预层的分级范畴
Graduated categories of presheaves
AI总结：
本文研究预层范畴中有限生成对象为有限可表现的条件，并刻画三类小范畴上预层范畴的分级性质，同时探讨降链条件与群预层的关系。
AI中文摘要：
我们研究小范畴 $\mathcal{A}$ 满足其预层范畴中每个有限生成对象都是有限可表现的条件。例如：对于群 $\mathcal{A}$，这成立当且仅当 $\mathcal{A}$ 是诺特群。对于序数 $\mathcal{A}=\alpha$，这成立当且仅当 $\alpha\leq \omega$，而对于 $\mathcal{A}=\alpha^{op}$ 则总是成立。集合函子的各种重要性质也适用于局部有限可表现范畴上的、具有分级（graduated）性质的端函子。这意味着每个有限可表现对象 $X$ 携带一个等级（在 $\mathbb{N}$ 中），且等级尊重 $X$ 的子对象和强商对象。我们刻画了对于三类小范畴 $\mathcal{A}$，预层范畴 $\mathbb{Set}^{\mathcal{A}^{op}}$ 是分级的条件。若 $\mathcal{A}$ 是偏序集，则所有元素的下集必须是有限的。对于群 $\mathcal{A}$，必须存在子群链长度的有限上界。在笛卡尔范畴 $\mathcal{A}$ 的情形，每个对象必须携带有限个筛子。例如：所有有限集合函子构成一个分级范畴，这里 $\mathcal{A}$ 是有限集合的对偶范畴。一个密切相关的概念是满足降链条件的局部有限可表现范畴（DCC范畴）：每个有限可表现对象只有有限的子对象或强商对象的降链。群 $G$ 上的预层范畴是 DCC 的当且仅当 $G$ 没有无限子群链。例如：群 $\mathbb{Z}$ 上的预层不是 DCC 的，但有限生成蕴含有限可表现。
英文摘要：
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.
