While the path integral is rarely mathematically well-defined, it is usually assumed to have some useful properties like sewing laws (relating the integral over a domain which decomposes into two subdomains to path integrals over the subdomains). These were included by Atiyah into the definition of the topological quantum field theory as a functor on the category of cobordisms. This is one of the many ways the modern mathematical language of categories and functors becomes relevant to physicists. In this lecture we will continue the introduction to categories, moving into the direction of the ones used in topological quantum field theories. We will show how various notions from different areas of mathematics get unified in the categorial framework. We will discuss categories with multiplication and some geometric functors (like homology or K-theory) arising whenever global effects of the spacetime play a role.