Definition of the shift C[n] in Complexes

Nobody has claimed this yet.

The function C[n] in Complexes, "Complex Array", shifts a complex C by setting C[n]_i = C_(n+i). I would argue for changing the definition to C[n]_i = C_(n-i). (The package also defines C[n] with a sign change of (-1)^n to the differentials of C, which is fine and not related to my comment.) Equivalently, in terms of cochain complexes, I'm recommending the definition C[n]^i = C^(n+i).

The convention I'm recommending has the effect that exact triangles in the derived category have the form A -> B -> C -> A[1] (rather than A[-1]). This agrees with the definition of triangulated categories by Verdier (Des categories derivees des categories abeliennes): his axioms say that exact triangles have the form A -> B -> C -> A[1]. (This is independent of whether you prefer to think in terms of chain complexes or cochain complexes.) The same convention for the meaning of C[n] is used in the standard books on derived categories by Gelfand-Manin (Methods of Homological Algebra) and Kashiwara-Schapira (Categories and Sheaves). I regret to say that Weibel's book on homological algebra uses the opposite convention, as Macaulay2 does now. But I think it's fair to say that there is stronger support for the convention I'm recommending.

Making this change would require a few edits to Complexes besides the definition, since shifts C[n] are used a few times. I would be happy to do that, but I am first checking what people think.

Start by inspecting the Complexes package and searching for the definition and uses of shifts C[n]. Compare the affected operations with the proposed convention, then update the definition and its dependent uses consistently; done means the package follows the new shift convention without conflicting existing behavior.

Written by the indexing model from the issue text.