What is the correct statement of Theorem 4.2 in Street's "Parity Complexes"?

Question

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:

1. 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.

2. 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 $\blacktriangleleft$).

3. $O(C)$ 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).

4. The notion of an atom is as defined in Section 4 of the paper.

5. 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, correcta: 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?

Answer 1

I hope I can offer some quick answers to your questions without errors.

Let's tackle the breaking down:

1. 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.

Yes, it is the notion of freeness of this article. Using some rephrasing, it means that there exists a polygraph/computad $P$ such that $O(C)$ is isomorphic to $P^*$, the free $\omega$-category on $P$.

2. 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 $\blacktriangleleft$).

It seems that the well-formed condition of parity complexes is badly reproduced in the nlab (condition 2.). Indeed, it not only a condition on the 1-cells but also on higher cells. Moreover, nlab's $<$ is Street's $<$ and nlab's $\prec$ is Street's $\triangleleft$.

3. $O(C)$ 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).

If I remember correctly, yes, the additions of the corrigenda is not required in order to obtain an $\omega$-category. So one can start from any parity complex.

4. The notion of an atom is as defined in Section 4 of the paper.

Yes.

5. 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.

Yes, this correta seems correct to me, in the sense that it should be the one deduced from the corrigenda.

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?

No, it can not be simplified to a globularity condition. The counter-example I gave in my article is a parity complex which satisfies the globularity condition. Still, Theorem 4.2 does not hold for this example.

By the way, it is no coincidence that Theorem 4.2 of Street's paper is not covered by Buckley, since it does not hold in its full generality with or without the corrigenda (but the counter-examples, like the already cited one, are very peculiar, so that most if not all the examples which use parity complexes in the literature should be fine).