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Spanier-Whitehead stabilization provides a way to extend a category $\bf E$ to a bigger one $\mathcal{SW}_\Omega(\bf E)$ where a given endofunctor $\Omega$ is invertible. The category $\mathcal{SW}_\Omega(\bf E)$ is constructed with

  1. Objects the pairs $(A,n)\in Ob(\mathbf C)\times\mathbb Z$;
  2. The set of morphisms $(A,n)\to (B,m)$ corresponds to the colimit set $$ \varinjlim_{k\in\mathbb N} \hom_{\bf E}(\Omega^{n+k}A, \Omega^{m+k}B) $$

It's a matter of bare computations to show that it defines a category, where $\bf E$ can be embedded via $A\mapsto (A,0)$, and where an entire family of functors $\bar\Omega^i\colon (A,n)\mapsto (A,n+1)$ can be defined; the functor $\bar\Omega^1$ plays the role of the initial endofunctor $\Omega$, and that's the end of the story.

But when you meet the formalism of stable $\infty$-categories you begin to wonder if there's a link between the two processes, the SW stabilization and what Lurie describes here (Def. 8.4). I'm aware that the SW construction is an "abstraction" (?) of the procedure exhibiting topological spectra, but I must confess I'm not able to go further (especially because I'm a "category theorist" slightly oriented to topology, not vice-versa).

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  • $\begingroup$ Should your $\overline{\Omega}^i$ be something like $\overline{\Omega}^{n-1}$? And careful using the word 'initial' :-) $\endgroup$
    – David Roberts ♦
    Commented Aug 22, 2013 at 2:26

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I think it's more typical to use this construction to invert the suspension functor, rather than the loop functor. So let me write $\Sigma$ where you wrote $\Omega$.

1) The category $SW_{\Sigma}(\mathcal{C})$ can be identified with the direct limit of the sequence $$ \cdots \rightarrow \mathcal{C} \stackrel{\Sigma}{\rightarrow} \mathcal{C} \stackrel{\Sigma}{\rightarrow} \mathcal{C} \stackrel{\Sigma}{\rightarrow} \cdots$$

2) The construction makes sense for $\infty$-categories as well as ordinary categories. Moreover, it commutes with passage to homotopy categories. That is, if $\mathcal{C}$ is an $\infty$-category, then the homotopy category of $SW_{\Sigma}(\mathcal{C})$ is $SW_{\Sigma}( h \mathcal{C} )$.

3) Let $\mathcal{C}$ be a small pointed $\infty$-category which admits finite colimits, and let $\Sigma: \mathcal{C} \rightarrow \mathcal{C}$ be the suspension functor. Then $Ind( SW_{\Sigma}( \mathcal{C} ) )$ is the stabilization of $Ind( \mathcal{C} )$.

You'd typically put these together by taking $\mathcal{C}$ to be something like the $\infty$-category of pointed finite spaces. Then $Ind( \mathcal{C} )$ is the $\infty$-category of pointed spaces, and its stabilization is the $\infty$-category of spectra. Your conclusion is that $SW_{\Sigma}(\mathcal{C} )$ can be identified with the $\infty$-category of finite spectra, so that $SW_{\Sigma}( h \mathcal{C} )$ is a model for the homotopy category of finite spectra (this is the classical construction).

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    $\begingroup$ To elaborate a little on step 3, the reason $SW_\Sigma(\mathcal{C})$ is the "wrong" construction in general is that it does not have sequential colimits. Since you've formally inverted the functor $\Sigma$, $SW_\Sigma(\mathcal{C})$ has objects $\Sigma^{-n} X$ for any $n\in\mathbb{Z}$ and any object $X$ in $\mathcal{C}$. But if you have a diagram $X_0\to \Sigma^{-1}X_1\to \Sigma^{-2}X_2\to\dots$, it will probably not have a colimit in $SW_\Sigma(\mathcal{C})$, even if $\mathcal{C}$ had all colimits. So to get a nicer category, we want to formally add these sequential colimits. $\endgroup$
    – Eric Wofsey
    Commented Aug 21, 2013 at 21:31
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    $\begingroup$ I couldn't hope for a better answer. The lack of filtered colimits explains why you referred to $Ind(\cal C)$, am I right? $\endgroup$
    – fosco
    Commented Aug 21, 2013 at 22:38
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    $\begingroup$ That's right. To put it differently: stabilization is for "big" $\infty$-categories, and $SW_{\Sigma}$ is for "small" $\infty$-categories; passage to $\infty$-categories of inductive limits intertwines the two constructions. The construction $SW_{\Sigma}$ is perhaps more concrete than stabilization because you can perform it at the level of homotopy categories: this is because passage from an $\infty$-category $\mathcal{C}$ to its homotopy category commutes with sequential direct limits, but not with sequential inverse limits. $\endgroup$
    – Jacob Lurie
    Commented Aug 21, 2013 at 23:52

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