# Injective model structure for simplicial presheaves

I am reading the paper by Jardine and Goerss, Localization theories for simplicial presheaves and having troubles with understand an argument. In this paper, the two authors considered $$\mathcal{C}$$ to be a small Grothendieck site (so in particular the underlying class of objects is a set and morphisms between two objects form a set). They proposed a list of axiom, called localization theories, so that whenever the site $$\mathcal{C}$$ satisfies this list, the category $$\mathbf{PreSh}(\mathcal{C},\Delta^{op}\mathbf{Sets})$$ of simplicial presheavse is a model category. For simplicity, we can think of weak equivalences are sectionwise equivalences and cofibrations are monomorphisms.

One of the axioms is:

Suppose that $$\gamma$$ is a limit ordinal and there is functor $$X: \gamma \longrightarrow \mathbf{PreSh}(\mathcal{C},\Delta^{op}\mathbf{Sets})$$ such that for each morphism $$i \leq j$$ in $$\gamma$$, $$X(i) \longrightarrow X(j)$$ is a trivial cofibration, then every canonical morphism $$X(i) \longrightarrow \operatorname{lim}_{j \in \gamma}X(j)$$ is a trivial cofibration.

As fas as I understand, this axiom was listed in order to make the small object argument work. Let me first pick a cardinal $$\alpha$$ bouding the set of morphisms of $$\mathcal{C}$$ and $$\beta$$ be some cardinal greater than $$2^{\alpha}$$ (this is where I do not understand the choice). Then the small object argument comes into the account: it factors any morphism $$f: X \longrightarrow Y$$ into $$X \longrightarrow X_{\beta} \longrightarrow Y$$ where $$X_{\beta}$$ is defined by a transfinite induction argument (basically, it is like what we should do for simplicial sets, we should fill all the holes $$\partial \Delta[n] \subset \Delta[n]$$), i.e. $$X_{\beta}$$ is some $$\beta$$-transfinite composition. What I do not understand is why $$X_{\beta} \longrightarrow Y$$ has to be a fibration. The authors proved this by using the lifting property. They proved that $$X_{\beta} \longrightarrow Y$$ has the right lifting property w.r.t all trivial cofibration $$i:U \longrightarrow V$$ with $$V$$ being $$\alpha$$-bounded (meaning that $$\left|V_n(A) \right| \leq \alpha$$ for every $$n\geq 0$$ and $$A \in \mathcal{C}$$).

Their argument is:

A morphism $$U \longrightarrow X_{\beta}$$ must factor through some stage $$X_{\gamma}$$ for some $$\gamma < \beta$$ because otherwise $$U$$ would have too many subobjects. Once the factorization is found, the lifting problem is solved.

I do not see why? Why $$U \longrightarrow X_{\beta}$$ does not factor implies that $$U$$ would have too many subobjects? And precisely, how many is too many?

To answer the question as it is stated: $$U$$ is an object in a locally presentable category, therefore $$U$$ is a small object, hence the corepresentable functor of $$U$$ preserves $$α$$-filtered colimits for some regular cardinal $$α$$. More precisely, the image of $$U$$ in such a colimit has a cardinality bounded by the product of cardinalities of $$U$$ and $$\cal C$$. Therefore, it must factor through some fixed stage of an $$α$$-filtered colimit if $$α$$ is greater than this cardinality.
However, that being said, the result discussed in the main post has been superseded by a considerably more general and easier to use theorem of Smith: for any left proper combinatorial model category, the left Bousfield localization at any set of morphisms $$S$$, and, more generally, at any class of morphisms $$S$$ that forms an accessible subcategory of $$C$$, exists. The conditions of Goerss and Jardine (especially, condition E7) simply guarantee that there is a set $$S$$ of such morphisms. For a reference, see, for example, Theorem 4.7 in Barwick's On left and right model categories and left and right Bousfield localizations.
• I understand what you commented, making an use of smallness, but what I am still not over is the choice of $\beta$, how can I prove that $U$ is $\beta$-small (let's ignore regularity of cardinals for simplicity). May 18, 2023 at 21:06
• @AlexeyDo: Take $β$ to be any regular cardinal larger than the total number of elements in all sets $U(X)$, $X∈C$. Now if $U→\mathop{\rm colim}_i Y_i$ is a morphism of simplicial presheaves, then the image of $U$ is a simplicial subpresheaf of the colimit, which has at most as many elements as $U$. Since colimits of simplicial presheaves are computed objectwise, every individual element in the image factors through some stage $Y_i$. Also, if restrictions of two sections of $U$ to the same object in $\cal C$ are equal, then the relevant elements become equal in some $Y_j$. May 18, 2023 at 21:22
• @AlexeyDo: Since $β$ exceeds the cardinality of the set of $i$ and $j$, all elements in the image $U$ as well as any relations between them factor through some fixed stage $Y_k$, which proves $U$ is $β$-small. May 18, 2023 at 21:23
• @AlexeyDo: If $β$ is a nonregular cardinal, then a category has $β$-filtered colimits if and only if it has $β^+$-filtered colimits, where $β^+$ is the successor cardinal of $β$, which is regular. Therefore, $β$-filtered colimits for nonregular cardinals $β$ are a redundant concept. Of course, if you only want to show an object to be $β$-small, there is no need to assume regularity. May 19, 2023 at 16:39