In the land of $\infty$-categories, we construct Spectra via starting from pointed topological space, and stabilizing via inverting loopspace.
In the hell of $1$-categories, the analog of pointed topolgical spaces is pointed Sets. The analog of Spectra is abelian groups. Then abelian groups should be the stabilization of the category of sets. That's probably true (a pointed set becomes a free abelian group with the pointed element being 0, and closing to cokernels we get all abelian groups).
Is there a construction though that is similiar to the infinity one? Of inverting loopspace (it's just that loopspace is degenerate in this case, so naively this doesn't work of course).
Why does this strategy (if inverting loopspace) work for infinity categories and not for 1-categories?