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probably that's still a remnant of what I edited?
Sorry for that. Yes, let's get rid of thosse codifferentials.
I think that the terminology ‘codifferential’ comes from situations where you have both chains and cochains interacting.
In manifold theory (which is where I first met these things in high school), you have a chain complex (consisting of formal linear combinations of certain submanifolds, which we really call ‘chains’) and a cochain complex of differential forms. And the operator on the chain complex takes the boundary, while the operator on the cochain complex of forms takes the differential (resulting in an exact form).
But then if you want to stress the duality between these two complexes (given by integrating a form on a chain), then you can say ‘cochain’ for a form, ‘coboundary’ for the differential operator or an exact form, and ‘codifferential’ for a boundary. Similarly, you can say ‘cocycle’ for a closed form.
Having come to homological algebra from manifold theory, I confess a like for keeping ‘differential’ with cochain complexes (where the differential has degree 1) and ‘boundary’ with chain complexes (where the boundary has degree −1). But even I don't see the point of saying ‘coboundary’ or ‘codifferential’ when the other name is available.
[Only minor grammatical editing done.]
Will you then skip saying cocycle (in nonabelian cohomology for example) and start saying cycle ? If one uses word cocycle then one uses cochain in the same context. If one talks about categories of complexes than it is not much important, but in real context, say when one talks about manifolds and various chains and cochains there than it points to some meaning when one says cochain and one says chain.
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