nLab cellular model

Cellular models

Cellular models


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



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.


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

It suffices to show that every monomorphism in EE can be written as an II-cell complex. Let f:XYf:X\to Y be a monomorphism and choose an surjection w:λ cCY(c)w : \lambda \to \coprod_{c\in C} Y(c) for some ordinal number λ\lambda. For each βλ\beta\le \lambda, let X βX_\beta be the smallest subobject of YY that contains XX and also w αw_\alpha for all α<β\alpha\lt\beta. Then X 0=XX_0 = X and X λ=YX_\lambda = Y, exhibiting ff as the transfinite composite of the inclusions X βX β+1X_\beta \hookrightarrow X_{\beta+1}, so it will suffice to show that each of these inclusions is a pushout of a morphism in II.

Now by the Yoneda lemma, w βY(c)w_\beta\in Y(c) corresponds to a map of presheaves y cYy_c \to Y, hence a map of sheaves ay cYa y_c \to Y. Let D βD_\beta be the image of ay cYa y_c \to Y in EE (which may not coincide with the image in [C op,Set][C^{op},Set], of course). Then X β+1=X βD βX_{\beta+1} = X_\beta \cup D_\beta as subobjects of YY, so since EE is a coherent category we can write it as a pushout X β+1=X β (X βD β)D βX_{\beta+1} = X_\beta \sqcup_{(X_\beta \cap D_\beta)} D_\beta. However, the inclusion X βD βD βX_\beta \cap D_\beta \hookrightarrow D_\beta is (isomorphic to) some morphism in II.

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

Last revised on February 18, 2019 at 17:23:13. See the history of this page for a list of all contributions to it.