On lifting stable diagrams in Frobenius categories

Abstract

Suppose given a Frobenius category $\mathcal{E}$, i.e. an exact category with a big enough subcategory $\mathcal{B}$ of bijectives. Let $\mathcal{E} := \mathcal{E/B}$ denote its classical stable category. For example, we may take $\mathcal{E}$, to be the category of complexes $C(\mathcal{A})$ with entries in an additive category $\mathcal{A}$, in which case $\underline{\mathcal{E}}$ is the homotopy category of complexes $K(\mathcal{A})$. Suppose given a finite poset $D$ that satisfies the combinatorial condition of being ind-flat. Then, given a diagram of shape $D$ with values in $\underline{\mathcal{E}}$ (i.e. stably commutative), there exists a diagram consisting of pure monomorphisms with values in $\mathcal{E}$ (i.e. commutative) that is isomorphic, as a diagram with values in $\underline{\mathcal{E}}$, to the given diagram.