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 I-spaces, which are diagrams indexed by finite sets and injections, and J-spaces, which are diagrams indexed by the Grayson-Quillen construction on the category of finite sets and bijections. We show that the category of I-spaces provides a convenient model for the homotopy category of spaces in which every E-infinity space can be rectified to a strictly commutative monoid. Similarly, the commutative monoids in the category of J-spaces model graded E-infinity spaces. Using the theory of J-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. 
Steffen Sagave and Christian Schlichtkrull
math.AT