Timeline for Stabilization of $\infty$-categories versus SW stabilization

5 events

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Aug 21, 2013 at 23:52	comment	added	Jacob Lurie		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.
Aug 21, 2013 at 22:38	comment	added	fosco		I couldn't hope for a better answer. The lack of filtered colimits explains why you referred to $Ind(\cal C)$, am I right?
Aug 21, 2013 at 22:34	vote	accept	fosco
Aug 21, 2013 at 21:31	comment	added	Eric Wofsey		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.
Aug 21, 2013 at 19:16	history	answered	Jacob Lurie	CC BY-SA 3.0