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Let C be a stable ∞-category in the sense of Lurie's DAG I. (In particular I do not assume that C has all colimits.) Then C does have all finite colimits, the suspension functor on C is an equivalence, and C is enriched in Spectra in a way I don't want to make too precise (basically the Hom functor Cop × C → Spaces factors through Spectra and there are composition maps on the level of spectra).

Now suppose instead that C is an ∞-category which has all finite colimits and comes equipped with an enrichment in Spectra in the above sense. One can show easily that C then has a zero object which allows us to define a suspension on C. Suppose it is an equivalence. Is C then a stable ∞-category? Moreover, is the enrichment on C the one which comes from the fact that it is a stable ∞-category?

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According to Corollary 8.28 in DAG I a pointed $\infty$-category is stable iff it has finite colimits and the suspension functor is an equivalence.

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Huh, so the enrichment in spectra is unnecessary: one only needs an enrichment in pointed spaces (to get a zero object). I wonder whether with the spectral enrichment, one can weaken one of the other hypotheses. But I don't see how. –  Reid Barton Jan 2 '10 at 2:40
    
I have the feeling that quategorifying an old argument due to Freyd one can get an enrichment in $E_\infty$-spaces simply out of the axiom "pull iff push" in the definition of stable $\infty$-category; but I've never made this precise, so I'd like to see a counterexample or a proof! –  tetrapharmakon Jun 21 at 19:58

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