Abstract

https://youtu.be/pD8M4u30dHM

The field of higher dimensional algebra can be described as the extension of ordinary algebra to situations which are “proof relevant”, which is to say, in which equations must themselves be regarded as structure and not merely properties of the underlying data. Perhaps the most well-known examples of this phenomenon arise from attempts at extending the notion of category to higher dimensions, a subject known generally as higher category theory. But many other examples are known from topology, and recent developments linking dependent type theory and homotopy theory have shown the relevance of these for formal proof assistants. In this talk, I will survey some of the ideas arising from this point of view and detail their applications in computer science. Time permitting, I will describe a small type theory called Catt which illustrates some of the main principles of the higher algebraic approach.