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What kind of definitions of t-structures on stable model categories have been investigated in the literature?

Of course, one can always define a t-structure on a stable model category as a t-structure on its homotopy category; this is analogous to how Lurie defines t-structures on stable quasicategories (Definition 1.2.1.4 in Higher Algebra).

However, in the setting of model categories one normally wants a more strict presentation that could be exploited to perform computations more easily.

In particular, one can conceive of various strictifications of k-connective objects C_{≥k}, k-coconnective objects C_{≤k}, and their truncating functors τ_{≥k} and τ_{≤k}.

For example, in the case of symmetric simplicial spectra one can say that a spectrum X is strictly connective if for each n the nth spectral level X_n is a simplicial set with exactly one k-simplex for all k

Using coskeletal simplicial sets one can also conceive of a similar picture for coconnective spectra and coconnective truncations.

Is there anything like this in the literature? Of course, I'm not just interested in symmetric simplicial sets, but also (say) in motivic symmetric spectra and other stable model categories.

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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.

The paper you 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". More info are also 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)

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  • $\begingroup$ Indeed, Bousfield's notion of a homotopy factorization system combined with t-structures provides one answer to the question. I didn't know that Bousfield already developed such a theory way back in 1975. $\endgroup$ – Dmitri Pavlov Dec 15 '15 at 18:19

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