The 2-group of symmetries of a split chain complex

TL;DR

This paper explicitly computes the 2-group of self-equivalences and chain homotopies for split chain complexes in any abelian category, revealing its dependence on homology and its triviality for split exact sequences.

Contribution

It provides a detailed description of the 2-group of self-equivalences for split chain complexes, generalizing the concept of the general linear 2-group to chain complexes.

Findings

The 2-group is split and depends only on the homology of the complex.

It is trivial for split exact sequences.

The results generalize the linear 2-group of 2-vector spaces to chain complexes.

Abstract

We explicitly compute the 2-group of self-equivalences and (homotopy classes of) chain homotopies between them for any {\it split} chain complex $A_{\bullet}$ in an arbitrary $\kb$-linear abelian category ($\kb$ any commutative ring with unit). In particular, it is shown that it is a {\it split} 2-group whose equivalence class depends only on the homology of $A_{\bullet}$, and that it is equivalent to the trivial 2-group when $A_\bullet$ is a split exact sequence. This provides a description of the {\it general linear 2-group} of a Baez and Crans 2-vector space over an arbitrary field $\mathbb{F}$ and of its generalization to chain complexes of vector spaces of arbitrary length.