# Are triangulated equivalence detected at compact level?

Suppose that $D$ and $E$ are compactly generated triangulated categories, even algebraic (i.e. equivalent to derived categories of small dg categories) if we want, and asume that their subcategories $D^c$ and $E^c$ of compact objects are triangle equivalent. Are $D$ and $E$ triangle equivalent?. By Theorem 9.2 of Keller's 'Deriving dg categories', we know that the answer is 'yes' when either $D$ or $E$ is the derived category of (the category of modules over) a small $k$-linear category, but I do not know the general answer. Any help would be highly appreciated.

• Welcome to MathOverflow, Manuel. The site supports basic LaTeX formatting ("MathJax", technically). I've added formatting to your question to make it a little easier to read: I hope you don't mind. – Jeremy Rickard Jun 23 '17 at 12:38
• Dear Jeremy, Thank you very mucy for adding TeX formatting to my question. – Manuel Saorín Jun 23 '17 at 18:26
• I fear that you cannot really write down a functor $D \to E$ because homotopy colimits are not functorial on the level of triangulated category. If you have instead an equivalence between stable infinity categories of compact object, you obtain an equivalence between $Ind(D^c)$ and $Ind(E^c)$, which agree with $D$ and $E$ when the latter are compactly generated stable infinity categories. – Lennart Meier Jun 26 '17 at 8:04
• Thank you Lennart. Your comment is very helpful for me. I had the impression that going to the context of infinity categories or that of (Grothendieck) derivators, the answer might be 'yes'. But still, without adding further structure, I would like to find examples beyond the derived categories of rings (with several objects), for which the answer to my question is positive. – Manuel Saorín Jun 27 '17 at 9:59

Consider the case where $D$ is the ordinary stable homotopy category of spectra, and $E$ is assumed to have all coproducts (which is clearly a necessary condition), and we also assume that $D^c\simeq E^c$ as tensor triangulated categories. It is an old conjecture of Margolis that this forces $D$ to be equivalent to $E$, and I believe that this is still open in full generality. In both $D$ and $E$ one can define a square-zero ideal of phantom maps. Hovey, Palmieri and I proved that $D/\text{Ph}_D\simeq E/\text{Ph}_E$ and that $\text{Ph}_D\simeq\text{Ph}_E$ as a module over this quotient. Schwede and Shipley proved in https://arxiv.org/abs/math/0108143 that $D\simeq E$ provided that $E\simeq\text{Ho}(E_0)$ for some Quillen model category $E_0$.