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• there are 3-morphisms $H_0,K,K',F,F'$ which can be pasted together to form a 3-wfps $S$, which satisfy $F \prec K \prec H_0 \prec K' \prec F'$
• there is a 4-morphism $\Delta$ with $\Delta$ going from "$F+F'$" to $F_0$ which can be pasted on $S$
• but if we paste and activate $\Delta$, we get a pasting scheme with generators $H_0,K,K',F_0$ with $F_0 \prec K \prec H_0 \prec K' \prec F_0$, so we get a 3-pasting scheme which is not decomposable
• note that $\{F,F'\}$ is not a segment for $\triangleleft_S$ since $F \triangleleft_S H_0 \triangleleft F'$. So this example is ruled out by Johnson loop-free axiom (iv)

• there are 3-morphisms $H,H',K,K',F,F'$ which can be pasted together to form a 3-wfps $S$, which satisfy $F \prec K \prec H \prec H' \prec K' \prec F'$
• there is a 4-morphism $\Delta$ with $\Delta$ going from "$F+F'$" to "$\tilde{F}+\tilde{F}'$" which can be pasted on $S$. $F$ and $F'$ are essentially copies of $F$ and $F'$
• but if we paste on $S$, we get a pasting scheme which is not decomposable. Otherwise, by taking its source, we would get a composition of $S$ where $F$ and $F'$ appear as a segment in the composition order. But it is not possible, since $K,H,H',K'$ must appear between $F$ and $F'$ according to $\triangleleft_S$
• note that $\{F,F'\}$ is not a segment for $\triangleleft_S$. So this example is ruled out by Johnson loop-free axiom (iv)

So, why do we need this segment condition ? Firstly, it is important to note that a formal composition of the generators of an $n$-surface $S$ should respect the order $\triangleleft_S$ to compose the generators : if $x \prec_S y$, then $x$ needs to be "activated" before $y$, because something in the source of $y$ is produced by $x$. Secondly, in an free $\omega$-category without loops, if a generator $x$ of dimension $m$ can be inserted in a context $E[\_]$ of dimension $n<m$ such that $E[x]$ exists, then there exists a formal decomposition of $\partial^-_n E[x]$ in which the $n$-generators of $\partial^-_n x$ appear as a segment in the decomposition (see figure 17).

So, why do we need this segment condition ? Firstly, it is important to note that a formal composition of the generators of an $n$-surface $S$ should respect the order $\triangleleft_S$ to compose the generators : if $x \prec_S y$, then $x$ needs to be "activated" before $y$, because something in the source of $y$ is produced by $x$. Secondly, in an free $\omega$-category without loops, if a generator $x$ of dimension $m$ can be inserted in a context $E[\_]$ of dimension $n<m$ such that $E[x]$ exists, then there exists a formal decomposition of $\partial^-_n E[x]$ in which the $n$-generators of $\partial^-_n x$ appear as a segment in the decomposition (see figure 17). So, with these two facts, if we want the pasting of an $m$-generator on a $n$-surface to be compatible with an $\omega$-categorical structure, we need to require that generators can only be pasted on segments of $n$-surface.

I can provide some pieces of knowledge for Street and Johnson, and, in a lesser extent, for Steiner. Since my reputation on mathoverflow is not important enough, I am not able to include too many links and I can only cite figures in a master thesis I made for an internship on this subject : Parity complexes and pasting schemes

I can provide some pieces of knowledge for Street and Johnson, and, to a lesser extent, for Steiner. Since my reputation on mathoverflow is not important enough, I am not able to include too many links and I can only cite figures in a master thesis I made for an internship on this subject : Parity complexes and pasting schemes