We give an operadic definition of a genuine symmetric monoidal |$G$|-category, and we prove that its classifying space is a genuine |$E_\infty $||$G$|-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power and Lack to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal |$G$|-categories to genuine permutative |$G$|-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When |$G$| is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal |$G$|-categories as input to an equivariant infinite loop space machine that gives genuine |$\Omega $|-|$G$|-spectra as output.

中文翻译：

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对称单调G分类及其严格化

我们给出真正对称单项式| $ G $ |的操作性定义。-类别，我们证明其分类空间是真实的| $ E_ \ infty $ | | $ G $ | -空间。我们通过发展一些非常普遍的分类连贯理论来做到这一点。我们结合Corner和Gurski，Power和Lack的结果，为操作数和单子上的伪代数开发了严格化理论。它专门用于严格限制真正的对称单项式| $ G $ | 类别到真正的置换| $ G $ | -类别。我们所有的工作都在一个通用的内部分类框架中进行，该框架具有许多完全不同的专业领域。当| $ G $ |是一个有限群，此处的理论与以前的工作相结合，可以推广从严格的空间级输入到相当普遍的类别级输入的等变无限循环空间理论。它需要真正的对称单曲面| $ G $ | -类别作为等式无限循环空间机器的输入，该机器给出真正的| $ \ Omega $ | - | $ G $ | -spectra作为输出。