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There's a fair amount of literature comparing different models for the homotopy theory of homotopy theories, or the homotopy theory of $(\infty,1)$-categories. Julie Bergner has a survey of this literature at

Does anyone know if similar comparisons have been made for different models of stable $(\infty,1)$-categories? In particular, have $A_{\infty}$-categories (in characteristic 0) been compared to the stable infinity categories that are conceived of as infinity categories with extra properties? (For example, to stable quasicategories?)

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Not all $A_\infty$-categories are stable, are they? – Mike Shulman Jan 20 '10 at 21:45
up vote 5 down vote accepted

There seems to be a little confusion in your question about what stable ($\infty$,1)-categories are, so I want to address that first.

The definition of a stable infinity category that I'm familiar with is Jacob Lurie's notion (definition 29 DAGI:Stable Infty-Categories). The prototypical examples of stable infinity categories which this definition is designed to capture are the category of spectra (in the topological setting) and the category of chain complexes in an abelian category (in the algebraic setting). Rather this notion is designed to also capture the derived localization of the category of chain complexes, which is a robust version of the derived category.

The advantage of Jacob's succinct definition is that it is expressed categorically. This means that if you have equivalent notions of ($\infty$, 1)-category, then they will yield equivalent notions of stable infinity category. So in this sense Julie Bergner's is also comparing stable infinity categories.

The only other model of stable ($\infty$,1)-categories that I know which doesn't quite fit this idea but is close is that of stable model categories. These are to stable ($\infty$,1)-categories as ordinary model categories are to ordinary ($\infty$,1)-categories. That is they are roughly equivalent to a particularly nice class of stable ($\infty$,1)-categories.

Your question also asks about $A_\infty$-categories as stable ($\infty$,1)-categories, and that is the part which is confusing me.

On the one hand, an infinity category can heuristically be defined as a category with topological spaces of morphisms and where composition is associative only up to higher coherence. All the various notions of $\infty$-category make this precise in one way or another. Since you mention characteristic zero, I take you are thinking of the algebraic/chain complex version of $A_\infty$-category. This can be related to topological $A_\infty$-categories via the Dold-Kan correspondence (assuming your complexes are connective).

One the other hand being enriched in chain complexes is like being enriched in topological abelian groups (or in HZ-spectra in the non-connective case). So maybe the concept you are after is that of spectral category, or category enriched in spectra? I know this has been studied, but I'm lean on references.

One last comment. Later in DAGI, Lurie shows that stable infinity categories are automatically enriched in the stable infinity category of spectra. This is totally analogous to the category of chain complexes in an abelian category being enriched in the category of chain complexes of abelian groups. However, just as being enriched in abelian groups is not enough for your category to be abelian, being enriched in spectra is not enough for your infinity category to be stable.

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See Charles Rezk's response to an earlier question of mine that is relevant to this answer. (This is to the op, not you, Chris.)… – Harry Gindi Jan 21 '10 at 2:28
Thanks! I was going to find this question and edit the link in later, but you beat me to it! – Chris Schommer-Pries Jan 21 '10 at 2:36
No problem, always glad to help! – Harry Gindi Jan 21 '10 at 2:43
Thanks! If I recall correctly, the chain complex version of $A_{\infty}$-categories does already have a precise notion of "associative up to higher homotopy". But I didn't realize there were two versions! Mike Shulman asked above if all $A_{infty}$ categories are actually stable, and I have to admit I don't know. I felt certain I'd seen a slogan somewhere... but only a slogan, if that. – Allison Smith Jan 21 '10 at 15:57

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