Continuous Forbenius Categories

We introduce the continuous Frobenius category. This category is constructed using representations of the circle over a discrete valuation ring. We show that it is Krull-Schmidt with one indecomposable object for each pair of not necessarily distinct points on the circle. By putting restrictions on these points we obtain various subquotient categories with good properties. The main purpose of our Frobenius categories is to give the precise triangulated structure of their stable categories which we call the continuous cluster categories. Cluster categories are categorifications of cluster algebras. Mutation of cluster variables is defined using the triangulated structure of these categories as we will explain in other papers.