AI 中文总结
本文研究Goodwillie微积分的乘积法则，证明导数函子是强对称幺半的，导数提取在(∞,2)-范畴层面保留多种结构，并通过实例展示其在计算导数操作子结构中的应用。
AI 中文摘要
本文证明了Goodwillie导数的乘积法则：给定一个稳定化等价于谱的∞-范畴Sp的可微∞-范畴C，我们证明导数函子∂_* ∶ Fun^ω(C, Sp) → RMod_{∂_*id_C}(SSeq(Sp))是强对称幺半的，其中源范畴配备逐点张量积，目标范畴配备Day卷积。由于∂_*id_C的Koszul对偶可以从Day卷积中作为共自同态操作子恢复，该乘积法则可用于在实例中计算操作子∂_*id_C。我们将该乘积法则作为更一般结论的推论导出：取导数操作在(∞,2)-范畴层面保持笛卡尔积。事实上，本文的核心主题是，当被视为(∞,2)-范畴的函子时，Goodwillie导数的提取保留了大量结构：除了乘积，它还保留余张量和某些拉回。我们通过确定带基点空间、操作子上的代数以及景上的层中的恒等函子导数的操作子结构，阐释了如何将结果用于计算Goodwillie导数。
英文摘要
In this paper, we prove a product rule for Goodwillie derivatives: given a differentiable $\infty$-category $\mathcal{C}$ whose stabilization is equivalent to the $\infty$-category $\mathrm{Sp}$ of spectra, we show that the derivatives functor $\partial_* \colon \mathrm{Fun}^ω(\mathcal{C}, \mathrm{Sp}) \to \mathrm{RMod}_{\partial_*\mathrm{id}_\mathcal{C}}(\mathrm{SSeq}(\mathrm{Sp}))$ is strong symmetric monoidal, where the source is equipped with the pointwise tensor product and the target with Day convolution. Since the Koszul dual of $\partial_*\mathrm{id}_\mathcal{C}$ can be recovered as a coendomorphism operad from Day convolution, this product rule is useful for calculating the operad $\partial_*\mathrm{id}_\mathcal{C}$ in examples. We derive the product rule as a consequence of the more general statement that taking derivatives preserves cartesian products on the $(\infty, 2)$-categorical level. In fact, the main theme of this paper is that the extraction of Goodwillie derivatives preserves a lot of structure when regarded as a functor of $(\infty, 2)$-categories: apart from products, it also preserves cotensors and certain pullbacks. We illustrate how our results can be used to calculate Goodwillie derivatives by determining the operad structure on the derivatives of the identity functor in pointed spaces, algebras over an operad and sheaves on a site.
Comments55 pages
