one can always define a t-structure on a stable model category as a t-structure on its homotopy category
This is a definition, but it's extremely unsatisfying and completely unenlightening.
Have a look atThe thispaper paper, where my advisor and I proveyou link from the nLab page proves that t-structures in a stable infinity-category correspond bijectively to suitable orthogonal factorization systems, called "normal torsion theories", i.e. to factorization systems $({\cal E},{\cal M})$ where
- Both classes $\cal E,M$ satify the 2-out-of-3 property; this implies [CHK] that the category $0/{\cal E} = \{X\mid 0\to X \in\cal E\}$ is coreflective, and dually that ${\cal M}/0 = \{ Y \mid Y\to 0\in\cal M\}$ is reflective. These categories play the role of the aisle and coaisle of your t-structure.
- The square $$ \begin{array}{ccc} \tau_{\ge 0}(X) &\to & X \\ \downarrow && \downarrow \\ 0 &\to & \tau_{<0}(X) \end{array} $$ is a homotopy pullback and pushout.
More info are also on the nLab page and in subsequent two papers.
Why should this address your question?
Well, we are able to characterize t-structures as a genuinely categorical gadget, living not on the level of homotopy categories, but in the "real" higher categorical world: since every model for stable categories has its own avatar of the semantics of factorization systems, you are able to speak about t-structures in every model:
- Quasicategories have quasicategorical FS, and hence "our" notion of normal torsion theory
- stable model categories, the setting you are interested in, have homotopy factorization systems, and hence "homotopy normal torsion theories".
- DG-categories have enriched factorization systems, and hence enriched normal torsion theories (I am not aware of anybody speaking of t-structures in enriched settings);
(I know, this is sketchy: it -especially the possibility to talk about these gadgets in stable derivators- is a work in progress with two colleagues)