Recall that a spectrum is called connective if it is $(-1)$-connected (that is, its homotopy is concentrated in nonnegative degrees).

However, this left me scratching my head a bit. Why "connective"? Is there some geometric intuition behind it that I'm missing?

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    I wasn't aware that the empty spectrum is a spectrum at all. What with the individual components being pointed and all... – Harry Gindi Apr 18 '11 at 5:54
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    Maybe just because (-1)-connected is a mouthful? And connected would not be the right term, since that would imply trivial $\pi_0$. Other than that, I don't know. Possibly a better question is: where did the term originate? I have no idea. – Dan Ramras Apr 18 '11 at 6:02
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    It is possible, even inevitable at times, to introduce a category of "unbased spectra". This is where the suspension spectra of unbased spaces live. The suspension spectrum of the empty space is the initial object, but I probably wouldn't call it empty. – Tom Goodwillie Apr 18 '11 at 13:10
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    It's a pretty trivial thing, but it does come up naturally in some things I think about. There are several equivalent constructions (which one are you thinking of?), and it's interesting to note that they're equivalent. I sometimes think of spectrum vs unbased spectrum (also of category of spectra vs category of unbased spectra) as analogous to vector spaces vs affine spaces. – Tom Goodwillie Apr 18 '11 at 14:49
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    My pleasure. I never miss a chance to talk about the empty set. – Tom Goodwillie Apr 18 '11 at 21:54
up vote 7 down vote accepted

Maybe just because (-1)-connected is a mouthful? And connected would not be the right term, since that would imply trivial $\pi_0$. Other than that, I don't know.

  • For spaces, $k$-connected has come to mean having trivial $\pi_j$ for $j\le k$ (basically -- let's ignore basepoint issues). In hindsight it might have been better to have a term for having trivial $\pi_j$ for $j<k$. Likewise for spectra. Not that "minus one" takes that much longer to say than "zero" ... – Tom Goodwillie Apr 18 '11 at 21:52

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