AI 中文总结
该研究在∞-范畴框架下发展动机范畴的内部同伦理论，建立相关准则，证明双有理动机同伦范畴具有特定拓扑斯性质，为动机同伦范畴的性质提供关键结论。
AI 中文摘要
我们为内部艾伦伯格-麦克莱恩对象、内部同伦群以及覆盖空间的内部概念建立了一个通用的∞-范畴框架，并研究它们在合适局部化下的行为。将此应用于普通动机局部化，我们将内部n-艾伦伯格-麦克莱恩对象与强𝔸¹-不变的群层（当n≥2时为阿贝尔群）对应起来，这为动机同伦范畴成为∞-拓扑斯提供了形式上的障碍。我们证明，取𝔸¹-局部化诱导了尼斯涅维奇局部空间的经典𝔸¹-覆盖与其动机局部化的内部动机覆盖之间的等价，从而为广义动机范坎彭定理提供了简化证明。在此过程中，我们建立了尼斯涅维奇局部到整体的𝔸¹-连通性准则：k-概型是𝔸¹-连通的，当且仅它存在由𝔸¹-连通概型构成的尼斯涅维奇覆盖，且每对交集都有一个（k-）点。最后，我们证明在一般拟紧拟分离（Qcqs）基上，过渡到双有理动机同伦范畴可恢复普通𝔸¹-情形中缺失的某些本质拓扑斯理论性质。事实上，对于具有有限多个 generic 点的概型，我们证明双有理动机同伦范畴是上同调维数为0的后斯涅维奇完备∞-拓扑斯。
英文摘要
We develop a general $\infty$-categorical framework for internal Eilenberg-MacLane objects, internal homotopy groups, and an internal notion of covering spaces, and study their behavior under suitable localizations. Applying this to the ordinary motivic localization, we identify internal $n$-Eilenberg-MacLane objects with strongly $\mathbb{A}^1$-invariant sheaves of (abelian when $n\geq 2$) groups, yielding a formal obstruction to the motivic homotopy category being an $\infty$-topos. We prove that taking $\mathbb{A}^1$-localizations induces an equivalence between classical $\mathbb{A}^1$-coverings of a Nisnevich local space and the internal motivic coverings of its motivic localization, providing a streamlined proof of a generalized motivic Van Kampen theorem. Along the way, we establish a Nisnevich-local-to-global $\mathbb{A}^1$-connectivity criterion: a $k$-scheme is $\mathbb{A}^1$-connected if and only if it admits a Nisnevich cover by $\mathbb{A}^1$-connected schemes such that each pairwise intersection has a ($k$-)point. Finally, we show that over a general Qcqs base, passing to the birational motivic homotopy category recovers certain essential topos-theoretic properties absent in the ordinary $\mathbb{A}^1$-setting. In fact, for a scheme with finitely many generic points, we show that the birational motivic homotopy category is a Postnikov-complete $\infty$-topos of cohomological dimension $0$.
