# Semisimple Extended Topological Quantum Field Theories

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## Relative modular categories

This section is devoted to the definition of the algebraic structures which are going to play the leading role in the construction, relative modular categories. They are a non-semisimple analogue of modular categories and, since their definition involves a lot of different ingredients, we first set the ground by recalling definitions for various concepts like group structures, traces on ideals induced by ambidextrous objects and group realizations.

### Group structures and homogeneous colored ribbon graphs.

We begin by introducing group structures on monoidal categories. In order to do so let us fix an abelian group G. If C is a strict monoidal category then a G-structure on C is given by a family of full subcategories fCg j g 2 Gg of C indexed by elements of G and satisfying V V 0 2 Ob(Cg+g0 ) for all V 2 Ob(Cg) and all V 0 2 Ob(Cg0 ). A strict monoidal category C equipped with a G-structure is called a G-category7 and G is called the structure group of C. For every g 2 G the subcategory Cg is called the homogeneous subcategory of index g. If an object V of C belongs to the homogeneous subcategory of index g then we will write i(V ) = g and we will say V is a homogeneous object of index g. We say a G-structure on a pivotal category C is compatible with the pivotal structure if V 2 Ob(C􀀀g) for all V 2 Ob(Cg). A pivotal G-category is then a pivotal category equipped with a compatible G-structure, and a ribbon G-category is a pivotal G-category which is ribbon. Let us fix for this subsection a ribbon G-category C. If is a 2-dimensional cobordism and if P is a ribbon set8 then a C-coloring V : P ! Ob(C) is G-homogeneous if V (p) is a homogeneous object of C for every vertex p 2 P. A C-colored ribbon set PV is G-homogeneous if the C-coloring V : P ! Ob(C) is G-homogeneous.
If M is a 3-dimensional cobordism with corners from to 0, if PV and P0V 0 0 are G-homogeneous C-colored ribbon sets and if T M is a ribbon graph from P to P0 then a C-coloring ‘ : T ! C extending V and V 0 is G-homogeneous, denoted , if ‘(e) is a homogeneous object of C for every edge e T. A C- colored ribbon graph T’ M from PV to P0V 0 is G-homogeneous if the C-coloring ‘ : T ! C extending V and V 0 is G-homogeneous. The ribbon G-category RibG C of G-homogeneous C-colored ribbon graphs is the subcategory of9 RibC whose objects are objects ((« 1; V1); : : : ; (« k; Vk)).

#### Group colorings on decorated cobordisms.

In this subsection we fix an abelian group G and we introduce a new piece of decoration for cobordisms: G-colorings. They consist of relative cohomology classes with coefficients in G together with discrete sets of specified base points which are needed in order to induce G-colorings on horizontal and vertical gluings. Their role will be to determine indices for surgery presentations of 3-dimensional manifolds. The notation we use for cobordisms is introduced in Appendix B.4. Definition 2.3.1. If 􀀀 is a 1-dimensional smooth manifold without boundary then a G-coloring A of 􀀀 is given by:
(i) a finite set A 􀀀 called the base set, whose elements are called base points.
(ii) a cohomology class 2 H1(􀀀;A;G).
These data satisfy the following condition: if 􀀀i is a non-empty connected component of 􀀀 then A \ 􀀀i = faig for exactly one point ai 2 􀀀i.
Definition 2.3.2. If is a 2-dimensional cobordism from 􀀀 to 􀀀0, if P is a ribbon set inside and if A and 0A0 are G-colorings of 􀀀 and 􀀀0 respectively then a G-coloring #B of (; P) extending A and 0A0 is given by:
(i) a finite set B ( r (P [ @)) called the base set, whose elements are called base points.
(ii) a cohomology class # 2 H1( r P;A [ B [ A0 ;G) where A := f􀀀(A); A0 := f+(A0).

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In this subsection we fix a pre-modular G-category C relative to (;X) and we define the symmetric monoidal 2-category CobC 3 of admissible decorated cobordisms of dimension 1+1+1. If is a connected 2-dimensional cobordism and if P is a ribbon set then a vertex p 2 P is said to be projective with respect to a C-coloring V of P if V (p) is a projective object of C. A C-coloring V : P ! Ob(C) is projective if there exists a projective vertex p 2 P with respect to V . If M is a connected 3-dimensional cobordism with corners and if T M is a ribbon graph then an edge e T is said to be projective with respect to a C-coloring ‘ of T if ‘(e) is a projective object of C. A C-coloring ‘ : T ! C is projective if there exists a projective edge e T with respect to ‘. If is a connected 2-dimensional cobordism and if P is a ribbon set then a simple closed oriented curve rP is generic with respect to a G-coloring #B of (; P) if h#; i 2 G r X as a homology class. A G-coloring #B of (; P) is generic if there exists a generic simple closed oriented curve rP with respect to #B. If M is a connected 3-dimensional cobordism with corners and if T M is a ribbon graph then an oriented knot K MrT is generic with respect to a G-coloring ! of (M; T) if h!;Ki 2 GrX as a homology class. A G-coloring ! of (M; T) is generic if there exists a generic oriented knot K M rT with respect to !.

Contents
Abstract
Keywords
Résumé
Mots-clés
Acknowledgements
Introduction
Why are TQFTs interesting?
What is a TQFT?
A brief history of quantum topology
Pleasures and pains of semisimple theories
Non-semisimple theories
What is an Extended TQFT?
Results
Préface
Pourquoi les TQFTs sont-elles intéressantes ?
Qu’est-ce qu’une TQFT?
Brève histoire de la topologie quantique
Joies et douleurs des théories semi-simples
Théories non semi-simples
Qu’est-ce qu’une TQFT Étendue ?
Résultats
Chapter 1. Semisimple Extended Topological Quantum Field Theories
1.1. Introduction
1.1.1. Main results
1.1.2. Outline of the construction
1.1.3. Structure of the exposition
1.2. Modular categories
1.2.1. Ribbon categories
1.2.2. Colored ribbon graphs
1.2.3. Main definitions
1.3. Decorated cobordisms
1.3.1. 2-Category of decorated cobordisms
1.3.2. Extended universal construction for decorated cobordisms
1.4. Surgery axioms
1.5. Connection Lemma
1.6. Witten-Reshetikhin-Turaev invariants
1.7. Skein modules
1.8. Morita reduction
1.9. Monoidality
1.10. TQFT
1.11. Universal linear categories
1.12. Universal linear functors
1.12.1. 2-Discs
1.12.2. 2-Pants
1.12.3. Examples of computations
Chapter 2. Non-Semisimple Extended Topological Quantum Field Theories
2.1. Introduction
2.1.1. Main results
2.1.2. Outline of the construction
2.1.3. Structure of the exposition
2.2. Relative modular categories
2.2.1. Group structures and homogeneous colored ribbon graphs
2.2.2. Traces on ideals and ambidextrous objects
2.2.3. Group realizations
2.2.4. Main definitions
2.3.1. Group colorings on decorated cobordisms
2.3.2. 2-Category of decorated cobordisms
2.3.4. Extended universal construction for admissible cobordisms
2.4. Surgery axioms
2.5. Connection Lemma
2.6. Costantino-Geer-Patureau invariants
2.8. Morita reduction
2.9.1. 2-Spheres
2.9.2. 3-Discs
2.9.3. 3-Pants
2.9.4. Suspension systems
2.10. Monoidality
2.11. Symmetry
2.14.1. 2-Discs
2.14.2. 2-Pants
2.14.3. 2-Cylinders
2.14.4. Examples of computations
2.14.4.1. Generic surfaces
2.14.4.2. Critical tori
Appendix A. Algebraic appendices
A.1. Monoidal categories
A.2. Enriched categories
A.3. 2-Categories
A.4. Monoidal 2-categories
A.5. Complete linear categories