What is the background of the terminology of spectra in homotopy theory? In what extend does the name "spectrum" fit to the definition and the properties? Also, are there relations to other spectra in mathematics (algebraic geometry, operator theory)?

PS: The title is an allusion to this question ;-)

  • $\begingroup$ Although I've been using spectra for some time now, I have never read or heard why this name was chosen, even in a history text like May's math.uiuc.edu/K-theory/0321/history.pdf . As described there, the concept of a spectrum has its origin in Lima's thesis. Perhaps one should have a look there. $\endgroup$ May 10 '10 at 16:57
  • $\begingroup$ To your other questions: there are of course articles, where several meanings of spectra are used, say in derived algebraic geometry or more generally in modern stable homotopy theory or also in K-Theory stuff, which can be defined by operators. But I don't think the topological meaning has a real mathematical relationship to the other meanings. $\endgroup$ May 10 '10 at 16:58
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    $\begingroup$ I always thought that the terminology "spectrum" in algebraic geometry and operator theory has been chosen because of the analogy between the Gelfand Naimark construction for commutative $C^*$ algebras and the way we represent varieties in terms of algebra hom's. Does the above construction of the spectra not realize a similar construction for the homotopy functor? $\endgroup$
    – Marc Palm
    Apr 19 '11 at 14:19

It seems reasonable to me that in operator theory the term "spectrum" comes from the Latin verb spectare (paradigm: specto, -as, -avi, -atum, -are), which means "to observe". After all in quantum mechanics the spectrum of an observable, i.e. the eigenvalues of a self adjoint operator, is what you can actually see (measure) experimentally.

Edit: after having a look to an online etymological dictionary, it seems the relevant Latin verb is another: spècere (or interchangeably spicere)= "to see", from which comes the root spec- of the latin word spectrum= "something that appears, that manifests itself, vision". Furthermore, spec- = "to see", -trum = "instrument" (like in spec-trum). Also the term "spectrum" in astronomy and optics has the same origin.

In algebraic geometry, I believe the term "spectrum", and the corresponding concept, has been introduced after the development of quantum mechanics became well known. In this context, the concept of spectrum as a space made of ideals is perfectly analogous of that in operator theory (think of Gelfand-Naimark theory, and that the Gelfand spectrum of the abelian C-star algebra generated by one operator is nothing but the spectrum of that operator).

I wouldn't be surprised if the term "spectral sequence" had something to do with "inspecting" [b.t.w. also "to inspect" comes from in + spècere...] step by step the deep properties of some cohomological constructions.

Maybe the term "spectrum" in homotopy theory and generalized (co)homology -but I don't know almost anything about these- has to do with "probing", "testing", a space via maps from (or to?) certain standard spaces such as the Eilenberg-MacLane spaces or the spheres. Does it sound reasonable?

Edit: The following paragraph from the wikipedia article on "primon gas" seems to support my guess:

"The connections between number theory and quantum field theory can be somewhat further extended into connections between topological field theory and K-theory, where, corresponding to the example above, the spectrum of a ring takes the role of the spectrum of energy eigenvalues, the prime ideals take the role of the prime numbers, the group representations take the role of integers, group characters taking the place the Dirichlet characters, and so on"

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    $\begingroup$ In 1920s people in topology considered "projective spectra" which were just what some now call inverse systems of toplogical spaces, whose index set is the natural numbers. Inverse system approximates its limit. Similarly spectral sequences approximate homologies. There are similarities of such definitions with the spectra in stable homotopy. $\endgroup$ Jan 20 '11 at 19:04
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    $\begingroup$ On the other hand the etypology above is a bit misleading. The spectrum of a system is a generalization of a spectrum of a light, related to energies the same way. For the light it is well known that the spectrum has been named by Newton who was doing the experiment with prism, which he could not explain as he followed corpuscular theory of light. The meaning was apparition, for its ghostly appearance from the prism. $\endgroup$ Jan 20 '11 at 19:16
  • $\begingroup$ @zoran: so the motivation would be different from the one I suggested (roughfly, "appearing" instead of "inspecting"), but the etymology could be the same. $\endgroup$
    – Qfwfq
    Mar 11 '11 at 14:24
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    $\begingroup$ Hi, does anyone know about the spectrum in topoi theory? I met someone named John Kennison who spoke about this at a conference recently and I wanted to figure out the origins of the term. If people think this would be better as a full question I can post it, but I didn't want another "What's so spectral" question after this great answer was given here. Kennison deals with the cyclic spectrum of a Boolean flow, which is a special case of a Cole Spectrum. He references P.T. Johnstone's "Topos Theory" which I can't find a copy of. Has anyone studied this kind of spectrum? $\endgroup$ May 20 '11 at 18:39

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