Diagram spaces and symmetric spectra

We present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra. The main motivating examples are the II-spaces  , which are diagrams indexed by finite sets and injections, and JJ-spaces  , which are diagrams indexed by Quillen’s localization construction Σ−1ΣΣ−1Σ on the category ΣΣ of finite sets and bijections.We show that the category of II-spaces provides a convenient model for the homotopy category of spaces in which every E∞E∞ space can be rectified to a strictly commutative monoid. Similarly, the commutative monoids in the category of JJ-spaces model graded E∞E∞ spaces.Using the theory of JJ-spaces we introduce the graded units of a symmetric ring spectrum. The graded units detect periodicity phenomena in stable homotopy and we show how this can be applied to the theory of topological logarithmic structures.