Cofibrancy of operadic constructions in positive symmetric spectra
Cofibrancy of operadic constructions in positive symmetric spectra
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DOI：
10.4310/hha.2016.v18.n2.a7
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发表时间：
2014-10
期刊：
影响因子：
--
通讯作者：
L. Pereira
中科院分区：
文献类型：
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作者：
L. Pereira
我们发现，当使用基本的正模型结构的对称谱得到一致性条件下温和得多的假设比一般类别的运算结构。我们的主要结果提供了这样一个分析的关键操作，“相对合成产品”$\circ_{\mathcal{O}}$之间的右和左$\mathcal{O}$-模在谱运算$\mathcal{O}$，作为一个结果，我们恢复（通常加强）以前的结果建立奎伦不变性的模型结构的范畴代数通过弱等价的操作，遗忘函子与上纤维化的相容性和棒结构的Reedy相容性。上述结果的关键是正协方差谱的n重粉碎幂的新的协方差结果（以及映射的相对陈述）。粗略地说，我们证明了这样的$n$倍幂满足一种（新的）类型的$\Sigma_n$-相关性，这种相关性可以被看作是“lax $\Sigma_n$-free/projective相关性”，因为它确定了一个更大的类的协纤维化仍然满足“真$\Sigma_n$-free/projective相关性”的关键技术性质。"
We show that when using the underlying positive model structure on symmetric spectra one obtains cofibrancy conditions for operadic constructions under much milder hypothesis than one would need for general categories. Our main result provides such an analysis for a key operation, the "relative composition product" $\circ_{\mathcal{O}}$ between right and left $\mathcal{O}$-modules over a spectral operad $\mathcal{O}$, and as a consequence we recover (and usually strengthen) previous results establishing the Quillen invariance of model structures on categories of algebras via weak equivalences of operads, compatibility of forgetful functors with cofibrations and Reedy cofibrancy of bar constructions. Key to the results above are novel cofibrancy results for $n$-fold smash powers of positive cofibrant spectra (and the relative statement for maps). Roughly speaking, we show that such $n$-fold powers satisfy a (new) type of $\Sigma_n$-cofibrancy which can be viewed as "lax $\Sigma_n$-free/projective cofibrancy" in that it determines a larger class of cofibrations still satisfying key technical properties of "true $\Sigma_n$-free/projective cofibrancy."