We prove two theorems about Goodwillie calculus and use those theorems to describe new models for Goodwillie derivatives of functors between pointed compactly generated $\infty$-categories. The first theorem says that the construction of higher derivatives for spectrum-valued functors is a Day convolution of copies of the first derivative construction. The second theorem says that the derivatives of any functor can be realized as natural transformation objects for derivatives of spectrum-valued functors. Together these results allow us to construct an $\infty$-operad that models the derivatives of the identity functor on any pointed compactly generated $\infty$-category. Our main example is the $\infty$-category of algebras over a stable $\infty$-operad, in which case we show that the derivatives of the identity essentially recover the same $\infty$-operad, making precise a well-known slogan in Goodwillie calculus. We also describe a bimodule structure on the derivatives of an arbitrary functor, over the $\infty$-operads given by the derivatives of the identity on the source and target, and we conjecture a chain rule that generalizes previous work of Arone and the author in the case of functors of pointed spaces and spectra.

中文翻译：

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Goodwillie 演算中的无穷操作数和 Day 卷积

我们证明了关于 Goodwillie 演算的两个定理，并使用这些定理来描述指向紧凑生成的函子之间的 Goodwillie 导数的新模型 $\infty$- 类别。第一个定理说，谱值函子的更高导数的构造是一阶导数构造副本的 Day 卷积。第二个定理说，任何函子的导数都可以实现为谱值函子的导数的自然变换对象。这些结果使我们能够构建一个$\infty$- 操作数，对任何指向紧凑生成的单位函子的导数进行建模 $\infty$-类别。我们的主要例子是$\infty$-稳定的代数范畴 $\infty$-operad，在这种情况下，我们表明身份的导数基本上恢复相同 $\infty$-operad，在Goodwillie calculus中精确地制作了一个众所周知的口号。我们还描述了任意函子的导数上的双模结构，在$\infty$- 由源和目标上的恒等式的导数给出的操作数，并且我们推测了一个链式规则，该规则概括了 Arone 和作者之前在指向空间和谱的函子的情况下的工作。