On the Goodwillie derivatives of the identity in structured ring spectra
On the Goodwillie derivatives of the identity in structured ring spectra
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发表时间：
2020-04
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通讯作者：
D. Clark
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作者：
D. Clark
本文件的目的有三：（i）我们构造了结构环谱上单位函子的导数上的一个自然发生的高度同伦相干运算结构，它可以描述为谱中运算$\mathcal{O}$上的代数，（ii）我们证明了每个连通$\mathcal{O}$-代数都有单位函子的导数的自然发生的左作用，（iii）证明了$\mathcal{O}$-代数上恒等式的导数与运算$\mathcal{O}$之间存在一个自然发生的高度同伦相干运算的弱等价.沿着的方式，我们引入的概念，$\mathbf{N}$着色运算的水平-通过建设-提供了一个精确的代数框架，工作和比较高度同伦相干的运算，运算，和它们的代数。
The aim of this paper is three-fold: (i) we construct a naturally occurring highly homotopy coherent operad structure on the derivatives of the identity functor on structured ring spectra which can be described as algebras over an operad $\mathcal{O}$ in spectra, (ii) we prove that every connected $\mathcal{O}$-algebra has a naturally occurring left action of the derivatives of the identity, and (iii) we show that there is a naturally occurring weak equivalence of highly homotopy coherent operads between the derivatives of the identity on $\mathcal{O}$-algebras and the operad $\mathcal{O}$. Along the way, we introduce the notion of $\mathbf{N}$-colored operads with levels which -- by construction -- provides a precise algebraic framework for working with and comparing highly homotopy coherent operads, operads, and their algebras.