  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.
