(2-)categorical constructions and the multiplicative equivariant Barratt-Quillen-Priddy theorem

The classical Barratt-Priddy-Quillen theorem states that the K-theory spectrum of the category of finite sets and isomorphisms is equivalent to the sphere spectrum. A more general statement is that for an unbased space X, the suspension spectrum \Sigma^\infty_+ X is equivalent to the spectrum associated to the free E_\infty space on X. In this talk we will present a categorical construction of the latter that is lax monoidal. This compatibility with multiplicative structures is necessary when using this functor to change enrichments, as in the work of Guillou-May. This is joint work with Bert Guillou, Peter May and Mona Merling.