∞-拓扑斯的形状理论：逆极限、乘积与（上）同调
Shape Theory of $\infty$-Topoi: Inverse Limits, Products, and (Co)homology
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中文总结 AI 辅助
该研究系统阐述∞-拓扑斯的形状理论，确立其函子性质、滤余极限保持性、乘积Künneth型公式及形状决定上同调与同调的条件。
中文摘要 AI 辅助
我们对∞-拓扑斯的形状理论进行了系统阐述，将∞-拓扑斯的形状视为其广义同伦型。我们确立了形状的基本函子性质，包括余极限保持、下降性质及同伦不变性；随后证明在紧性与完备性假设下形状保持滤余极限，并建立了乘积的Künneth型公式；最后给出∞-拓扑斯的形状决定其上同调与同调的条件。
英文摘要
We give a systematic account of the shape theory of $\infty$-topoi, viewing the shape of an $\infty$-topos as its generalized homotopy type. We establish the basic functorial properties of the shape, including preservation of colimits, descent, and homotopy invariance. We then prove that shape preserves cofiltered limits under compactness and perfectness hypotheses and establish Künneth-type formulas for products. Finally, we give conditions under which the shape of an $\infty$-topos determines its cohomology and homology.
