Stable infinity categories vs dg-categories - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:59:55Z http://mathoverflow.net/feeds/question/114251 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114251/stable-infinity-categories-vs-dg-categories Stable infinity categories vs dg-categories Jan Weidner 2012-11-23T15:29:25Z 2012-11-24T06:19:26Z <p>What is the relation between dg-categories and stable $\infty$-categories?</p> <p>Given a dg-category one can form its dg-nerve and get a $\infty$-category (which will be stable if the dg-category is?). Can one turn a stable $\infty$-category into a dg-category or $A_\infty$-category somehow?</p> <p>I have heard the statement that at least over a field of characteristic zero the theories of stable $\infty$-categories and dg-categories are "equivalent".</p> <p>What would be a precise formulation of this statement and what would be a reference?</p> http://mathoverflow.net/questions/114251/stable-infinity-categories-vs-dg-categories/114315#114315 Answer by Dylan Wilson for Stable infinity categories vs dg-categories Dylan Wilson 2012-11-24T06:19:26Z 2012-11-24T06:19:26Z <p>Here are a few observations...</p> <ol> <li>I think there exist stable infinity categories that are not the dg-nerve (resp. $A_\infty$-nerve) of a dg-category (resp. $A_\infty$ category). In particular, the category of spectra should not arise in this way. I think Keller has a paper on differential graded categories that answers this question; he notes at some point that the homotopy category of spectra is not "algebraic" but that homotopy categories of differential graded categories are (and in fact sort of encompass all such algebraic categories.) Basically it comes down to something like the existence of Hopf maps. Now- could one define somehow the "closest dg-category approximation" to a given stable infty category? Probably. I don't know how. Or maybe I could come up with how, but I'm not sure how useful this would be if the functor wasn't an equivalence? </li> <li>To answer Fernando's question, see DAG X.5 or DAG VII.6.2. That is, a stable $\infty$-category over a field $k$ is a presentable, stable $\infty$-category "equipped with an action of the monoidal $\infty$-category of $k$-module spectra". Unless I'm mistaken I think this basically implies that it is enriched and tensored over k-module spectra.</li> <li>Here would be a precise formulation of the statement about categories over a field of characteristic zero: The dg-nerve functor induces an equivalence of $\infty$-categories between the $\infty$-category underlying the model category of dg-categories over k and the $\infty$-category of stable, k-linear $\infty$-categories. (I don't mean to overwhelm with "infinities", I would state this in the perhaps friendlier world of model categories, but I'm not sure what precisely the model category is that corresponds to stable, k-linear $\infty$-categories.) I don't know of a reference for a proof, though Lurie alludes to this a lot. It would be great if someone wrote this down!</li> </ol>