@Aaron: I left the setup intentionally vague so that it can be (hopefully) modified to give a correct answer. I guess what I am really asking is weather or not something non-canonical can be done canonically in some sense.

@Fernando: yes, I believe so but I would like to know what the precise statements are.

The answer to your first question is yes if A is a hereditary category (meaning the higher Ext groups vanish), but it is not true in general.

@Jim: this sounds like what I was looking for. Do you know of any references for these aspects of the theory?

@John: this is more or less the kind of result I was looking for. Do you know if we can say anything about $\mu_n(x_1,\dots,x_n)$ if the product $\langle x_1,\dots, x_n\rangle$ is not defined? For example, does this force $\mu_n(x_1,\dots,x_n)=0$?