Equivariant iterated loop space theory and permutative \(G\)-categories
It has been a classical problem solved in different ways by Boardman-Vogt, May and Segal how to recognize whether a space is an (infinite) loop space and to construct a delooping if it is. \textit{K. Shimakawa} has developed in [Publ. Res. Inst. Math. Sci. 25, No. 2, 239--262 (1989; Zbl 0677.55013)] a \(G\)-equivariant analogue of Segal's approach. The present paper considers an operadic approach to equivariant (infinite) loop space machines. The authors begin with a space-level treatment. Let us call an operad in \(G\)-spaces a \(G\)-operad. Given a \(G\)-representation \(V\), we can consider the associated little disk \(G\)-operad, where \(G\) acts on the centers of the disks. One defines an \textit{\(E_V\)-operad} to be a \(G\)-operad weakly equivalent to the little disk operad. The authors' preferred model is the Steiner operad. Given an \(E_V\)-space \(Y\), a bar construction gives a potential delooping \(\mathbb{E}_V Y\). It is shown that \(Y \to \Omega^V\mathbb{E}_V\) is a group completion if \(V\) contains the trivial representation \(\mathbb{R}^2\) and actually an equivalence if \(Y\) is \(G\)-connected. To produce \(G\)-spectra, one has to use \(G\)-\(E_\infty\)-operads, i.e.\ \(G\)-operads whose \(j\)-th space is a universal \(G\times\Sigma_j\)-space for the family of subgroups \(H\) with \(H\cap \Sigma_j = \{e\}\). Examples are the linear isometry operad and the infinite Steiner operad, both for a complete universe. If \(\mathcal{C}_G\) is a \(G\)-\(E_\infty\)-operad and \(Y\) a \(\mathcal{C}_G\)-space, the authors construct an associated orthogonal spectrum, whose zeroth space is group-completing \(Y\). An example of such a \(Y\) is the classifying space of the category of finite free \(R\)-modules for a ring \(R\) with a \(G\)-action, leading to a genuinely equivariant algebraic \(K\)-theory spectrum (as studied in detail in [\textit{M. Merling}, Math. Z. 285, No. 3--4, 1205--1248 (2017; Zbl 1365.19007)] and, using different techniques, by Barwick, Glasman and Shah). For this and other examples, the authors develop a theory of suitable categorical input so that the classifying space becomes a \(G\)-\(E_\infty\)-space. Using an equivariant version of the Barratt-Eccles operad, one of the main results of the present paper identifies the \(G\)-fixed points of the free \(G\)-\(E_\infty\)-space \(\mathbb{P}_GY\) on a space \(Y\) as \[ \prod_{(H)} \mathbb{P}(EW_H \times_{WH} X^H)_+, \] where \((H)\) runs over all conjugacy classes of subgroups of \(G\), the symbol \(\mathbb{P}\) denotes the free \(E_\infty\)-space and \(W_H\) the Weyl group. Using that the orthogonal spectrum associated with \(\mathbb{P}_GY\) is equivalent to \(\Sigma^\infty_G Y\), this gives an alternative proof of the tom Dieck splitting.
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