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I'm having trouble filling in a detail in Goerss and Jardine's book Simplicial homotopy theory. Their Proposition 4.2 claims that two classes of monomorphisms $\mathbf{B}_2\subset\mathbf{B}_3$ in the category of simplicial sets define the same saturated class, in their notation, $M_{\mathbf{B}_2}=M_{\mathbf{B}_3}$.

How do you prove $M_{\mathbf{B}_2}=M_{\mathbf{B}_3}$?

I'm following their proof suggestion. The first few paragraphs are to introduce notation; there is a specific point that I don't see how to deal with (see the highlighted bit at the end).


$\mathbf{B}_2$ is the class of all inclusions $$\Delta^1\times\partial\Delta^n\underset{e\times\Delta^n}{\cup}e\times\Delta^n\;\hookrightarrow\; \Delta^1\times\Delta^n$$ (simply written $\Delta^1\times\partial\Delta^n \cup e\times\Delta^n\hookrightarrow\Delta^1\times\Delta^n$) with $e=0,1$, while

$\mathbf{B}_3$ is the class of all inclusions $$\Delta^1\times A\cup e\times X\;\hookrightarrow\; \Delta^1\times X$$ again, $e=0,1$, where $A\hookrightarrow X$ is an arbitrary monomorphism.


Obvisouly $M_{\mathbf{B}_2}\subset M_{\mathbf{B}_3}$. The text provides an indication as to how to prove the reverse inclusion: for every inclusion $A\subset X$, $X$ is obtained from $A$ by adjoining cells. Inspired by the proof of Corollary 4.6, I tried to prove that the class $\cal C$ of all monomorphisms $A\hookrightarrow X$ such that $\left(\Delta^1\times A\cup e\times X\;\hookrightarrow\; \Delta^1\times X\right)\in M_{\mathbf{B}_2}$ is a saturated class (minus the stability by retracts part).

By definition of $M_{\mathbf{B}_2}$, $\cal C$ contains all boundary inclusions $\partial\Delta^n\subset\Delta^n$. Also, every inclusion $A\hookrightarrow X$, $X$ can be recovered as a colimit of complexes $X^{(n)}$, $n\geq -1$, with $X^{(-1)}=A$ and $X^{(n)}$ is obtained from $X^{(n-1)}$ by adjoining cells, i.e. as a pushout diagram $$\begin{matrix} \coprod\partial\Delta^n & \hookrightarrow & \coprod\Delta^n\\ \downarrow &&\downarrow\\ X^{(n-1)} &\rightarrow & X^{(n)} \end{matrix}$$ (the coproduct is taken over all non-degenerate $n$-simplices in $X$ that aren't already in $A$) Then stability by transfinite composition of $\cal C$ would guarantee that the inclusion $$A\hookrightarrow\mathrm{colim}_n X^{(n)}$$ is in $\cal C$, but this inclusion is isomorphic to the original inclusion $A\hookrightarrow X$ and so $\cal C$ would be the class of all monomorphisms, proving $M_{\mathbf{B}_3}\subset M_{\mathbf{B}_2}$


I can prove that $\cal C$ contains all isomorphisms, is closed under pushouts and arbitrary coproducts, but I don't see why it is stable under countable compositions (I don't think I need stability under retracts).

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Ok, the solution isn't that bad. Consider an infinite sequence of monomorphisms in $\cal C$ $$A=A_1\overset{i_1}{\hookrightarrow}A_2\overset{i_2}{\hookrightarrow}A_3\overset{i_3}{\hookrightarrow}\cdots\hookrightarrow\mathrm{colim}_n\,A_n=X$$ We need to prove that the inclusion $A\hookrightarrow X$ is still in $\cal C$, that is, that the inclusion $$\Delta^1\times A\cup e\times X\;\hookrightarrow\;\Delta^1\times X$$ is in $M_{\mathbf{B}_2}$, for $e=0,1$. But this is because for any $n$, the square below is a pushout: $$\begin{matrix} \Delta^1\times A_n\cup e\times A_{n+1} & \hookrightarrow & \Delta^1\times A_{n+1}\\ \downarrow&&\downarrow\\ \Delta^1\times A_n\cup e\times X &\to&\Delta^1\times A_{n+1}\cup e\times X \end{matrix}$$ and since the top morphism is in $M_{\mathbf{B}_2}$ by hypothesis, so is the monomorphism $$\underbrace{\Delta^1\times A_n\cup e\times X}_{B_n} \;\hookrightarrow\;\underbrace{\Delta^1\times A_{n+1}\cup e\times X}_{B_{n+1}}$$ By construction, $B_1=\Delta^1\times A\cup e\times X$, and $\mathrm{colim}_n\,B_n=\Delta^1\times X$, and the stability by countable compositions of $M_{\mathbf{B}_2}$ implies that the inclusion $$\left(\Delta^1\times A\cup e\times X\,\hookrightarrow\,\Delta^1\times X\right)\in M_{\mathbf{B}_2}$$ i.e. $\left(A\hookrightarrow X\right)\in\cal C$.


Everything about this proof still works for a general inclusion $K\subset L$ rather than the two inclusions $d^0,d^1:\Delta^0\hookrightarrow\Delta^1$, or for a collection of inclusions.

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