Continuous Frobenius categories

TL;DR

This paper introduces continuous Frobenius categories constructed from circle representations over a discrete valuation ring, providing a clear description of their stable categories' triangulated structure and conditions for cluster structures.

Contribution

It defines a new class of topological Frobenius categories and characterizes their stable categories, linking them to cluster structures under specific parameters.

Findings

Categories are Krull-Schmidt with indecomposables for each circle point pair.

Subcategories obtained by restricting points on the circle.

Stable categories exhibit cluster structures under certain conditions.

Abstract

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.