Sharma, Amit2016-10-252016-10-252016-08https://hdl.handle.net/11299/182756University of Minnesota Ph.D. dissertation. August 2016. Major: Mathematics. Advisor: Alexander Voronov. 1 computer file (PDF); viii, 146 pages.In the first part of this thesis we propose a model for additive $\infty$-categories based on $\gSs$ and construct the archetype example of an additive $\infty$-category, namely the (quasi)-category of higher Picard groupoids, $\pic$. The goal of the second part of this thesis is to categorify Dijkgraaf-Witten (DW) theory, aiming at providing foundation for a direct construction of DW theory as an Extended Topological Quantum Field Theory. The main tool is cohomology with coefficients in a Picard groupoid, namely the Picard groupoid of hermitian lines.enCategory TheoryInfinite loop space theoryMathematical PhysicsTopologyThesis or Dissertation