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?


1 Answer 1


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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    $\begingroup$ 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. $\endgroup$ Jan 2, 2010 at 2:40
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    $\begingroup$ 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! $\endgroup$
    – fosco
    Jun 21, 2014 at 19:58
  • $\begingroup$ @ReidBarton Note that any full subcategory of a stable $\infty$-category is still spectrally enriched, and in particular, the full subcategories spanned by connective spectra and by shifts of the sphere spectrum are spectrally enriched. It does not seem to be easy to find any significant weakening. $\endgroup$
    – Z. M
    Aug 5, 2022 at 15:58

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