Suppose given a Frobenius category E, i.e. an exact category with a big enough subcategory B of bijectives. Let E := E/B denote its classical stable category. For example, we may take E to be the category of complexes C(A) with entries in an additive category A, in which case E is the homotopy category of complexes K(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 E (i.e. stably commutative), there exists a diagram consisting of pure monomorphisms with values in E (i.e. commutative) that is isomorphic, as a diagram with values in E , to the given diagram.
Homology, Homotopy and Applications, Vol. 9 (2007), No. 1, pp.163-183.
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