Diagram Spaces, Diagram Spectra, and Fsp’s

This is one of several papers in which we explore the interrelationships among certain categories that can be taken as models for spectra and (highly structured) ring spectra. In this paper, we study certain categories of “diagram ring spectra” that are isomorphic to corresponding categories of “functors with smash product (FSP’s)”. The notion of an FSP was introduced by Bökstedt [2], and his use of FSP’s to define topological Hochschild homology established their convenience and importance in stable homotopy theory. Versions of FSP’s had been defined earlier: in different language, what we call FSP’s in the category of symmetric spectra were defined by Gunnarson [7], and what we call FSP’s in the category of orthogonal spectra were defined by the second author and others [18, 17] in the early 1970’s. As we shall see, FSP’s are defined in terms of “external smash products”. It is a crucial insight of Jeff Smith that external smash products can be internalized. We show that each category of generalized FSP’s is isomorphic to the category of monoids in an associated symmetric monoidal category of diagram spectra. Such monoids are what we mean by diagram ring spectra. Smith introduced the category of symmetric spectra, showed that it is symmetric monoidal, and observed that its externally defined FSP’s and internally defined monoids give isomorphic categories. We study a general form of the construction. In fact, the relevant categorical framework was already in place by 1970, in work of Day [5]. Hovey, Shipley, and Smith [8] have studied the category of symmetric spectra and its homotopy theory, and the papers [21] and [23] go further with the homotopical study of its ring spectra. There is a coordinate-free analogue of the category of symmetric spectra, which we call the category of orthogonal spectra; it was introduced by May [17, §5] (who called its objects I∗-prespectra). The names come from the fact that actions by symmetric groups and by orthogonal groups are built into the