Ross Street's 1991 paper Parity Complexes (apologies; I don't know how to find DOI links for Cahiers papers) develops some very useful tools for working with free strict $\omega$-categories. There is a corrigenda to the paper. I find it a bit difficult to assemble together these two ingredients to be sure what are the correct statements of the main theorems in Street's paper. A useful later reference for this material is Buckley's Formalizing Parity Complexes.
I am interested in understanding Theorem 4.2, which unfortunately is not covered in Buckley's treatment. The theorem states
Theorem 4.2 The $\omega$-category $O(C)$ is freely generated by the atoms.
Let's break that down:
As indicated in the paper, the notion of "free generation" comes from Street's earlier The Algebra of Oriented Simplices. I believe this notion is to be read as-is without change from the corrigenda.
I believe that as originally written (but see (5) below), $C$ was intended to be an arbitrary parity complex, a notion defined in Section 1 of the paper; I believe this definition is faithfully reproduced at the linked nlab page (the nlab's $<$ being Street's $\triangleleft$ and the nlab's $\prec$ being Street's $\blacktriangleleft$).
$O(C)$ is the $\omega$-category defined at the beginning of Section 3. It is proven in Theorem 3.6 that for any parity complex $C$, $O(C)$ is an $\omega$-category. I believe that Theorem 3.6 is understood to be true as stated -- the corrigenda does not indicate that the definition of $O(C)$ (or the subsidiary notions of cells or well-formed subsets of $C$) need be changed, nor does it indicate that any additional hypothesis on the parity complex $C$ is needed to ensure that $O(C)$ is an $\omega$-category (Thm 3.6).
The notion of an atom is as defined in Section 4 of the paper.
I believe the corrigenda indicates that the statement of Thm 4.2 should be changed as follows. On p. 1 of the corrigenda, it is indicated that for every element $x \in C_p$ of the parity complex $C$, we need to assume throughout Section 4 (including, apparently, in the statement of Thm 4.2) that the sets $\mu(x)$ (defined at the beginning of Section 4, with the definition corrected at the beginning of the corrigenda) are tight in the sense defined further down p. 1 of the corrigenda.
Therefore, I believe the correct statement of Theorem 4.2 is:
Theorem 4.2, correctus: Let $C$ be a parity complex. Assume that for every $p \in \mathbb N$ and every $x \in C_p$, the sets $\mu(x)$ (as defined in the corrigendum, not as defined in the paper) are tight (as defined in the corrigendum). Then the $\omega$-category $O(C)$ is freely generated by the atoms.
Question 1: Do I have that right?
The corrigendum also defines a notion of globularity at the beginning of p.2, and I believe the corrigendum asserts that for every parity complex $C$, and for every relevant $x \in C_p$ (as defined in Section 4), the globularity condition holds for $x$. I believe that Prop 5.2 of the corrigendum gives a criterion ensuring that the globularity condition holds for all $x \in C_p$, not just for the relevant $x \in C_p$. This condition uses the corrected definition of $\mu(x)$ as well as the corrected definition of $\pi(x)$.
Question 2: Can the corrected statement of Theorem 4.2 be simplified by assuming something about the "globularity condition" rather than explicitly assuming something about tightness?