发表机构
University of Cape Town; Stellenbosch University; National Institute for Theoretical and Computational Sciences (NITheCS)（开普敦大学; 斯泰伦博斯大学; 理论与计算科学国家研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过引入分配性形式，将Grothendieck拓扑刻画为筛子形式的外延表示，并应用于多种拓扑结构。
AI 中文摘要
范畴上的Grothendieck拓扑既决定了覆盖筛子的一个子形式，也决定了通过识别局部等价筛子而得到的商形式。我们将这两种构造置于一个短正合序列中。为此，我们引入了分配性形式：带指标的交半格，配备有特殊的指标并，使得有限指标交在其上分配，以及一个强版本，其中这些并还满足Beck--Chevalley条件。在固定基上，所得范畴具有所有小极限，并且相对于纤维上常顶态射的闭理想具有核和余核。它们的单态射是纤维上单射态射，并且它们的相对余核与顶反射态射一起构成一个正交分解系统。余核可复合但不必在拉回下稳定，而核则不必可复合。我们称具有给定中间项的短正合序列为外延表示，并证明范畴上的Grothendieck拓扑恰好是其最大分配性筛子形式的外延表示。进一步的应用恢复了泛乘积闭算子、Lawvere--Tierney拓扑、函子性非阿基米德群拓扑以及交换环上的函子性线性拓扑。
英文摘要
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.
Comments41 pages
