
With the topological space structure on (PBordVn )k1 ,...,kn 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 (PBordVn )k1 ,...,kn is given by tuples
(M, I(s) = (I0i (s) ≤ · · · ≤ Iki i (s))i=1,...,n )s∈|∆l |
such that:
(1) I(s) = (I0i (s), . . . , Iki i (s))s∈|∆l | is an l-simplex of Intnk1 ,...,kn ;
(2) M is a closed and bounded (n + l)-dimensional submanifold of V × B(I(s))s∈|∆l | × |∆l | such that
(∞, 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 pS the composition of π with the projection on the S-coordinates, for any S ⊂
i
{1, . . . , n}, for every 1 ≤ i ≤ n and 0 ≤ ji ≤ ki , at every point x ∈ p−1
i (Iji ) 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 Ms of M → |∆l | determines an element of (PBordVn )k1 ,...,kn
(Ms ) = (Ms ⊂ V × B(I(s)), I(s)),
and we will use the notation πs : Ms → B(I(s)) for the composition of the embedding with the projection.
Let us now construct (PBordVn )k1 ,...,kn 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 (PBordVn )k1 ,...,kn to an m-simplex in the following way:
• the m-simplex in Intnk1 ,...,kn is obtained by precomposing with |f |
f ∆ ((I0i (s) ≤ · · · ≤ Iki i (s))s∈|∆l | ) = (I0i (|f |(s)) ≤ · · · ≤ Iki 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 |:
[
M|f |(s) .
f ∆M =
s∈|∆m |
The proof of proposition below follows directly from the way (PBordVn )k1 ,...,kn is constructed as a simplicial
space, so it is left to the reader.
Proposition 3.8. (PBordVn )k1 ,...,kn 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,
PBordn = lim PBordVn .
−→∞
V ⊂R
Our purpose now is to make the collection of spaces (PBordn )•,...,• 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 [ki ] → (PBordn )k1 ,...,kn to a functor from ∆op , for 1 ≤ i ≤ n, in
the following way: for every 1 ≤ i ≤ n, consider a morphism g : [mi ] → [ki ] ∈ ∆([mi ], [ki ]) and let g = Πgi ;
then we define the map g ∗ : (PBordn )k1 ,...,kn → (PBordn )m1 ,...,mn that sends an element
(M ⊂ V × B(I), I = (I0i ≤ · · · ≤ Iki i )i=1,...,n )
to
i
g ∗ M = π −1 (B(g ∗ (I))) ⊂ V × B(g ∗ (I)), g ∗ (I) = (Ig(0) ≤ · · · ≤ Ig(m
)
.
i ) i=1,...,n
In this way, we have obtained a simplicial set structure on PBordn . We can also define the map g ∗ on
l-simplices: the image of an element
(M ⊂ V × B(I(s))s∈|∆l | × |∆l |, I(s) = (I0i ≤ · · · ≤ Iki i )i=1,...,n (s))
is
(g ∗ M = π −1 (B(g ∗ (I(s))s∈|∆l |×|∆l | )) ⊂ V × B(g ∗ (I(s))s∈|∆l | × |∆l |),
i
g ∗ (I)(s) = (Ig(0) ≤ · · · ≤ Ig(m
)
(s)).
i ) i=1,...,n
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 (PBordn )•,...,• are compatible.
Proposition 3.10. Given any f : [l] → [p] and gi : [mi ] → [ki ] for 1 ≤ i ≤ n, the induced maps f ∆ and g ∗
commute, which means that we have an n-fold simplicial space structure on (PBordn )•,...,• .
Proposition 3.11. (PBordn )•,...,• is an n-fold Segal space.
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
∼
(PBordn )k1 ,...,ki +ki0 ,...,kn → (PBordn )k1 ,...,ki ,...,kn
×
(PBordn )k1 ,...,0,...,kn
(PBordn )k1 ,...,ki0 ,...,kn .
Moreover, the second step of the proof is to show that, for every i and every k1 , . . . , ki−1 , the (n−i)-fold Segal
space (PBordn )k1 ,...,ki−1 ,0,•,...,• is essentially constant: to do so, one shows that the degeneracy inclusion
map
(Pbordn )k1 ,...,ki−1 ,0,0,...,0 ,→ (PBordn )k1 ,...,ki−1 ,0,ki+1 ,...,kn
admits a deformation retraction and thus is a weak equivalence. For the construction of such a retraction,
see [CS15, Proposition 5.20].
3.3. The (∞, n)-category Bordn and its symmetric monoidal structure. In the previous subsection
we constructed (PBordn )•,...,• as an n-fold Segal space. Since our model for (∞, n)-categories are complete
n-fold Segal space we give the following definition.
\ n of PBordn , which
Definition 3.12. The (∞, n)-category of cobordisms Bordn is the completion PBord
is a complete n-fold Segal space.
Remark 3.13. For n = 1, 2, PBordn is already complete, thanks to the classification theorems for 1- and
2-manifolds. On the other hand, for n ≥ 6, PBordn is not complete, see [Lur09, Warning 2.2.8] for more
details.
In Definition 1.1 the ordinary category Bordn 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 (PBordVn [m]) which form a Γ-object in SsSpn and
thus, by Lemma 2.39, endow Bordn with a symmetric monoidal structure.
Definition 3.14. Consider a finite dimensional vector space V and fix n ≥ 1. For every k1 , . . . , kn ≥ 0 and
m ≥ 1, let (PBordVn [m])k1 ,...,kn be the collection of tuples
(M1 , . . . , Mm , (I0i ≤ · · · ≤ Iki i )i=1,...,n ),
where, for each j = 1, . . . , m, (Mj , (I0i ≤ · · · ≤ Iki i )i=1,...,n ) is an element of (PBordVn )k1 ,...,kn and
M1 , . . . , Mm 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 Bordn [m]. It is clear
that Bordn [1] = Bordn .
Proposition 3.15. There is a functor
Γ −→ CSeSpn ,
hmi −→ Bordn [m]
endowing Bordn with a symmetric monoidal structure.
Proof. By Lemma 2.39, to prove the above result it suffices to define a functor Γ → SeSpn , sending hmi to
PBordn [m]. Firstly, given a morphism f : hmi → hli ∈ Γ(hmi, hli) we define the morphism
PBordn [m] → PBordn [l],
G
G
(M1 , . . . , Mm , I) → (
Mα , . . . ,
α∈f −1 (1)
Mα , I).
α∈f −1 (l)
Thus, we get a functor Γ → SeSpn and we are left to show that the property in Definition 2.37 holds, i. e.
we need to show that the map
Y
γβ : PBordn [m] → (PBordn [1])m
1≤β≤m
Q
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
Q a path from a generic element
(M1 , . . . , Mm , I) ∈ (PBordn [1])m to an element in the image of the map 1≤β≤m γβ , which induces a strong
homotopy equivalence between the two spaces. First, there is a path to an element for which all (Mα ) 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 Mα ,→ V × B(I) with the embedding
into R × V × B(I) given by the map V → R × V, v 7→ (sα, v).
(∞, n)-CATEGORIES, FULLY EXTENDED TQFTS AND APPLICATIONS
15
3.4. Cobordisms with additional structure. In this subsection, we construct an enriched version of
the n-fold Segal space Bordn : 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 3.16. 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)
the set of (X, E)-structured manifolds of dimension n.
n
Example 3.17. Consider a topological group G with a continuous homomorphism e : G → O(n). By abuse
of notation, we will denote with e : BG → BGL(Rn ) the induced map between the classifying spaces. Let
us consider the vector bundle (Rn × 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 ManG
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 = IdO(n) ; in this case, given that the group Diff(Rn ) 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 3.18. Note that, given two (X, E)-structured manifolds M and N , it is possible to define morphisms
. For the sake of brevity,
into a category Man(X,E)
between them, so that we can turn the set Man(X,E)
n
n
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 cobor. The steps of the construction are almost the same as for Bordn : we define the simplicial
disms Bord(X,E)
n
and spacial structures, which turn out to be compatible and then we apply the completion functor.
Definition 3.19. Fix a finite dimensional vector space V , n ≥ 1 and k1 , . . . , kn ≥ 0. We then let
(PBord(X,E),V
)k1 ,...,kn be the collection of tuples (M, f, φ, (I0i ≤ · · · ≤ Iki i )i=1,...,n ), where:
n
(1) (M, (I0i ≤ · · · ≤ Iki i )i=1,...,n ) is an element of (PBordVn )k1 ,...,kn ,
(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 3.20. An l-simplex in (PBordn(X,E),V )k1 ,...,kn is given by a tuple (M, f, φ, I(s) = (I0i (s) ≤ · · · ≤
Iki i (s))s∈|∆l | ) such that
• I = (I0i ≤ Iki i )i=1,...,n → |∆l | is an l-simplex of Intnk1 ,...,kn ,
• 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 pS the composition of π with the projection on the S-coordinates, for any S ⊂
i
{1, . . . , n}, for every 1 ≤ i ≤ n and 0 ≤ ji ≤ ki , at every point x ∈ p−1
i (Iji ) the map p{i,...,n} is
submersive.
Remark 3.21. From the previous definitions, we see that the only difference between PBordVn 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 PBordVn 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)
= lim PBord(X,E),V
.
n
n
−→∞
V ⊂R
16
ANDREA TIRELLI
Then, we take the completion and get the following definition.
\
Definition 3.22. The (∞, n)-category of (X, E)-structured cobordisms is Bord(X,E)
= PBord(X,E)
, which
n
n
can be endowed with a symmetric monoidal structure by using a (X, E)-structured version of Definition
3.14. In particular, when (X, E) = (∗, Rn × ∗), we will use the notation Bordfnr .
3.5. 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 3.23. 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 Bordn and target C,
F : Bordn −→ C.
We will also be interested in fully extended TQFTs that have as source the (∞, n)-category of (X, E)structured cobordisms.
Definition 3.24. 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)
−→ C.
n
Remark 3.25. The most important cases of the previous definition are when (X, E) is equal to either
(∗, Rn × ∗) - such theories are called f ramed - 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 : Bordfnr → 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(Bordfnr , C) → C : Z 7→ Z(∗)
factors through the fully dualizable sub-∞-groupoid (C fd )0 of C and the induced functor
Fun(Bordn , C) −→ (C fd )0
is a Dwyer-Kan equivalence.
Remark 3.27. 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 = τ0 (C), where τ0 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.
4.1. Fully dualizable objects in Bordf2 r : 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
Bordf2 r of framed 2-bordisms is fully dualizable. We first give the proof of this result by using the heuristic
description of the Bordf2 r , 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 ⊕ Rn−k . Heuristically, we could define
the (∞, n)-category HBordfnr as follows:
• objects: n-framed 0-manifolds;
• 1-morphisms: n-framed bordisms between objects;
(∞, 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 # Rn 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 ⊕ Rn−k ∼
= T M ⊕ ν(ι) ∼
= Rn .
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 Bordf2 r 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, Rn ), where l is a section of the normal bundle of N
pointing into the bulk of the manifold M and s ⊂ ν(M, Rn ) 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 R2 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 π1 (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 Bordf2 r . 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+ t pt− → ∅ and a coevaluation bordism coev : ∅ → pt− t 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
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 Bordf2 r 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: evL is a 2-framed
1-bordism from ∅ to pt+ t pt− , evL : ∅ → pt+ t 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 2morphisms of the adjuction evL a ev. The unit u1 is a 2-morphism Id∅ → ev ◦ evL and it is given by the
following 2-framed 2-bordism.
:
Id∅
→
Note that in the picture of the 2-manifold u1 , 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 v1 : evL ◦ ev →
Idpt+ tpt− and it is represented by the following 2-framed surfaces with cuspidal corners.
Remark 4.5. Comparing the two 2-morphisms v1 and u1 , 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 u1
means that the source and the target of u1 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 v1 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 evR 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 evR and allowing it
to self-intersect as follows.
In this case, the unit on the adjunction ev a evR are the 2-bordism u2 : ∅ → ev ⊗ evR shown in the following
picture
and the 2-bordism v2 : ev ⊗ evR → ∅ drawn as follows.
4.2. Fully dualizable objects in Bordf2 r : 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 Bordf2 r . 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].
(∞, n)-CATEGORIES, FULLY EXTENDED TQFTS AND APPLICATIONS
19
We first note that PBordf2 r = Bordf2 r . Thus, an object in Bordf2 r , 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 (Bordf2 r )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 evpt+ 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 h2 (Bordfnr ): it is depicted as follows.
The coevaluation 1-morphism coevpt+ 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 evpt+ . 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.
20
ANDREA TIRELLI
In a similar flavour, the unit of the adjunction is a saddle with the framing coming from the one on the
torus which turns by 2π along each of the two fundamental loops.
The fact that unit and counit satisfy the adjunction identity, for a statement of which we refer to [ML98,
Chapter 4], is proved by the following picture.
We now want to compare the two proofs of full dualizability and explain how the first argument can be
made completely rigorous. Essentially, this is done by carefully analysing how a fully dualizable object in
an (∞, 2)-category is defined.
First, notice that this reduces to doing computations in a ordinary categorical context: indeed, the definition
of fully dualizable object in an (∞, n)-category, as spelled out in Subsection 2.5, is given in terms of certain
homotopy 2-categories. Secondly, given the above sketchy definition of the (∞, 2)-category HBordf2 r , we can
extract from it a rigorously defined 2-category, which turns out to be the setting in which the computations
involving normally framed immersions are done. Then, our claim is the following.
Claim. This bicategory is equivalent to the homotopy 2-category h2 (Bordf2 r ), where Bordf2 r is now regarded
as a complete 2-fold Segal space
Remark 4.6. Note that a precise formulation of this claim, which is given in Proposition 4.10, and a proof
of such a statement would give a rigorous justification to the heuristic arguments presented in Subsection
4.2. Moreover, this equivalence of 2-categories is exactly the property we want a good model for (∞, n)categories to satisfy, i.e. a good model for (∞, n)-categories should be such that all the computations that
are based on the heuristic definition of (∞, n)-category can be formalised in the homotopy categories that
we can naturally associate to our model.
The proof of the claim boils down to giving the right definition of the bicategory in which the computations
explained above are carried out. Before giving such a definition, let us point out that this was first defined
by Chris Schommer-Pries in his PhD thesis [SP09], to which we refer for more details. We will use the
notation of [CS15]
Definition 4.7. Given n ≥ 2, the bicategory nBordext is defined as follows:
• the objects are (n − 2)-dimensional smooth closed manifolds,
• the 1-morphisms are (n − 1)-dimensional 1-bordisms between objects;
• the 2-morphisms are isomorphism classes of n-dimensional 2-bordisms between 1-morphisms,
where
(1) a 1-morphism from an object Y0 to an object Y1 is an (n − 1)-dimensional 1-bordism as in Definition 1.1, i.e. a smooth compact (n − 1)-dimensional manifold with boundary W , together with a
(∞, n)-CATEGORIES, FULLY EXTENDED TQFTS AND APPLICATIONS
21
decomposition and isomorphism
∂W = ∂in W t ∂out W ∼
= Y0 t Y1 ;
(2) for two 1-bordisms W0 and W1 from Y0 to Y1 , a 2-bordism from W0 to W1 is an n-dimensional
2-bordism S together with
- a decomposition and isomorphism
∼
∂0 S = ∂0,in S t ∂0,out S −→ W0 t W1
- a decomposition and isomorphism
∼
∂1 S = ∂1,in S t ∂1,out S −→ Y0 × [0, 1] t Y1 × [0, 1];
(3) two 2-bordisms S and S 0 are isomorphic if there is a diffeomorphism f : S → S 0 compatible with
the boundary data.
Remark 4.8. Note that compositions of 2-morphisms are defined by gluing manifolds after choosing a collar.
This is well defined because we are taking isomorphism classes of 2-bordims. On the other hand, composition
of 1-morphisms requires the use of a choice of a collar and, thus, is associative only up to non-canonical
isomorphism of 1-bordisms. It is however rather easy to see that the above definition defines a bicategory.
Remark 4.9. Note that nBordext can be endowed with a symmetric monoidal structure by taking disjoint
unions and that we can modify its definition in the obvious way to obtain the framed version nBordext
fr .
We can finally state the result that gives a proof of the previous claim. Note that it is a particular case of a
r
.
more general theorem, the statement of which involve the construction of the (∞, 2)-category Bord(∞,2),f
n
fr
This is a modified version of the usual (∞, n)-category Bordn and it is defined in [CS15, Proposition 8.15],
where the proof, omitted here for the sake of brevity, of the following proposition is given.
Proposition 4.10. There is an equivalence of symmetric monoidal bicategories between h2 (Bordf2 r ) and
2Bordext
fr .
Remark 4.11. Note that the first proof of full dualizability is carried out in the bicategory 2Bordext
f r whilst
fr
the second one is given in the setting of the bicategory h2 (Bord2 ). Thus, the previous proposition gives
an equivalence between the two arguments.
4.3. An ∞-model for the Morita bicategory of a monoidal category. We now want to give another
example in which we use Proposition 2.46 to prove full dualizability of certain objects in an (∞, 2)-category.
Moreover, our aim is to find a model for the classical Morita bicategory associated to a general monoidal
category. More specifically, given a monoidal category S, the associative algebra objects and their bimodules
form a bicategory as follows.
Definition 4.12. The bicategory Alg1 (S) is defined as follows:
• objects: associative algebras in S,
• 1-morphisms from A to B: (A, B)-bimodules,
• 2-morphisms: bimodule homomorphisms,
where composition of 1-morphisms is given by taking tensor product and composition of 2-morphisms is the
usual composition of homomorphisms.
There is a generalisation of this construction to the ∞-categorical context. Namely, Scheimbauer and
Johnson-Freyd, in [JS15], construct, given a symmetric monoidal (∞, n)-category S, an (∞, n + d)-category
Alg∞
d (S), called the “even higher” Morita (∞, n+d)-category of Ed -algebras of S. The following proposition,
whose validity was conjectured by Scheimbauer in [Sch14, §3.4.1], explains why this is a good extension of
the previous definition.
Proposition 4.13. Let S be a symmetric monoidal category, considered as an (∞, 1)-category. Then, there
is an equivalence of symmetric monoidal categories between h2 (Alg∞
1 (S)) and Alg1 (S), where the latter is
the bicategory introduced in Definition 4.12.
Remark 4.14. It is worth pointing out that the construction of the (∞, n + d)-category Alg∞
d (S) is quite
involved and it is carried out in [JS15] using a rather technical machinery. Moreover, the previous proposition
assures that, in order to prove full dualizabaility of certain objects in Alg∞
d (S), we only need to look at the
usual Morita bicategory, at least in the case when S is an (∞, 1)-category and d = 1. We will explain such
computations in the following subsection.
22
ANDREA TIRELLI
4.4. Full dualizability in Alg∞
1 (S). The goal of this subsection is to give a rigorous proof of the folklore
statement that an algebra is fully dualizable if and only if it is smooth and proper. Although it is possible
to prove that this result, once stated rigorously, is valid in the (∞, 2)-category Alg∞
2 (S), for a general
symmetric monoidal (∞, 1)-category with colimits, we will restrict our attention to the case of ordinary
(dg)-algebras over a field k, which means taking S to be either Vectk or Chk , where the latter denotes
the category of chain complexes of k-vector spaces, where the symmetric monoidal structure is given by
tensor product. Applying Proposition 4.13, we get that, in this case, Alg1 (S) is the bicategory where
objects k-(dg)-algebras, 1-morphisms are (dg)-bimodules over those algebras, 2-morphisms are bimodule
homomorphisms. Thus, from now on, we will only work with these bicategories, with the awareness that
what we will prove can be transferred, thanks to the previous result, to the higher categorical context of
Alg∞
1 (S).
Remark 4.15. In what follows we will use the term algebra to identify an associative monoid object in either
of the aforementioned categories. It is clear that: in the first case it will be an ordinary k-algebra; in the
second case it will be a dg-k-algebra.
Let us start by proving that any algebra A has a dual which is the opposite algebra Aop . As customary, we
will denote with Ae the tensor product A ⊗k Aop , the enveloping algebra of A.
Lemma 4.16. Let A be an algebra. Then, its dual is the opposite algebra Aop , and as evaluation and coevaluation maps we can take A, considered as an (Ae , k)-bimodule and as a (k, Aop ⊗ A)-bimodule respectively.
Proof. First, let us note that the proof we are going to give is valid in both the ordinary and the dg case.
Secondly, let us denote by ev and coev the evaluation and coevaluation bimodules, respectively. What we
need to prove is that the following compositions of 1-morphisms
IdA ⊗coev
ev⊗IdA
A∼
= A ⊗ k −−−−−−→ A ⊗ Aop ⊗ A −−−−−→ k ⊗ A ∼
=A
and
coev⊗Id
Id
op ⊗ev
A
A
Aop ∼
= k ⊗ Aop −−−−−−−−→ Aop ⊗ A ⊗ Aop −−−−−−→ Aop ⊗ k ∼
= Aop
are isomorphic to the identity morphism on A and the identity morphism on Aop respectively. We will show
the computations only for the first case, as the second is completely similar. First, note that IdA is just A,
considered as an (A, A)-bimodule. To prove that ev = A⊗Aop Ak and coev = k AAop ⊗A , we then simply need
to show that the following tensor product
O
Λ = A A ⊗ AA⊗Aop ⊗A
A⊗Aop ⊗A A ⊗ AA
op
A⊗Aop ⊗A
is isomorphic to A AA is an (A, A)-bimodules. Using the standard facts that, for any ring B, right B-module
M and left B-module N ,
M ⊗B ∼
= M,
B
B⊗N ∼
=N
B
we obtain that
Λ∼
= A A ⊗ AA ,
Aop
which is clearly isomorphic to A AA as an (A, A)-bimodule.
Alg∞
2 (S)
Thus, thanks to Proposition 2.46, the previous lemma proves that the (∞, 2)-category
has duals
for objects. Despite this, we will see that not every object is fully dualizable. In order to explain which
algebras satisfy this condition, we need to introduce the notions of properness and smoothness for an algebra,
with which we can give a rigorous meaning of the folklore statement that fully dualizable objects are those
ones that satisfy certain finiteness conditions.
Definition 4.17. Let A be an algebra in S. Then A is
• smooth if it is a dualizable object in Ae -Mod,
• proper if it is a dualizable object in k-Mod.
Remark 4.18. Note that Ae -Mod and k-Mod have different meaning depending on whether A is a monoid
object in the category Vectk or in the category Chk . We will characterise later what properness and
smoothness mean in each of the contexts.
We are now ready to state and prove the full dualizabality result.
Proposition 4.19. Let A be an algebra in S. Then A is fully dualizable is and only if A is smooth and
proper.
(∞, n)-CATEGORIES, FULLY EXTENDED TQFTS AND APPLICATIONS
23
Proof. By the previous lemma, we know that any algebra A has Aop as dual. Thus, by Proposition 2.46,
proving the above statement is equivalent to showing that A is smooth and proper if and only the evaluation
morphism ev = Ae Ak has both right and left adjoints. Thus, let us suppose that ev has both left and right
adjoints, which we call evL and evR respectively. In [Zha13, Theorem 4.2.6] it is proved that evL and evR
are the duals of A in Ae -Mod and k-Mod respectively:
evL = A∨ = HomAe (A, Ae ),
evR = A∗ = Homk (A, k).
So, if A is fully dualizable, then it has duals in Ae -Mod and k-Mod, which is the same as saying that A is
smooth and proper. Conversely, let us suppose that A is smooth and proper. Then, its duals in Ae -Mod
and k-Mod can be taken as left and right adjoints of the evalutation Ae Ak , respectively. For example, let
S be the dual of A in Ae -Mod, then, is it possible to check that k SAe is the left adjoint of Ae Ak , by using
the evaluation and coevaluations maps that witness this duality, see [Zha13, Theorem 2.4.6]. The same
argument holds for the right adjoint.
As promised, we now explain the meaning of smoothness and properness for an associative monoid object in
Vect(k) and Ch(k). Note that the latter case compromises also the first one: indeed, given any ordinary kalgebra A, this can be viewed as a dg-k-algebra A• , where A is concentrated in degree 0 and the differential
is the zero map. This gives in fact an immersion of Alg(k) in dgAlg(k) as a full subcategory. Thus,
once we have given a characterisation of smooth and proper dg-k-algebra, we can deduce the corresponding
characterisation for k-algebras.
Proposition 4.20. Let A = A• an associative dg-k-algebra and M an A• -module. The following condition
are equivalent:
(1) M is dualizable in A• -Mod.
(2) There exist n < ∞ and k1 , . . . , kn ∈ Z such that there is a commutative diagram of A-linear maps
Ln
M
i=1 A[ki ]
.
IdM
M
(3) M is finitely generated and projective as an A-module.
Proof. See [Zha13, Proposition 4.2.9].
The case S = Vect(k). Consider a k-algebra A. Then, combining Proposition 4.20 and Proposition 4.19,
we get that A is fully dualizable in Alg∞
1 (Vect(k)) if it is finitely generated and projective in Vect(k) and
Ae -Mod. From the definition of projective module over a ring we have that, the first condition means that
A is a finite dimensional vector space. Furthermore, it is possible to prove that the second condition means
that A has to be a finite dimensional separable algebra, see [Wei94, Theorem 9.2.11]. Moreover, it is a
known result that, when the field k is perfect, separability is exactly the same as semisimplicity. In fact, this
is a well known characterisation: an algebra over a perfect field is fully dualizable if and only if it is finite
dimensional and semisimple.
The case S = Ch(k). Let A = A• be a dg-k-algebra. Then, we can apply the previous proposition
and we get that A• is fully dualizable in Alg∞
1 (S) is and only if it is finitely generated and A• -projective
as a dg-k-module and as an Ae -module. In the case of dg-algebras this condition can be stated in the more
general context of dg-categories. In particular, a dg-A• -module which is finitely generated and projective
is usually called perfect. In [Kel06], Keller proved that an object X in a dg-category D is perfect if and
only if it is compact, which, loosely speaking, means that the functor HomD (X, −) commutes with arbitrary
colimits, for a precise definition see [Kel06]. Thus, A• is fully dualizable if and only if it is perfect as a
dg-Ae -module and as a dg-k-module. This last condition can be made even more explicit:
P indeed, it is
possible to prove that A• is a perfect dg-k-module if and only if its cohomology is finite, i H i (A) < ∞.
4.5. Applications to fully extended TQFTs. We will now briefly discuss how the results proved previously in this section can be applied to exhibit two examples of 2-dimensional fully extended TQFTs.
First, consider the case C = Bordf2 r . Then, showing that any object in C is fully dualizable gives a
proof of on direction of Theorem 3.26; in fact, given a fully extended TQFT
Z : Bordf2 r → C,
24
ANDREA TIRELLI
the image of pt+ under the induced functor on the homotopy bicategories
h2 (Z) : 2Cobext
f r −→ h2 (C)
is fully dualizable. The arguments in Subsections 4.1 and 4.2 prove this result not only for the identity functor on Bordf2 r , but for an arbitrary TQFT, since the image of a fully dualizable object under a
symmetric monoidal functor is still fully dualizable: indeed, given a symmetric monoidal functor between
(∞, n)-categories C → D, the image of C fd ,→ C → D consists of fully dualizable objects by the universal
property of Dfd .
In the second case, i.e. when C = Alg∞
1 (S), where S is either Vect(k) or Ch(k), we invoke the Cobordism
Hypothesis, Theorem 3.26, which assures that, given a smooth and proper algebra A in S, there exists a
unique, up to equivalence, fully extended 2-dimensional framed TQFT
Z : Bordfnr −→ Alg∞
1 (S),
such that, the induced functor on the homotopy bicategories
h2 (Z) : 2Cobext
f r −→ Alg1 (S),
satisfies the following equation
h2 (Z)(pt+ ) = A.
Moreover, even if the proof of Theorem 3.26 is not constructive, one can infer some properties of Z by
knowing the duals of pt+ and A and the adjoints of the (co)evaluation morphims in each of the two cases.
1
Theorem 4.21. Consider the fully extended TQFT Z defined above. Let S+
denote the 2-framed 1-manifold
1
S with the framing given by normal bundle in the outgoing direction (as depicted in Example 4.2). Then,
we have:
1 ∼
Z(S+
) = HH• (A, A),
•
where HH (A, A) denotes the Hochschild cohomology of A with values in A itself.
1
Proof. The key point to show the above result is to understand how S+
decomposes into simple pieces. In
L
1 ∼
Subsection 4.1 we showed that, in fact S+ = evA ◦ evA . Thus, since Z is symmetric monoidal, we have
1
Z(S+
) = Z(evA ) ◦ Z(evL
A ).
Since composition in Alg1 (S) is given by tensor product, we get
1
e
Z(S+
) = Z(evL
A ) ⊗Ae Z(evA ) = HomAe (A, A ) ⊗Ae A,
and, since A is compact as an Ae -module, we have
Z(S 1 ) ∼
= HomAe (A, A) ∼
= HH• (A, A),
+
which is exactly the formula that we were required to prove.
Si1
Remark 4.22. Now, denote with
the 2-framed 1-manifold given by a circle with i twists and framing as
depicted in Example 4.2. For example, S21 can be depicted as follows.
Then, similarly to the previous theorem, one can compute Z(Si1 ) as Z(Si1 ) ∼
= HH• (A, (A∗ )⊗A i ).
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E-mail address: [email protected]