I've seen a bunch of definitions of spectra in the literature, and the fanciest seems to be the $(\infty, 1)$-category of spectra obtaining by "stablizing" the higher category of spaces, as in DAG I. I don't really understand this stabilization procedure yet and would like to connect this idea to the more concrete notions that I've heard about, such as:

- The Boardman category of spectra: here a spectrum is a bunch of spaces (say, CW complexes) $E_n$ with closed cellular imbeddings $SE_n \to E_{n+1}$, and morphisms are defined via cofinal subspectra.
- Symmetric or orthogonal spectra, where one just has spaces and morphisms $SE_n \to E_{n+1}$, but there is some additional equivariance condition (and this way we get an honest symmetric monoidal category).

Presumably from one of these other constructions one can still recover the $\infty$-category of spectra.

For concreteness, I still like to think of a higher category as a topologically (or simplicially) enriched category, or even more concretely a set of objects together with 1-morphisms, 2-morphisms, etc. and various ways of composing them. The 1-morphisms in all these concrete categories are spectra are known (e.g. they're equivariant morphisms in the symmetric or orthogonal case). How should I think of the higher morphisms?