Cellular models

# Cellular models

## Definition

Let $E$ be a Grothendieck topos, or more generally a locally presentable category. A cellular model for $E$ is a set of monomorphisms $I$ in $E$ that cofibrantly generates a weak factorization system whose left class is precisely the monomorphisms.

## Existence

###### Theorem

A cellular model exists in any Grothendieck topos.

For presheaf toposes this is Cisinski 06, prop. 1.2.27; the result for general toposes is stated without proof as Cisinski 03, prop. 1.2.2. The proof below is a direct generalization of the presheaf version.

###### Proof

Let $E=Sh(C)$ be the category of sheaves on some small site $C$, let $a : [C^{op},Set] \to Sh(C)$ be the associated sheaf functor, and for each $c\in C$ let $y_c = C(-,c)$ denote the representable presheaf. Let $I$ be a set of representatives for isomorphism classes of all monomorphisms $i:A\to B$ such that there exists an epimorphism $a y_c \to B$ in $E$ for some $c\in C$. Since toposes are both well-powered and well-copowered, $I$ is a small set.

It suffices to show that every monomorphism in $E$ can be written as an $I$-cell complex. Let $f:X\to Y$ be a monomorphism and choose an surjection $w : \lambda \to \coprod_{c\in C} Y(c)$ for some ordinal number $\lambda$. For each $\beta\le \lambda$, let $X_\beta$ be the smallest subobject of $Y$ that contains $X$ and also $w_\alpha$ for all $\alpha\lt\beta$. Then $X_0 = X$ and $X_\lambda = Y$, exhibiting $f$ as the transfinite composite of the inclusions $X_\beta \hookrightarrow X_{\beta+1}$, so it will suffice to show that each of these inclusions is a pushout of a morphism in $I$.

Now by the Yoneda lemma, $w_\beta\in Y(c)$ corresponds to a map of presheaves $y_c \to Y$, hence a map of sheaves $a y_c \to Y$. Let $D_\beta$ be the image of $a y_c \to Y$ in $E$ (which may not coincide with the image in $[C^{op},Set]$, of course). Then $X_{\beta+1} = X_\beta \cup D_\beta$ as subobjects of $Y$, so since $E$ is a coherent category we can write it as a pushout $X_{\beta+1} = X_\beta \sqcup_{(X_\beta \cap D_\beta)} D_\beta$. However, the inclusion $X_\beta \cap D_\beta \hookrightarrow D_\beta$ is (isomorphic to) some morphism in $I$.

More generally, cellular models exist in any locally presentable coherent category whose monomorphisms are closed under transfinite composition. The proof is similar, using a strong generating set instead of the representables, and the fact that every locally presentable category is well-powered and well-copowered; see Beke00, Proposition 1.12.

• Tibor Beke, Sheafifiable homotopy model categories. Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 447–475