Hello! I would appreciate if someone explain to me this cochain complex. I know it is a sequence of abelian groups connected by homomorphisms such such that the composition of any two consecutive maps is zero

here is the cochain complex

But I would like someone explain to me what these indices mean? on wikipedia it says that n corresponds to the degree or dimension. Dimenson of an abelian group? (nr of elements). How the dimension of an abelian group can be -2 for example? How it is related to d? I guess d must be the exterior derivative or differential form.
For example we have $\displaystyle d^0, d^1, d^2....$ which would mean 0-form, 1-form, 2-form and so on but when I learn abour r-forms we used to write the number next to d as the low index....
the other guess it is the exterior derivative.....$\displaystyle d^0, d^1, d^2....$ or a degree: 0,1,2 and so on? but if we go the other way does it exist an exterior derivative with the degree -1, -2????

Please could someone give me a detailed explanaition of interpretation this cochain?