Graduated categories of presheaves

@misc{adamek2026graduated,
  title = {{Graduated categories of presheaves}},
  author = {Jiri Adamek and Lurdes Sousa},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30444},
  eprint = {2609.30444},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30444},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Graduated categories of presheaves
AU  - Jiri Adamek
AU  - Lurdes Sousa
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30444
UR  - https://arxiv.org/abs/2609.30444
AB  - We study conditions on a small category 𝒜 under which every finitely generated object of its presheaf category is finitely presentable. Example: this holds for a group 𝒜 iff 𝒜 is Noetherian. For ordinals 𝒜=α this holds iff α≤ ω, whereas this always holds for 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 ℕ) and grades respect subobjects and strong quotients of X. We characterize presheaf categories 𝕊et^𝒜^op which are graduated for three types of small categories 𝒜. If 𝒜 is a poset, all down sets of elements must be finite. For a group 𝒜, a finite bound on the length of a chain of subgroups must exist. In the case of cartesian categories 𝒜, every object must carry a finite number of sieves. Example: all finitary set functors form a graduated category, here 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 ℤ are not DCC, but finite generation implies finite presentation.
ER  -