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arXiv 2007.054402020-07math.AT

空间上恒等函子的分划偏序复形与古德里导数

The partition poset complex and the Goodwillie derivatives of the identity in spaces

Duncan A. Clark

备注：: 47 pages, substantial expansion from v2

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AI总结：

该研究通过上单纯形对象配对构造空间恒等函子导数的典范同伦协调 operad 结构，证明其与 Ching 描述的结构一致，还揭示谱上交换余代数的导出本原元为该 operad 上的代数。

摘要：

我们通过上单纯形对象的配对，在空间上恒等函子的导数上构造了一个典范的高度同伦协调的 operad 结构，提供了 Ching 首次描述的这类对象上 operad 结构的新表述。我们证明这两种结构一致：二者均为带权树上带窗割系统的 operad 代数的限制，该 operad 的分量空间可缩，限制沿的结构 operad 映射均为 levelwise 弱等价。此外，我们证明谱上交换余代数的导出本原元构成该 operad 上的代数。

英文摘要：

We produce a canonical highly homotopy-coherent operad structure on the derivatives of the identity functor in spaces via a pairing of cosimplicial objects, providing a new description of an operad structure on such objects first described by Ching. We prove that the two structures agree: both are restrictions of a single algebra over an operad of windowed cut systems on weighted trees, whose component spaces are contractible, along maps of structuring operads which are all levelwise weak equivalences. In addition, we show the derived primitives of a commutative coalgebra in spectra form an algebra over this operad.

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