TITLE: Why are spectra indexed over the natural numbers?
QUESTION [5 upvotes]: A spectrum is a sequence $X_0,X_1,...$ of spaces together with structure morphisms $\Sigma X_n\to X_{n+1}$. To get the usual model for the stable homotopy category based on the category of spectra, one "inverts" the suspension functor (or the shift functors) which is not an isomorphism in the category of spectra. It seems to me that this is a kind of allowing the objects to be indexed over the integers. So why does one not define a spectrum to be indexed over the integers or at least bounded below?

REPLY [2 votes]: The reason may be slightly historical, the first examples of spectra were most easily seen to be indexed over the natural numbers: $MU$, $HR$, and $\mathbb{S}$. Now though, we know we ought to be using finite dimensional inner product subspaces of $\mathbb{R}^\infty$! Also, we want there to be an underlying "point set" category of spectra that we do something to do in order to get the stable homotopy category. What we really care about is some spectrum, not the image of that spectrum in the stable homotopy category. I guess I am trying to say that even though we will end up inverting something later in order to work on it does not mean we should throw out the original and only remember the localization.