TITLE: The relation between t-structures and derived category
QUESTION [10 upvotes]: Let $\mathcal{D}$ be a triangulated category and a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$.  The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap \mathcal{D}^{\geq 0}$, is an abelian category.  
I know that in general there is not a natural functor from the derived category of the heart and the triangulated category .
But it's seem to be very linked ...
But if you consider stable infinite category or Grothendieck derivator is there such functor ?

REPLY [5 votes]: I had reason to think about this a few years ago. When $\mathcal D$ arises as the derived category of an abelian category (with a possibly exotic $t$-structure), a construction of a realization functor $D^b(A) \to \mathcal D$ can be found already in Beilinson-Bernstein-Deligne-Gabber. (For them $\mathcal D$ is the derived category of constructible sheaves, equipped with the perverse $t$-structure, so $A$ is the category of perverse sheaves.) 
Their construction of the realization functor can in fact be imitated in the $\infty$-categorical setting, too, and in this case it works more generally for an arbitrary stable $\infty$-category $\mathcal D$. The way B-B-D-G construct the functor is they use the filtered derived category $\mathcal{D}F$. If $\mathcal D$ is the derived category of an abelian category $B$, then $\mathcal D F$ is the category of complexes in $B$ with a bounded filtration, localized at filtered quasi-isomorphisms. There is an induced $t$-structure on $\mathcal D F$ from a $t$-structure on $\mathcal D$, such that if the $t$-structure on $\mathcal D$ has heart $A$ then the heart of the $t$-structure on $\mathcal D F$ is isomorphic to the abelian category $\mathrm{Ch}^b(A)$ of bounded chain complexes in $A$. Thus we can consider the composition $\mathrm{Ch}^b(A) \to \mathcal D F \to \mathcal D$ where the first is the inclusion of the heart and the second forgets the filtration. A spectral sequence argument shows that this composition takes quasi-isomorphisms in $\mathrm{Ch}^b(A)$ to equivalences in $\mathcal D$, so there is an induced functor $D^b(A) \to \mathcal D$ which is the one we want. 
The point of the above is that for a general triangulated category $\mathcal T$ there is no sensible triangulated category $\mathcal T F$ of filtered objects in $\mathcal T$, but if $\mathcal T$ happens to be the derived category of an abelian category we can write down the filtered derived category by hand. This is not a problem in the world of stable $\infty$-categories.
I believe that an analogous argument works also for complexes that are not necessarily bounded (using instead filtrations that are unbounded to the left or right or both). But if we want there to exist a functor $\mathcal DF \to \mathcal D$ that forgets the filtration then we need $\mathcal D$ to have sequential limits or colimits, and one should be careful with the spectral sequence argument.