A 型 Soergel 双模上的编织：半严格性与自然性
Braiding on type A Soergel bimodules: semistrictness and naturality
AI总结：
本文在具体 dg-模型中显式构造了 A 型 Soergel 双模链复形范畴上的编织结构，建立其半严格幺半 2-范畴框架，并给出编织自然性的高阶同伦数据。
摘要：
我们考虑对称群 S_n 在其 gl(n)-实现中的 Soergel 双模范畴（对所有 n），并将它们组装成一个局部线性幺半双范畴。Soergel 双模的链复形同样构成一个局部 dg-幺半双范畴，该范畴可配备编织结构，其数据包括 shuffle 辫子的 Rouquier 复形。该编织及其唯一性结果已在最近与 Yu Leon Liu、Aaron Mazel-Gee 和 David Reutter 的合作工作中于无穷范畴框架下建立。在本文中，我们在态射范畴的具体 dg-模型中显式构造该编织，并描述其所必需的相干自然性结构。为此，我们首先将 Elias-Khovanov-Williamson 图解 Hecke 范畴及其上的链复形范畴组装成局部线性半严格幺半 2-范畴。在此过程中，我们证明了某些标准范畴构造的严格性结果，这些结果可能具有独立的研究价值。第二步，我们针对 Elias-Khovanov-Williamson 图解演算的生成态射，给出编织自然性的显式（高阶）同伦。相当令人惊讶的是，我们由此观察到，对于生成态射的高度移动关系，高阶同伦已经出现。最后，我们利用上同调消失论证将编织的同伦相干自然性数据扩展到所有链复形。
英文摘要：
We consider categories of Soergel bimodules for the symmetric groups S_n in their gl(n)-realizations for all n and assemble them into a locally linear monoidal bicategory. Chain complexes of Soergel bimodules likewise form a locally dg-monoidal bicategory which can be equipped with the structure of a braiding, whose data includes the Rouquier complexes of shuffle braids. The braiding, together with a uniqueness result, was established in an infinity-categorical setting in recent work with Yu Leon Liu, Aaron Mazel-Gee and David Reutter. In the present article, we construct this braiding explicitly and describe its requisite coherent naturality structure in a concrete dg-model for the morphism categories. To this end, we first assemble the Elias-Khovanov-Williamson diagrammatic Hecke categories as well as categories of chain complexes thereover into locally linear semistrict monoidal 2-categories. Along the way, we prove strictness results for certain standard categorical constructions, which may be of independent interest. In a second step, we provide explicit (higher) homotopies for the naturality of the braiding with respect to generating morphisms of the Elias-Khovanov-Williamson diagrammatic calculus. Rather surprisingly, we observe hereby that higher homotopies appear already for height move relations of generating morphisms. Finally, we extend the homotopy-coherent naturality data for the braiding to all chain complexes using cohomology-vanishing arguments.
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