Completely connected vs local

## Trigger 旧HP Problems 4.0.3（theme: Completely connected topoi）から移住する公開問題。出典は ryuya-hora-homepage の commit `1b8b74c`、`scripts/site.js` 849–858行。`created: 2026-03-30` は取り下げ済みPR #33の移住候補から引き継いだ代理値で、正確な着想日は不明。関連する公開論文は *Grothendieck topoi with a left adjoint to a left adjoint to a left adjoint to the global sections functor*。 ## Idea A presheaf topos \(\mathbf{PSh}(\mathcal{C})\) is completely connected if and only if \(\mathbf{PSh}(\mathcal{C}^{\mathrm{op}})\) is local. Is there a broader duality between completely connected topoi and local topoi? For example, is the category of Lawvere distributions on a local topos completely connected as in the presheaf case? If not, is the obstruction described by the monoidal closed structure of presentable categories? ## Goal presheaf caseを越えたcompletely connected topoiとlocal topoiの双対性を明らかにし、旧原文が挙げるLawvere distributionsの例とpresentable categoriesのmonoidal closed structureによる障害の可能性を検討する。 ## Personal context 旧タグは local topos、Lawvere distribution、presentable category。旧一覧以後の進展はこの移住では未確認であり、棚卸し待ちとして `status: dormant` とする。 ## References - [Ryuya Hora, *Grothendieck topoi with a left adjoint to a left adjoint to a left adjoint to the global sections functor*](https://arxiv.org/abs/2503.04317) - [Lawvere distribution](https://ncatlab.org/nlab/show/Lawvere+distribution) - [Tensor product of presentable categories](https://ncatlab.org/nlab/show/Pr%28infinity%2C1%29Cat#tensor_product) ## Provenance - 旧ID: 4.0.3 - 旧分類: informal question - [旧出典 `scripts/site.js` 849–858行](https://github.com/hora-algebra/ryuya-hora-homepage/blob/1b8b74c2b3447bdc744a0cac256c5d5806588adc/scripts/site.js#L849-L858) - Ideaの2段落は旧HPのstatement / descriptionをそのまま保持した。