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fosco
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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.

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

  1. 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.
  2. 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)

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 at this paper, where my advisor and I prove 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

  1. 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.
  2. 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 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)

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)

Source Link
fosco
  • 13.6k
  • 2
  • 28
  • 77

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 at this paper, where my advisor and I prove 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

  1. 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.
  2. 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 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)