Grothendieck Topologies Are Extensional Presentations of the Form of Sieves

@misc{ferguson2026grothendieck,
  title = {{Grothendieck Topologies Are Extensional Presentations of the Form of Sieves}},
  author = {Roy Ferguson and Zurab Janelidze},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14671},
  eprint = {2609.14671},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14671},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Grothendieck Topologies Are Extensional Presentations of the Form of Sieves
AU  - Roy Ferguson
AU  - Zurab Janelidze
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14671
UR  - https://arxiv.org/abs/2609.14671
AB  - A Grothendieck topology on a category determines both a subform of covering sieves and a quotient form obtained by identifying locally equivalent sieves. We place these two constructions in a single short exact sequence. For this purpose we introduce distributivity forms: indexed meet-semilattices equipped with distinguished indexed joins over which finite indexed meets distribute, and a strong version in which these joins also satisfy Beck--Chevalley. Over a fixed base, the resulting categories have all small limits and have kernels and cokernels relative to the closed ideal of fibrewise constant-top morphisms. Their monomorphisms are the fibrewise injective morphisms, and their relative cokernels, together with the top-reflecting morphisms, form an orthogonal factorization system. Cokernels compose but need not be stable under pullback, whereas kernels need not compose. We call a short exact sequence with prescribed middle term an extensional presentation, and prove that Grothendieck topologies on a category are precisely the extensional presentations of its maximally distributive form of sieves. Further applications recover universal productive closure operators and Lawvere--Tierney topologies, functorial non-Archimedean group topologies, and functorial linear topologies on commutative rings.
ER  -