Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves

2. Frohman-Nicas TQFT’s over Z..............................7 3. Lefschetz Decompositions and Specht Modules............. 11

The motivation for this paper was to construct approximations to a conformal version of homotopy quantum field theory using 2-categories. A homotopy quantum field theory, as defined by Turaev in [9], is a variant of a topological quantum field theory in which manifolds come equipped with a map to some auxiliary space X.

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We show that the Reshetikhin-Turaev-Walker invariant of 3-manifolds can be normalized to obtain an invariant of 4-dimensional thickenings of 2-complexes. Moreover when the underlying semisimple tortile category comes from the Lie family (“quantum groups”) over the ring Z (p)[v] where v is a primitive prime root of unity, the 0-term in the Ohtsuki expansion of this invariant depends only on the spine and is the Z/pZ invariant of 2-complexes defined in [10]. As a consequence it is shown that when the Euler characteristic is greater or equal to 1, the 2-complex invariant depends only on homology. The last statement doesn’t hold for the negative Euler characteristic case.

Abstract. In this article, we introduce and study Hopf monads on monoidal categories with duals. Hopf monads generalize Hopf algebras to a non-braided (and non-linear) setting. We extend many fundamental results of the theory of Hopf algebras (such as the decomposition of Hopf modules, the existence of integrals, Maschke’s criterium of semisimplicity, etc...) to Hopf monads. We also introduce and study quasitriangular and ribbon Hopf monads (again

Abstract: We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the intersection homology of U(1)-representation varieties on the one side and the combinatorially constructed Hennings-TQFT based on the quasitriangular Hopf algebra N = Z/2 ⋉ ∧ ∗ R 2 on the other side. We find that both TQFT’s are SL(2, R)-equivariant functors and also as such isomorphic. The SL(2, R)-action in the Hennings construction comes from the natural action on N and in the case of the Frohman-Nicas theory from the Hard-Lefschetz decomposition of the U(1)-moduli spaces given that they are naturally Kähler. The irreducible components of this TQFT, corresponding to simple representations of SL(2, Z) and Sp(2g, Z), thus yield a large family of homological TQFT’s by taking sums and products. We give several examples of TQFT’s and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion,

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Abstract: In [5] Frohman and Nicas define a topological quantum field theory via the intersection homology of U(1)-representation varieties J(X) = Hom(π1(X), U(1)). We show that this TQFT is equivalent to the combinatorially constructed Hennings-TQFT based on the quasitriangular Hopf algebra N = Z/2 ⋉ ∧ ∗ R 2. The natural SL(2, R)-action on N is identified with the SL(2, R)action for the Lefschetz decomposition of H ∗ (J(Σ)) implied by the Kähler structure on J(Σ) for a surface, Σ. We compare peculiarities of both theories, such as the Z/2-projectivity and vanishing phenomena due to non-semisimplicty. This equivalence induces a graded Hopf algebra structure on H ∗ (J(Σ)), which is isomorphic to the canonical one but at the same time compatible with the Hard-Lefschetz decomposition. We discuss generalizations to higher rank gauge theories and a

We introduce a self-dual, noncommutative, and noncocommutative Hopf algebra HGT which takes for certain Hopf categories (and therefore braided monoidal bicategories) a similar role as the Grothendieck-Teichmüller group for quasitensor categories. We also give a result which highly restricts the possibility for similar structures for higher weak n-categories (n ≥ 3) by showing that these structures would not allow for any nontrivial deformations. Finally, give an explicit description of the elements of HGT. 1 The Hopf algebra HGT In [Dri] Drinfeld introduced the Grothendieck-Teichmüller group by considering the (formal) reparametrizations of the data (commutativity and associativity isomorphisms) of a quasitensor category. Consider now braided (weak) monoidal bicategories arising from the representations of a Hopf category (as defined in [CF]) on 2-vector spaces (see [KV]), i.e. on certain module categories. Let us assume, in addition, that the Hopf category itself is given