I was reading through "A closed model structure for $n$-categories, internal $Hom$, $n$-stacks and generalized Seifert Van-Kampen" by Simpson and was struggling to piece together the argument he made for decomposing morphisms into a trivial cofibration followed by a fibration (on pp. 45-46). Briefly, I'm failing to understand why the argument by Jardine in Lemma 2.4 in "Simplicial Presheaves" can be applied in this setting.
I'll try to summarise the important points for discussion to get to my main issues.
- the model structure is over the category of presheaves on $\Theta^n$ (called prenerves), which is a quotient of the $n$-fold product of the simplex category $\Delta$ by declaring all objects of the form $(m_1,\dots,m_k,0,*,\dots,*)\in\Delta^n$ to be the same. (It's defined explicitly on p. 5)
- cofibrations in this model structure are those that are levelwise monomorphisms, except at the top level. This is defined precisely on p. 12, but the point for this post is that (trivial) cofibrations are not necessarily monic
On pp. 45-46, Simpson argues that the trivial cofibrations are generated by those whose domain and codomain are both bounded (where $A:(\Theta^n)^{\mathrm{op}}\to\mathbf{Set}$ are said to be bounded if $A_M$ is countable for every $M\in\Theta^n$). The argument starts by demonstrating that for any trivial cofibration $A\to C$ and any bounded subpresheaf $B\subseteq C$, there are bounded subpresheaves $B\subseteq B_\omega\subseteq C$ and $A_\omega\subseteq A\times_CB_\omega$ (he writes $A\times_BB_\omega$ but I imagine this is a typo) such that the induced map $A_\omega\to B_\omega$ is also a trivial cofibration.
From here, he appeals to the argument made by Jardine in Lemma 2.4, saying that "the rest of [it] works." This is where I fail to make the connection, and it might possibly be because I don't fully understand his argument either, so I will recount it below (doing my best to fill in details):
Jardine was working with simplicial presheaves over a small site $\mathcal C$, and here the cofibrations were given by the (levelwise) monomorphisms. As the presheaves were over a small category, we can find a cardinal $\alpha > 2^{|\mathcal C_1|}$, from which we can call a simplicial presheaf $X$ $\alpha$-bounded if $|X_n(U)|<\alpha$ for every $U\in\mathcal C$ and $n\geq0$. Lemma 2.4 demonstrates that the trivial cofibrations are then generated by those whose domains and codomains are both $\alpha$-bounded.
The argument starts by showing that for any trivial cofibration $i:A\to C$ and any $\alpha$-bounded subpresheaf $B\subseteq C$, there is an $\alpha$-bounded subpresheaf $B\subseteq B_\omega\subseteq C$ such that $A\times_CB_\omega\to B_\omega$ is a trivial cofibration. (Since $i$ is monic, we get that $A\times_CB_\omega$ is also $\alpha$-bounded.) I can see that Simpson sets up the analogous result for prenerves, including $A_\omega$ to account for the possibility that $A\times_CB_\omega$ is too big in his case.
Jardine then shows that if $p:X\to Y$ has the right lifting property against all trivial cofibrations with $\alpha$-bounded domain and codomain, then it will have the right lifting property against any trivial cofibration $i:A\to C$. To do so, he uses Zorn's lemma on "partial lifts" which are diagrams
with $i'$ a trivial cofibration and $B\neq A$. (Honestly, I'm not sure why we need to assert $B\neq A$ here.) Edit: I don't think Jardine makes this assertion explicit, but it's also important that $j$ is monic. The ordering on partial lifts is given by monomorphisms $B\hookrightarrow B'$ which respect the maps $i$, $j$, $\theta$ (and they are necessarily unique from respecting $j$, as it is monic). An upper bound to any chain $(B_\lambda)_\lambda$ is then just the colimit $\varinjlim_\lambda B_\lambda$.
To show that there is at least one such partial lift, Jardine takes any $\alpha$-bounded subobject $B'\subseteq C$ not contained in $A$ such that $A\times_CB'\to B'$ is a trivial cofibration, and pushes this out along $A\times_CB'\to A$ to obtain another trivial cofibration $i':A\to B$. The map $p:X\to Y$ admits a lift against $A\times_CB'\to B'$ by assumption, and the lift will factor uniquely through the pushout to give $\theta:B\to X$. (If we didn't assert $B\neq A$ earlier, $B=A$ would have also demonstrated this fact.) The map $j:B\to C$ is induced by the pushout, and is monic because $B$ is precisely the union (as a subobject) of $B'$ and $A$, as the latter's intersection is precisely $A\times_CB'$.
By Zorn's lemma, there must then be a maximal partial lift $B^*$ and the claim is that $B^*=C$ (thus providing a lift against $i$). The reasoning is that $C$ is a filtered colimit of its $\alpha$-bounded subpresheaves (and we can restrict to those $B'$ for which $A\times_CB'\to B'$ is a trivial cofibration). Thus, a maximal element would form a cocone for the partial lifts these subpresheaves induce, and this must factor uniquely through $C$, giving a map $C\to B^*$, which (I think) is inverse to $j^*:B^*\to C$ (but even if it isn't, we can use $C\to B^*\xrightarrow{\theta^*}X$ as a lift anyway). $\square$
When trying to port this to Simpson's case, the only change I think needs to be made is that when showing that the system of partial lifts is nonempty, you replace the trivial cofibration $A\times_CB'\to B'$ in the above argument with the trivial cofibration $A_\omega\to B_\omega$. I imagine the idea is to push this out along $A_\omega\subseteq A\times_CB_\omega\to A$ to obtain the desired trivial cofibration $A\to B$. However, I fail to see why the induced map $j:B\to C$ will be monic.
To be explicit, the trivial cofibration $A_\omega\to B_\omega$ comes from
where I've indicated which maps are trivial cofibrations and which are monic. Note that in Simpson's case, trivial cofibrations are not necessarily monic (otherwise there would have been no need for $A_\omega$).
From this, the pushout I thought would be used gives the diagram
where the map $B_\omega\to B$ is monic because in a topos (and we are in a presheaf topos), monics are stable under pushout. In Jardine's case, all of the (undashed) arrows in the above diagram were monic, so we could conclude the same for $j$. Now, $A_\omega$ need not inject into $B_\omega$, and as a result I am fairly sure $j$ need not be a monomorphism at all.
