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 $(m_1,dots,m_k,0,*,dots,*)inDelta^n$ 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 $A:(Theta^n)^{mathrm{op}}tomathbf{Set}$ are said to be *bounded* if $A_M$ is countable for every $MinTheta^n$).

The argument starts by demonstrating that for any trivial cofibration $Ato C$ and any bounded subpresheaf $Bsubseteq C$, there are bounded subpresheaves $Bsubseteq B_omegasubseteq C$ and $A_omegasubseteq Atimes_CB_omega$ (he writes $Atimes_BB_omega$ but I imagine this is a typo) such that the induced map $A_omegato B_omega$ 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 $X$ $alpha$–boundedif $|X_n(U)|<alpha$ for every $Uinmathcal C$ and $ngeq0$. 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 $i:Ato C$ and any $alpha$-bounded subpresheaf $Bsubseteq C$, there is an $alpha$-bounded subpresheaf $Bsubseteq B_omegasubseteq C$ such that $Atimes_CB_omegato B_omega$ is a trivial cofibration. (Since $i$ is monic, we get that $Atimes_CB_omega$ 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 $Atimes_CB_omega$ is too big in his case.

Jardine then shows that if $p:Xto Y$ 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 $i:Ato C$.

To do so, he uses Zorn’s lemma on “partial lifts” which are diagramswith $i’$ a trivial cofibration and $Bneq A$. (Honestly, I’m not sure why we need to assert $Bneq A$ here.)

The ordering on partial lifts is given by monomorphisms $Bhookrightarrow B’$ which respect the maps $i$, $j$, $theta$ (though I am uneasy about this since such monomorphisms are not necessarily unique).

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 $Atimes_CB’to B’$ is a trivial cofibration, and pushes this out along $Atimes_CB’to A$ to obtain another trivial cofibration $i’:Ato B$.

The map $p:Xto Y$ admits a lift against $Atimes_CB’to B’$ by assumption, and the lift will factor uniquely through the pushout to give $theta:Bto X$. (If we didn’t assert $Bneq A$ earlier, $B=A$ would have also demonstrated this fact.)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 $Atimes_CB’to B’$ 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 $Cto B^*$, which (I think) is inverse to $j^*:B^*to C$ (but even if it isn’t, we can use $Cto B^*xrightarrow{theta^*}X$ 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 $Atimes_CB’to B’$ in the above argument with the trivial cofibration $A_omegato B_omega$, and push this out along $A_omegasubseteq Atimes_CB_omegato A$, then the rest of the argument should follow through virtually unchanged.

What makes me uneasy is that if the above were true, then I think the same argument can be used to “prove” the false claim that a map $f:Xto Y$ is bijective if any $Cto Y$ factors through $f$ (this property is satisfied by any surjective map, modulo the axiom of choice).

The reason would be that the only part of Jardine’s/Simpson’s argument that uses properties of trivial cofibrations is when constructing $A_omegato B_omega$ (where for Jardine, $A_omega=Atimes_CB_omega$), so I could use the rest of the argument for any class of morphisms so long as I can provide a substitute for this construction.

In particular, I could replace “trivial cofibrations” with “arbitrary maps of $mathbf{Set}$” and say that for any map $Ato C$ of sets, and any countable subset $Bsubseteq C$, then I can produce another map of sets $A_omegato B_omega$ where I take $Bsubseteq B_omegasubseteq C$ to be $B_omega := B$ and $A_omega := varnothing$.

Tracing out this argument, it would follow that a map of sets has the RLP against all maps (making it a bijection) if and only if it has lifts against the maps $varnothingto B$ for $B$ countable, because the only lifts that are ever needed in the rest of Jardine’s argument are given against those $A_omegato B_omega$.

This line of reasoning is clearly false (since the conclusion is), but I’m struggling to see at which point my understanding breaks down. Is there a nuance to Jardine’s argument that I’m overlooking, or somewhere else where the “trivial” part of a trivial cofibration is important that I missed?