Let be a Grothendieck topos, or more generally a locally presentable category. A cellular model for is a set of monomorphisms in 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 be the category of sheaves on some small site , let be the associated sheaf functor, and for each let denote the representable presheaf. Let be a set of representatives for isomorphism classes of all monomorphisms such that there exists an epimorphism in for some . Since toposes are both well-powered and well-copowered, is a small set.
It suffices to show that every monomorphism in can be written as an -cell complex. Let be a monomorphism and choose an surjection for some ordinal number . For each , let be the smallest subobject of that contains and also for all . Then and , exhibiting as the transfinite composite of the inclusions , so it will suffice to show that each of these inclusions is a pushout of a morphism in .
Now by the Yoneda lemma, corresponds to a map of presheaves , hence a map of sheaves . Let be the image of in (which may not coincide with the image in , of course). Then as subobjects of , so since is a coherent category we can write it as a pushout . However, the inclusion is (isomorphic to) some morphism in .
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.
Last revised on February 18, 2019 at 17:23:13. See the history of this page for a list of all contributions to it.