Generalized HRS Tilting Overview

• Generalized HRS tilting is a framework that refines classical tilting theory using intervals of torsion pairs and extended hearts in both abelian and extriangulated categories.
• It establishes derived equivalences via exact sequences and bijections between torsion pairs, unifying tilting, cotilting, and non-splitting cases.
• In geometric settings, it decomposes 3D rotations into heading, roll, and sagittal tilts, offering a smooth, well-structured alternative to traditional axis-based models.

Generalized HRS tilting refers to categorical and geometric frameworks that extend the classical Happel–Reiten–Smalø (HRS) tilting construction. In the categorical setting, generalized HRS tilting provides new equivalences between derived (or extriangulated) categories via refined torsion-theoretic data, often using intervals of torsion pairs and extended hearts. In geometric and control-theoretic contexts, as in the tilt phase space formalism, Generalized HRS Tilting introduces a globally smooth, well-structured linear model for decomposing 3D rotations into heading, roll, and sagittal tilt components, with advantages over conventional axis-based parameterizations.

1. Foundational Definitions and Frameworks

Classical HRS tilting originates in the context of an abelian category $\mathcal{A}$ equipped with a torsion pair $(\mathcal{T},\mathcal{F})$, producing a new heart $\mathcal{B}$ (the HRS-tilt) in the derived category $D^b(\mathcal{A})$. The heart $\mathcal{B}$ is characterized as

$$\mathcal{B} = \big\{ X\in D^b(\mathcal{A}) \mid H^{-1}(X)\in \mathcal{F},\; H^0(X)\in\mathcal{T},\; H^i(X)=0\text{ for } i\ne -1,0 \big\}$$

and realizes a new $t$-structure on $D^b(\mathcal{A})$. The realization functor $R\colon D^b(\mathcal{B})\to D^b(\mathcal{A})$ is a $t$-exact triangle functor, compatible with the inclusion of $\mathcal{B}$ into $D^b(\mathcal{A})$.

Generalized HRS tilting extends this construction to:

• Intervals of torsion pairs in extriangulated or triangulated categories with negative first extensions, yielding extended hearts that interpolate between classical hearts and their shifts (Chen et al., 15 Dec 2025).
• Torsion-theoretic bijections between subintervals, and between hearts of bounded $t$-structures and their generalizations (Chen et al., 2018, Parra et al., 2020).
• Multidimensional tilt parameterizations in geometric settings, notably the vector-space model for rotations in the Heading–Roll–Sagittal (HRS) decomposition (Allgeuer et al., 2018).

2. Categorical Generalization: Intervals and Extended Hearts

In an extriangulated category $(C,\mathbb{E},\mathfrak{s})$ with a bifunctor $\mathbb{E}^{-1}$ (negative first extension), one can define torsion pairs $(U,V)$ and $s$-torsion pairs (additionally requiring $\mathbb{E}^{-1}(U,V)=0$). For torsion pairs $t_1=(U_1,V_1)$, $t_2=(U_2,V_2)$ with $t_1\preccurlyeq t_2$, the interval $\tors[t_1,t_2]$ comprises all torsion pairs $t$ with $t_1\preccurlyeq t\preccurlyeq t_2$.

The heart of the interval $\mathcal{H}_{[t_1,t_2]} = V_1\cap U_2$ is itself extriangulated with a naturally induced structure; a bijection (Theorem 3.8) relates torsion pairs in the interval $\tors[t_1,t_2]$ with certain torsion pairs in the heart, refining the functoriality known from triangulated categories.

The construction extends to $m$-extended hearts $\mathcal{H}^{[-(m-1),0]} = H[m-1]*H[m-2]*\dots*H$, facilitating the study of hearts beyond the classical abelian case (for $m=1$). These extended hearts support torsion-theoretic structures not visible in the conventional setup, giving rise to genuinely new (quasi-)abelian subcategories (Chen et al., 15 Dec 2025).

3. Derived Equivalence via Generalized HRS Tilting

The central result for derived equivalence asserts that, for an abelian category $\mathcal{A}$ with torsion pair $(\mathcal{T},\mathcal{F})$ and associated HRS-tilt $\mathcal{B}$, a realization functor $R$ is a triangle-equivalence if and only if each $A\in \mathcal{A}$ admits an exact sequence

$$0 \to F^0 \to T^0 \to A \to F^1 \to T^1 \to 0 \quad\text{with}\quad F^0,F^1 \in \mathcal{F},\; T^0,T^1\in \mathcal{T}$$

such that the corresponding $[F^0 \to T^0 \to A \to F^1 \to T^1]$ class in the Yoneda $3$-extension group $Yext^3_{\mathcal{A}}(T^1,F^0)$ vanishes. This unifies classical tilting (when $F^0=F^1=0$), cotilting ($T^0=T^1=0$), and splitting torsion pairs (when every $A\cong T\oplus F$) as special cases (Chen et al., 2018).

When $R$ is dense, it is automatically fully faithful, making the derived equivalence criterion both necessary and sufficient. The classical results of Happel–Reiten–Smalø are recovered for tilting or cotilting torsion pairs, and new equivalences outside this framework are exhibited for certain silting or non-splitting cases.

4. Torsion Pair Bijections, Grothendieck Hearts, and Module Categories

Generalized HRS-tilting establishes bijections between torsion pair data in abelian, Grothendieck, and module categories. Specifically, an abelian category with a 1-tilting torsion pair corresponds, via HRS-tilt, to an abelian category (AB3) with a projective generator where the corresponding torsion pair is faithful; dually, cotilting torsion pairs correspond to categories with an injective cogenerator and co-faithful torsion pairs.

A Grothendieck heart $\mathcal{H}_t$ arises precisely when the torsion pair $t$ is of finite type (i.e., the torsionfree class is closed under direct limits); all structural AB5/generator properties are inherited in this case. The general theory recovers module categories as HRS-hearts when projective generators are self-small, thus encompassing classical tilting theory (Brenner–Butler theorem) (Parra et al., 2020).

5. Geometric Generalization: HRS Tilting in Tilt Phase Space

Beyond the categorical context, "Generalized HRS Tilting" also refers to a geometric model for rotating bodies, where rotations are decomposed into orthogonal heading (H), roll-axis (R), and sagittal (S) tilt components. In this framework, the tilt phase space (TPS) is equipped with a vector space structure: $$P_{\rm HRS} = (p_H, p_R, p_S) = (\psi, \lambda_R p_x, \lambda_S p_y) \in \mathbb{R}^3$$ with $\psi$ the heading, $p_x$, $p_y$ the lateral and sagittal tilts, and $\lambda_R,\lambda_S$ tunable gains for anisotropy. Addition, inversion, and scaling are globally well-defined, rendering the HRS decomposition smooth, axisymmetric, and free of gimbal-lock. This structure supports independent design and tuning of H, R, S controllers for applications requiring nuanced balance and reactivity, e.g., legged robots or UAVs (Allgeuer et al., 2018).

6. Applications, Examples, and New Phenomena

Categorical instances include:

• Module categories constructed via one-point extensions, yielding classical reflection functors (APR/Happel–Wakamatsu).
• Non-splitting, non-tilting torsion pairs producing new derived equivalences outside two-term tilting.
• Two-term silting subcategories yielding equivalences when suitable vanishing conditions (e.g., $\mathrm{Hom}(P,\Sigma^i P)=0$ for $i<0$) are satisfied.

The formalism of intervals of torsion pairs in extriangulated categories underpins new phenomena, such as:

• Bijections between posets of torsion pairs inside intervals and in the associated hearts.
• Extension and gluing principles for $t$-structures between subcategories and ambient triangulated categories (Chen et al., 15 Dec 2025).
• In the geometric context, HRS phase space tilting yields controllers with smooth anisotropy and no singularities, outperforming traditional yaw-pitch-roll schemes (Allgeuer et al., 2018).

7. Further Developments and Open Problems

Key directions for further research include:

• Characterization and uniqueness of realization functors, with implications for the classification of derived equivalences (the "standard" derived equivalence question).
• Generalization of homological criteria, such as the vanishing of specific higher Yoneda classes, to broader classes of $t$-structures beyond HRS-tilts.
• Extensions to unbounded derived categories, dg-enhancements, and connections to Bridgeland stability conditions and wall-crossing phenomena.
• In the geometric domain, further formalization of weighted HRS tilting and its impact on control design remains an open field.

These developments collectively exemplify the unifying power of generalized HRS tilting in both categorical algebra and geometric analysis, providing robust methods for constructing equivalences, decomposing morphisms, and interpolating between rigid algebraic and flexible geometric frameworks (Chen et al., 2018, Chen et al., 15 Dec 2025, Parra et al., 2020, Allgeuer et al., 2018).