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Given a(n $\infty$-)category, there is a process called "stabilitazion" which spits out a stable $\infty$-category (as one can read about in either Higher Algebra or the nlab).

The famous example is Spectra, which is the stabilization of Top (and I guess the various enhancements of derived categories, out of categories of modules).

If I start instead with the plain old category of pointed sets, what is its stabilization? And if I start with finite pointed sets?

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If you take the nerve of the category of pointed sets, then the suspension (and the based loops) of any object is the terminal object. So the stabilization is the trivial, stable oo-category with a single object (up to equivalence): The zero object. – Hiro Lee Tanaka Aug 24 '14 at 6:05
So it's like saying: In the space of pointed oo-categories, pointed sets (and strict categories in general) are very singular points. Their tangent space is zero. – Hiro Lee Tanaka Aug 24 '14 at 6:06
That's an interesting thought, but stabilizing a strict category doesn't magically create something with homotopically meaningful structure. For instance, the oo-category of chain complexes is not the stabilization of abelian groups. You need a separate construction (e.g., define the notion of a chain complex) to create homotopical structures. – Hiro Lee Tanaka Aug 24 '14 at 15:59
The "nonabelian" analogue of a chain complex is generally considered to be a simplicial object. So in your example, you would consider simplicial objects in pointed sets, a.k.a. pointed simplicial sets, and then stabilize to get the usual category of spectra. – Mike Shulman Aug 25 '14 at 5:52
@tetrapharmakon The other model structures are not relevant: they are models for homotopy $(-1)$-types or $(-2)$-types, so the corresponding quasicategories are weakly categorically equivalent to either $\Delta^1$ or $\Delta^0$. – Zhen Lin Aug 25 '14 at 10:45
up vote 4 down vote accepted

The answer is in the comments: the stabilization is the trivial stable category.

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