This question was inspired by the Homotopy Type Theory Book.

Might we define a weak $\omega$-category as described below?

Is any similar approach already considered in the literature?

Let $\def\Ob{{\rm Ob}\,} \Ob\mathcal C$ be a set [class] with partial functions $\def\dom{\rm dom} \def\cod{\rm cod} \dom,\cod$ defined on a subset $\mathcal C'\subseteq\Ob\mathcal C$ (i.e. $\dom,\cod:\mathcal C'\to\Ob\mathcal C$). We write of course $f:A\to B$ for $\dom(f)=A,\ \cod(f)=B$ and elements of $\mathcal C'$ are also called *arrows*.

We require a *composition* of consecutive arrows and *identity arrows*, satisfying associativity and unit constraintments with coherence isomorphisms such that the coherence diagrams commute *up to equivalence*.

(Objects $A$ and $B$ are *equivalent* if there is a binary tree of objects $X_{\bf t}$ and arrows $f_{\bf t}$, ${\bf t}\in\{0,1\}^*$ such that
$X_0=A,\ \ X_1=B,\ \ \ f_{{\bf t}0}:X_{{\bf t}0}\to X_{{\bf t}1},\ \
f_{{\bf t}1}:X_{{\bf t}1}\to X_{{\bf t}0}$
where objects $X_{{\bf t}00},\ X_{{\bf t}01},\ X_{{\bf t}10},\ X_{{\bf t}11}$ are defined to be $f_{{\bf t}1}\circ f_{{\bf t}0} $, $\ 1_{X_{{\bf t}0}}$,
$f_{{\bf t}0}\circ f_{{\bf t}1} $, $\ 1_{X_{{\bf t}1}}$; so that $A\simeq B \iff \exists f:A\to B,\ g:B\to A\,$ s.t. $\,g\circ f\simeq 1_A,\ f\circ g\simeq 1_B$)

To formulate these, we also need a *horizontal composition* functor, from the full substructure of $\mathcal C'\times\mathcal C'$ generated by consecutive pairs of arrows as objects, to $\mathcal C'$.

(For first,) globularity is assumed (i.e., $\alpha:f\to g$ with $f,g\in\mathcal C'$ implies $\cod(f)=\cod(g)$ and $\dom(f)=\dom(g)$).

Objects are $0$-cells, and an arrow $f:A\to B$ is regarded an $n+1$-cell whenever $A$ and $B$ are $n$-cells. [Note that an $n$-cell is automatically also $k$-cell for any $k<n$.]

More details can be found in my note.