# Degenerating $0$ in Triangulated Categories

## TL;DR

This paper explores the phenomenon of the zero object degenerating in triangulated categories, revealing that such degenerations can generate all other degenerations and are linked to Grothendieck group properties.

## Contribution

It systematically studies zero object degeneration in triangulated categories, showing its role in generating all degenerations via homotopy pullback and its relation to Grothendieck group zero images.

## Key findings

- Zero object degeneration can induce all other degenerations.
- Degeneration of zero is linked to, but not equivalent to, zero image in Grothendieck group.
- Zero object degeneration is systematically characterized in triangulated categories.

## Abstract

In previous work, based on work of Zwara and Yoshino, we defined and studied degenerations of objects in triangulated categories analogous to degeneration of modules. In triangulated categories it is surprising that the zero object may degenerate. We study this systematically. In particular we show that the degeneration of the zero object actually induces all other degenerations by homotopy pullback, that degeneration of $0$ is closely linked, but not equivalent, to having zero image in the Grothendieck group.