class of maps, but this is not enough: In order to have some structure in place to talk about one higher smoothness inductively, we need to consider above again for the tangent map $Tf: (x,v)\mapsto (f(x), Df(x)v), E_1\times E_0\mapsto F_1\times F_0$ in the place of $f: E_0\to F_0$; a natural choice of data analogous to $E_1$ would be $E_2\times E_1\subset E_1\times E_0$ with a compact dense inclusion. The first factor of the map $Tf$ is $f:E_1\to F_1$, so we need the assumptions in the above discussion to hold with subscripts shifted up by 1. Then we consider $T(Tf), \cdots, T(T^{k-1}f), \cdots$. So in general, we need $E_0\supset E_1\supset E_2\supset E_3\supset\cdots\supset E_{k+1}\supset\cdots$ a nested sequence of Banach spaces with the next one compactly densely included into the previous one, this $(E_m)_{m\in\mathbb{N}_0}$ is precisely the **sc-structure** (for an sc-Banach space). All of the above assumptions are summarized as [@Polyfoldanalysis] 2.1. However, an equivalent definition which is more direct and easier to use and check is provided in [@PolyfoldI] 2.13 as the official definition of $sc^1$, the new smooth notion of a map. The amazing fact is that this is not the order HWZ discovered it: They found the official definition first and 