We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3dimensional 3-framed local field theories. We also show that all finite tensor categories are 2-dualizable, and yield categorified 2-dimensional 3-framed local field theories. On the other hand, topological properties of 3-framed manifolds determine algebraic equations among functors of tensor categories. We show that the 1-dimensional loop bordism, which exhibits a single full rotation, acts as the double dual autofunctor of a tensor category. We prove that the 2-dimensional belt-trick bordism, which unravels a double rotation, operates on any finite tensor category, and therefore supplies a trivialization of the quadruple dual. This approach produces a quadruple-dual theorem for suitably dualizable objects in any symmetric monoidal 3-category. There is furthermore a correspondence between algebraic structures on tensor categories and homotopy fixed point structures, which in turn provide structured field theories; we describe the expected connection between pivotal tensor categories and combed fixed point structures, and between spherical tensor categories and oriented fixed point structures.

We provide a model of the String group as a central extension of finite-dimensional 2-groups in the bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This bicategory is a geometric incarnation of the bicategory of smooth stacks and generalizes the more naive 2-category of Lie groupoids, smooth functors and smooth natural transformations. In particular this notion of smooth 2-group subsumes the notion of Lie 2-group introduced by Baez and Lauda [5]. More precisely we classify a large family of these central extensions in terms of the topological group cohomology introduced by Segal [56], and our String 2-group is a special case of such extensions. There is a nerve construction which can be applied to these 2-groups to obtain a simplicial manifold, allowing comparison with the model of Henriques [23]. The geometric realization is an A 1 -space, and in the case of our model, has the correct homotopy type of String.n/. Unlike all previous models [58; 60; 33; 23; 7] our construction takes place entirely within the framework of finitedimensional manifolds and Lie groupoids. Moreover within this context our model is characterized by a strong uniqueness result. It is a canonical central extension of Spin.n/. 57T10, 22A22, 53C08; 18D10

We axiomatise the theory of ( ∞ , n ) (\infty ,n) -categories. We prove that the space of theories of ( ∞ , n ) (\infty ,n) -categories is a B ( Z / 2 ) n B(\mathbb {Z}/2)^n . We prove that Rezk’s complete Segal Θ n \Theta _n spaces, Simpson and Tamsamani’s Segal n n -categories, the first author’s n n -fold complete Segal spaces, Kan and the first author’s n n -relative categories, and complete Segal space objects in any model of ( ∞ , n − 1 ) (\infty , n-1) -categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of ( Z / 2 ) n (\mathbb {Z}/2)^n .

The balanced tensor product M ⊗ A N of two modules over an algebra A is the vector space corepresenting A-balanced bilinear maps out of the product M × N . The balanced tensor product M C N of two module categories over a monoidal linear category C is the linear category corepresenting C-balanced right-exact bilinear functors out of the product category M × N . We show that the balanced tensor product can be realized as a category of bimodule objects in C, provided the monoidal linear category is finite and rigid.Tensor categories are a higher dimensional analogue of algebras. Just as modules and bimodules play a key role in the theory of algebras, the analogous notions of module categories and bimodule categories play a key role in the study of tensor categories (as pioneered by Ostrik [Ost03]). One of the key constructions in the theory of modules and bimodules is the relative tensor product M ⊗ A N . As first recognized by Tambara [Tam01], a similarly important role is played by the balanced tensor product of module categories over a monoidal category M C N (for example, see [Müg03, ENO10, JL09, GS12b, GS12a, DNO13, FSV13]). For C = Vect, this agrees with Deligne's [Del90] tensor product K L of finite linear categories. Etingof, Nikshych, and Ostrik [ENO10] established the existence of a balanced tensor product M C N of finite semisimple module categories over a fusion category, and Davydov and Nikshych [DN13, §2.7] outlined how to generalize this construction to module categories over a finite tensor category. 1 We give a new construction of the balanced tensor product over a finite tensor category as a category of bimodule objects.Recall that the balanced tensor product M ⊗ A N of modules is, by definition, the vector space corepresenting A-balanced bilinear functions out of M × N . In other words, giving a map M ⊗ A N → X is the same as giving a billinear map f : M × N → X with the property that f (ma, n) = f (m, an). If the balanced tensor product exists, it is certainly unique (up to unique isomorphism), but the universal property does not guarantee existence. Instead existence is typically established by an explicit construction as a quotient of a free abelian group on the product M ×N . We now describe the balanced tensor product M C N of module categories over a tensor category. Again this should be universal for certain bilinear functors, however when passing from algebras to tensor categories, the analogue of the equality f (ma, n) = f (m, an) is a natural system of isomorphisms η m,a,n : F(m ⊗ a, n) → F(m, a ⊗ n) satisfying some natural coherence properties. A bilinear functor F together with a natural coherent system of isomorphisms η m,a,n is called a C-balanced functor. Thus the balanced tensor product M C N is defined to be the linear category corepresenting C-balanced right-exact bilinear functors out of M × N . In other words, giving a right-exact functor M C N → X is the same as giving a right-exact bilinear functor M × N → X together with isomorphisms η m,a,n : F(m ⊗ a, n) → F(m, a ⊗ n) sa...

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A d-dimensional invertible topological field theory is a functor from the symmetric monoidal (∞, n)-category of d-bordisms (embedded into R ∞ and equipped with a tangential (X, ξ)-structure) which lands in the Picard subcategory of the target symmetric monoidal (∞, n)-category. We classify these field theories in terms of the cohomology of the (n − d)-connective cover of the Madsen-Tillmann spectrum. This is accomplished by identifying the classifying space of the (∞, n)-category of bordisms with Ω ∞−n M T ξ as an E ∞ -spaces. This generalizes the celebrated result of Galatius-Madsen-Tillmann-Weiss [Gal+09] in the case n = 1, and of Bökstedt-Madsen [BM14] in the n-uple case. We also obtain results for the (∞, n)-category of d-bordisms embedding into a fixed ambient manifold M , generalizing results of Randal-Williams [Ran11] in the case n = 1. We give two applications: (1) We completely compute all extended and partially extended invertible TFTs with target a certain category of n-vector spaces (for n ≤ 4), and (2) we use this to give a negative answer to a question raised by Gilmer and Masbaum in [GM13].7 Examples and applications 52 7.

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Abstract. These lecture notes form an expanded account of a course given at the Summer School on Topology and Field