论文标题
稳定的功率操作
Stable power operations
论文作者
论文摘要
对于任何$ e_ \ infty $ ring spectrum $ e $，我们表明有一个代数$ \ mathrm {pow}（e）$稳定的电源操作，它们自然地在任何$ e $ e $ algebra的基础频谱上。此外，我们表明有环$ e \ to \ mathrm {pow}（e）\ to \ mathrm {end}（e）$的地图，后者决定了从空间共同体中的电源操作到稳定操作的限制。如果$ e $是mod- $ p $ eilenberg-mac泳道频谱，这实现了曼德尔（Mandell）代数的广义steenrod操作代数的自然商，以示为mod- $ $ $ p $ steenrod代数。更笼统地，这是从$ \ infty $ -Category $ \ Mathcal {C} $分类到Spectra的代表函子的内态性分类的一部分，尤其要注意$ \ Mathcal {c} $是$ \ Mathcal {o \ Mathcal {O} $} $ - MONIODAL $ - MONIODAL $ \ iffty $ -CATERTY。
For any $E_\infty$ ring spectrum $E$, we show that there is an algebra $\mathrm{Pow}(E)$ of stable power operations that acts naturally on the underlying spectrum of any $E$-algebra. Further, we show that there are maps of rings $E \to \mathrm{Pow}(E) \to \mathrm{End}(E)$, where the latter determines a restriction from power operations to stable operations in the cohomology of spaces. In the case where $E$ is the mod-$p$ Eilenberg-Mac Lane spectrum, this realizes a natural quotient from Mandell's algebra of generalized Steenrod operations to the mod-$p$ Steenrod algebra. More generally, this arises as part of a classification of endomorphisms of representable functors from an $\infty$-category $\mathcal{C}$ to spectra, with particular attention to the case where $\mathcal{C}$ is an $\mathcal{O}$-monoidal $\infty$-category.
