Fibration theorems for TQ-completion of structured ring spectra
Fibration theorems for TQ-completion of structured ring spectra
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发表时间：
2020-01
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通讯作者：
Nikolas Schonsheck
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作者：
Nikolas Schonsheck
这篇简短论文的目的是在适当的条件下建立Bousfield-Kan“纤维化引理”的谱代数模拟。我们工作的背景下，代数结构，可以被描述为代数运算$\mathcal{O}$在对称谱。我们的主要结果是，完成关于拓扑Quillen同调（或TQ-完成，简称）保持同伦纤维化序列的基础和总$\mathcal{O}$-代数是连接的。我们的论证本质上归结为证明从同伦纤维到其TQ-完备塔的自然映射是亲$\pi_*$同构。更一般地，我们还表明，类似的结果仍然成立，如果我们取代“同伦纤维化序列”与“同伦拉回广场。"
The aim of this short paper is to establish a spectral algebra analog of the Bousfield-Kan "fibration lemma" under appropriate conditions. We work in the context of algebraic structures that can be described as algebras over an operad $\mathcal{O}$ in symmetric spectra. Our main result is that completion with respect to topological Quillen homology (or TQ-completion, for short) preserves homotopy fibration sequences provided that the base and total $\mathcal{O}$-algebras are connected. Our argument essentially boils down to proving that the natural map from the homotopy fiber to its TQ-completion tower is a pro-$\pi_*$ isomorphism. More generally, we also show that similar results remain true if we replace "homotopy fibration sequence" with "homotopy pullback square."