We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.

We show that simple coverings of B 4 branched over ribbon surfaces up to certain local ribbon moves coincide with orientable 4-dimensional 2-handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S 3 branched over links, in terms of local moves. This result generalizes to coverings of any degree the ones by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of our equivalence theorem to possibly non-simple coverings of S 3 branched over embedded graphs.

Abstract. Lagrangian cobordisms are three-dimensional compact oriented cobordisms between once-punctured surfaces, subject to some homological conditions. We extend the Le–Murakami–Ohtsuki invariant of homology three-spheres to a functor

Category theory provides a uniform method of encoding mathematical structures and universal constructions with them. In this article we apply the method of additional structures on the objects of a category to deform a comonoid structure, used implicitly in all categories. To deform this comultiplication we consider internal categories in a monoidal category with some special properties. Then we consider structures over comonoids and show that deformed internal categories form a 2-category. This provides the possibility to study, in a uniform way, different types of generalized multiplicative and comultiplicative structures of 2-dimensional Topological Quantum Field Theory.