  We introduce continuous Frobenius categories. These are topological
categories which are constructed using representations of the circle over a
discrete valuation ring. We show that they are 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 Frobenius
subcategories. The main purpose of constructing these Frobenius categories is
to give a precise and elementary description of the triangulated structure of
their stable categories. We show in arXiv:1209.1879 for which parameters these
stable categories have cluster structure in the sense of [1] and we call these
continuous cluster categories.
