A geometric model of tube categories

TL;DR

This paper introduces a geometric model for tube categories using arcs in an annulus, linking extension groups to geometric intersection numbers, and extends the approach to type A-double-infinity quivers.

Contribution

It provides a novel geometric interpretation of extension groups in tube categories and related quiver representations, connecting algebraic and geometric perspectives.

Findings

Extension groups correspond to negative geometric intersection numbers.

The model applies to finite dimensional representations of type A-double-infinity quivers.

Provides a geometric framework for understanding extension groups in these categories.

Abstract

We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative geometric intersection numbers between corresponding arcs, giving a geometric interpretation of the description of an extension group in the cluster category of a tube as a symmetrized version of the extension group in the tube. We show that a similar result holds for finite dimensional representations of the linearly oriented quiver of type A-double-infinity.