Filter Quotients and Non-Presentable $(\infty,1)$-Toposes

Dual tangent structures for infinity-toposes

We describe dual notions of tangent bundle for an infinity-topos, each underlying a tangent infinity-category in the sense of Bauer, Burke and the author. One of those notions is Lurie's tangent bundle functor for presentable infinity-categories, and the other is its adjoint. We calculate that adjoint for injective infinity-toposes, where it is given by applying Lurie's tangent bundle on infinity-categories of points.

具有共良基原子范畴的原子拓扑斯

Atomic Toposes with Co-Well-Founded Categories of Atoms

本文给出原子位点的一般准则，使原子拓扑斯的原子可类似 Schanuel 拓扑斯描述，并证明其局部有限可表示，同时通过 Malitz-Gregory 例子反驳相关猜想，并刻画层为保拉回函子的组合条件。

Left determined model structures for locally presentable categories

We extend a result of Cisinski on the construction of cofibrantly generated model structures from (Grothendieck) toposes to locally presentable categories and from monomorphism to more general cofibrations. As in the original case, under additional conditions, the resulting model structures are "left determined" in the sense of Rosicky and Tholen.

Exponentiable Higher Toposes

We characterise the class of exponentiable $\infty$-toposes: $\mathcal X$ is exponentiable if and only if $\mathcal S\mathrm{h}(\mathcal X)$ is a continuous $\infty$-category. The heart of the proof is the description of the $\infty$-category of $\mathcal C$-valued sheaves on $\mathcal X$ as an $\infty$-category of functors that satisfy finite limits conditions as well as filtered colimits conditions (instead of limits conditions purely); we call such functors $ω$-continuous sheaves. As an application, we show that when $\mathcal X$ is exponentiable, its $\infty$-category of stable sheaves $\m…

将拓扑斯的语言表示为 span 范畴的商

Representing the language of a topos as quotient of the category of spans

本文通过 span 范畴的商刻画拓扑斯语言，并分析由逻辑关系诱导的商，进而证明以逻辑函子为态射时 Boolean 拓扑斯范畴构成拓扑斯范畴的反射子范畴。

离散动力系统的商拓扑斯

Quotient toposes of discrete dynamical systems

本文分类了离散动力系统在有限极限和小余极限下封闭的类，证明其与N×N的理想一一对应，并提供了处理Lawvere商拓扑斯开放问题的理论框架。