spaces. Specifically, we produce a symmetric monoidal product on a subcategory of ${{\mathscr{T}}}$ such that monoids for this product are precisely ${A_{\infty}}$-spaces; this allows us to work with models of ${{GL_{1}}R}$ which are strict monoids for the new product. The observation that one could carry out the program of [@EKMM] in the setting of spaces is due to Mike Mandell, and was worked out in the thesis of the second author [@Blumberg-thesis] (see also the forthcoming paper [@Blumberg-Cohen-Schlichtkrull]).

The relevant category of spaces for this product is $*$-modules, the space-level analogue of EKMM’s $S$-modules. Precisely, let ${\mathbb{L}}$-spaces be the category of spaces with an action of the $1$-space ${{\mathcal{L}}}(1)$ of the linear isometries operad. On this category we can define an operadic product ${\times_{\mathcal{L}}}$ which is associative and commutative but not unital. The category of $*$-modules is the subcategory of ${\mathbb{L}}$-spaces for which the unit $*{\times_{\mathcal{L}}}X \to X$ is an isomorphism. The category of $*$-modules is Quillen equivalent to the category of spaces and admits an operadic symmetric monoidal product.

In this setting, we can form a model of ${{GL_{1}}R}$ which is a group-like monoid, and then model $E{{GL_{1}}R} \to B{{GL_{1}}R}$ as a quasi-fibration with an action of ${{GL_{1}}R}.$ Given 