Abstract

We will formulate and proof the Fundamental Theorem of Category Theory. This theorem (and related theorems) lie at the heart of many applications of Category Theory to other fields of mathematics. It is also an important technical tool in various subfields of Category Theory itself. We shall present some of the examples and point out where we have encountered the Fundamental Theorem in secret on the example sheets already.

The fundamental theorem stresses once more the importance of functor categories of the form [C^op, Set] for a small category C and the accompanying Yoneda embedding. We can hence ask the question on necessary and sufficient conditions for a category E to be equivalent to a functor category of this form. In particular, we need to ask when we can recover C from [C^op,Set]. It turns out that C can be recovered iff it is Cauchy complete. In the second part of this talk we shall present various equivalent descriptions of the Cauchy completion of a category.

If there is time, we shall consider metric spaces as (enriched) categories and sketch why the Cauchy completion of a metric space considered as a category is the familiar Cauchy completion of a metric spaces to a complete metric space as encountered in Analysis, Functional Analysis and Topology.