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Abstract 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. 

We use the C2-equivariant Adams spectral sequence to compute part of the C2-equivariant stable homotopy groups π^{C2}_{n,n}. This allows us to recover results of Bredon and Landweber on the image of the geometric fixed-points map π^{C2}_{n,n}→π_0. We also recover results of Mahowald and Ravenel on the Mahowald root invariants of the elements 2^k.