I am trying to make sense of section 3 of the following paper. I quote Remark 3.5 below, which is causing me some confusion:
We implicitly formulate the theory of graded groups in such a way that the zero elements in different degrees are distinct. Thus, the notation $|u|$ is meaningful even if $u = 0$.
Of course, there is only one zero element and it is contained in each component of the grading so has undefined degree. With this idea in mind, how does one decide what the degree of 'a' zero element is? I just can't make sense of this explicitly. Is there a clearer/neater way to understand this?
When dealing with graded objects, homotopy theorists often consider only homogeneous elements. Indeed, while some people view a graded vector space (for example) as being a direct sum of vector spaces, one for each grading, others (and I've seen this in homotopy theory) view a graded vector space as a collection of vector spaces, one for each grading — not a direct sum, just a collection of independent objects. With the latter point of view, there are many zero elements, one for each graded piece.