The class $\mathcal{CW}_n$ of finite $n$-dimensional CW complexes can be defined recursively:
$\mathcal{CW}_0$ consists of finite sets;
If $X \in \mathcal{CW}_n$, $\phi:S^n \amalg \cdots \amalg S^n \to X$ is continuous, then the pushforward $Y$ in \begin{array}{ccc} S^n\amalg \cdots \amalg S^n & \to & X \\ \downarrow&& \downarrow \\ D^{n+1}\amalg \cdots\amalg D^{n+1} & \to & Y \end{array} is in $\mathcal{CW}_{n+1}$.
i.e. $n$-dimensional CW complexes are built by recursively gluing $n$-dimensional cells onto $(n-1)$-dimensional skeletons. My question is whether there is a name for spaces achieved by the same construction, except that $n$-dimensional cells may be glued onto skeletons of any dimension.
To be precise, let $\mathcal A$ be the smallest class such that:
$\mathcal A$ contains finite sets, and
If $X \in \mathcal{A}$, $\phi:S^n \amalg \cdots \amalg S^n \to X$ is continuous, then the pushforward $Y$ in \begin{array}{ccc} S^n\amalg \cdots \amalg S^n & \to & X \\ \downarrow&& \downarrow \\ D^{n+1}\amalg \cdots\amalg D^{n+1} & \to & Y \end{array} is in $\mathcal{A}$.
Question: does the class $\mathcal A$ have a name?
Background: These spaces might not be useful for anything, but I have a related noncommutative construction and I'm looking for a name to give it, which could be inspired by the name for $\mathcal A$.