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I see that (from long, long time ago) one section of the entry graded vector space defines “pre-graded” to mean $\mathbb{Z}$-graded and “graded” to be $\mathbb{N}$-graded.
I am not sure if that is a good terminology, mainly because it seems not to be common. I came here from the entry dg-Lie algebra, wondering what that entry might actually mean by a “pre-graded” Lie algebra. (I should have commented on this long ago, of course).
Weird. I’ve never ever heard that terminology. Where does it come from?
Tim will know more, I suppose. (?) But I’d suggest that even if it comes from somewhere, we should deprecate it. And at dg-Lie algebra we should accordingly edit to make it clear that in general in fact dg-Lie algebras are $\mathbb{Z}$-graded.
Those pages were initially my attempt to understand Daniel Tanré’s lecture notes. I struck with his terminology at the time. I now find it awkward! $\mathbb{Z}$-graded is much clearer.
Is the formula $(f\otimes g) (v\otimes w) = (-1)^{|g||f|}(f(v) \otimes g(w))$ the intended one (last section on the tensor product)? For me the formula $(f\otimes g) (v\otimes w) = (-1)^{|g||v|}(f(v) \otimes g(w))$ is the standard one, and the associated category is not monoidal, but rather supermonoidal.
Thanks for the heads-up. This formula is from revision 9.
I am not sure what the intention of this category of “pre-gvs” really is.
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