分布截面范畴的新界限及其在分布同伦距离中的应用
New Bounds on Distributional Sectional Category and Applications to Distributional Homotopic Distance
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中文总结 AI 辅助
本文针对分布截面范畴（dsecat）建立新界限，证明Jauhari猜想并得到不同系数下的下界，应用于分布同伦距离，推导相关等价表述与新不等式。
中文摘要 AI 辅助
本文针对分布截面范畴（$\boldsymbol{\text{dsecat}}$）建立了若干新的界限。首先证明了Jauhari猜想，从而得到任意系数下$\text{dsecat}$的上同调下界；接着通过构造对角包含诱导的上同调映射到对称幂的自然分裂，得到有理系数下的类似下界。作为应用，给出了$\text{dsecat}$的若干计算结果，还建立了$\text{dsecat}$的乘法积不等式与复合不等式。最后将这些结果应用于Jauhari和Oprea新近提出的分布同伦距离，给出其基于分布同伦的等价表述，并推导得到新的纤维化不等式与乘法三角不等式。
英文摘要
In this paper, we establish several new bounds for the distributional sectional category ($\mathrm{dsecat}$). We first prove Jauhari's conjecture, thereby establishing a cohomological lower bound for $\mathrm{dsecat}$ with arbitrary coefficients. We then obtain an analogous lower bound with rational coefficients by constructing a natural splitting of the map induced on cohomology by the diagonal inclusion into symmetric powers. As applications, we provide several computations of the distributional sectional category. We also establish multiplicative product and composition inequalities for $\mathrm{dsecat}$. Finally, we apply these results to the distributional homotopic distance recently introduced by Jauhari and Oprea, giving an equivalent formulation in terms of distributed homotopies and deriving new fibration and multiplicative triangle inequalities.
