The 2-group of symmetries of a split chain complex

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.

arXiv comment23 p