Higher-dimensional Algebra - Double Groupoids, Fundamental Groupoids, 2-categories, Categorical QFTs and TQFTs

Double Groupoids, Fundamental Groupoids, 2-categories, Categorical QFTs and TQFTs

In higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions, and the latter groupoid can be considered as a special case of a category with all invertible arrows, or morphisms.

Double groupoids are often used to capture information about geometrical objects such as higher-dimensional manifolds (or n-dimensional manifolds). In general, an n-dimensional manifold is a space that locally looks like an n-dimensional Euclidean space, but whose global structure may be non-Euclidean. A first step towards defining higher dimensional algebras is the concept of 2-category of higher category theory, followed by the more 'geometric' concept of double category. Other pathways in HDA involve: bicategories, homomorphisms of bicategories, variable categories (aka, indexed, or parametrized categories), topoi, effective descent, enriched and internal categories, as well as quantum categories and quantum double groupoids. In the latter case, by considering fundamental groupoids defined via a 2-functor allows one to think about the physically interesting case of quantum fundamental groupoids (QFGs) in terms of the bicategory Span(Groupoids), and then constructing 2-Hilbert spaces and 2-linear maps for manifolds and cobordisms. At the next step, one obtains cobordisms with corners via natural transformations of such 2-functors. A claim was then made that, with the gauge group SU(2), "the extended TQFT, or ETQFT, gives a theory equivalent to the Ponzano-Regge model of quantum gravity"; similarly, the Turaev-Viro model would be then obtained with representations of SU_q(2). Therefore, one can describe the state space of a gauge theory – or many kinds of quantum field theories (QFTs) and local quantum physics, in terms of the transformation groupoids given by symmetries, as for example in the case of a gauge theory, by the gauge transformations acting on states that are, in this case, connections. In the case of symmetries related to quantum groups, one would obtain structures that are representation categories of quantum groupoids, instead of the 2-vector spaces that are representation categories of groupoids.