In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures.
Supercategories were first introduced in 1970,[9] and were subsequently developed for applications in theoretical physics (especially quantum field theory and topological quantum field theory) and mathematical biology or mathematical biophysics.
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";[18] similarly, the Turaev–Viro model would be then obtained with representations of SUq(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,[16] instead of the 2-vector spaces that are representation categories of groupoids.