space. Technical complexities with spectra One of the canonical complexities while working with spectra and defining a category of spectra comes from the fact each of these categories cannot satisfy five seemingly obvious axioms concerning the infinite loop space of a spectrum sendinga pair of adjoint functors , the and the smash product in both the category of spaces and the category of spectra. If we let denote the category of based, compactly generated, weak Hausdorff spaces, and denote a category of spectra, the following five axioms can never be satisfied by the specific model of spectra: is a symmetric monoidal category with respect to the smash product The functor is left-adjoint to The unit for the smash product is the sphere spectrum Either there is a natural transformation or a natural transformation which commutes with the unit object in both categories, and the commutative and associative isomorphisms in both categories. There is a natural weak equivalence for which that there is a commuting diagram:where is the unit map in the adjunction. Because of this, the study of spectra is fractured based upon the model being used. For an overview, check out the article cited above. History A version of the concept of a spectrum was introduced in the 1958 doctoral dissertation of Elon Lages Lima. His advisor Edwin Spanier wrote further on the subject in 1959. Spectra were adopted by Michael Atiyah and George W. Whitehead in their work on generalized homology theories in the early 1960s. The 1964 doctoral thesis of J. Michael Boardman gave a workable definition of a category of spectra and of maps (not just homotopy classes) between them, as useful in stable homotopy theory as the category of CW complexes is in the unstable case. (This is essentially the category described above, and it is still used for many purposes: for other accounts, see Adams (1974) or Rainer Vogt (1970).) Important further theoretical advances have however been made since 1990, improving vastly the formal properties of spectra. Consequently, much recent literature uses modified definitions of spectrum: see Michael Mandell et al. (2001) for a unified treatment of these new approaches. See also Ring spectrum Symmetric spectrum G-spectrum Mapping spectrum Suspension (topology) Adams spectral sequence References Introductory Modern articles developing the theory Historically relevant articles External links Spectral Sequences - Allen Hatcher - contains excellent introduction to spectra and applications for constructing Adams spectral sequence