In this talk I will introduce C*-categories as the dimension 2 component of a 4 dimensional fully extended topological field theory. Factorization Homology will then be introduced as the principle by which one associates C*-cateogories to surfaces, enforcing the strong locality feature that characterizes fully extended theories. As a main source of examples and inspiration, I will say a few words about my ongoing project with quantization of non-compact topological gauge theories, where the theory of locally compact quantum groups naturally enters the picture.
