Graduated categories of presheaves

Abstract

We study conditions on a small category $\mathcal{A}$ under which every finitely generated object of its presheaf category is finitely presentable. Example: this holds for a group $\mathcal{A}$ iff $\mathcal{A}$ is Noetherian. For ordinals $\mathcal{A}=α$ this holds iff $α\leq ω$, whereas this always holds for $\mathcal{A}=α^{op}$. Various important properties of set functors also apply to endofunctors on locally finitely presentable categories which are graduated. This means that every finitely presentable object $X$ carries a grade (in $\mathbb{N}$) and grades respect subobjects and strong quotients of $X$. We characterize presheaf categories $\mathbb{Set}^{\mathcal{A}^{op}}$ which are graduated for three types of small categories $\mathcal{A}$. If $\mathcal{A}$ is a poset, all down sets of elements must be finite. For a group $\mathcal{A}$, a finite bound on the length of a chain of subgroups must exist. In the case of cartesian categories $\mathcal{A}$, every object must carry a finite number of sieves. Example: all finitary set functors form a graduated category, here $\mathcal{A}$ is the dual of finite sets. A closely related concept is a locally finitely presentable category with the descending chain condition (a DCC category): every finitely presentable object has only finite descending chains of subobjects or strong quotients. The presheaf category on a group $G$ is $DCC$ iff $G$ has no infinite chain of subgroups. Example: presheaves on the group $\mathbb{Z}$ are not $DCC$, but finite generation implies finite presentation.