The Geometry of Interaction (GoI) program was introduced by Girard in late 80's in a series of influential papers. GoI is a new kind of semantics which captures the dynamics of computation (cut-elimination). This is in contrast to the extant denotational semantics which is static and syntax which is blurred with unnecessary bureaucracy. In this talk we look at the role played by category theoretical approaches to GoI. In particular, we discuss the importance of the notion of trace in a categorical setting. Our emphasis will be on the class of partially additive categories of Manes and Arbib and their extensions.