Surely this must be known, but where is there a reference? If one considers any abelian category A, one can from a 2-category as follows: objects: differential graded objects in A (i.e. chain complexes) 1-arrows: chain maps 2-arrows: equivalence classes of chain homotopies where two chain homotopies a,b:F--->G are equivalent if there exits a map x of degree -2 from the common source of F and G to the common target of F and G such that a - b = xd - dx. The two dimensional composition of chain homotopies is addition, the one dimensional is given by if a:F--->G, b:H--->K (source(H)=target(G)) then a *1 b = aH + Gb (composites in diagrammatic order). Also, can one play the game again, and get a 3-category, etc.? A related question: does anyone know of a notion of "weak n-category" in which associativities are strict, but the middle-four-interchange holds only up to higher dimensional data? (That's what happens here if one uses chain homotopies instead of equivalence classes.) --David Yetter ==========================================================================