Silting Subcategory in Triangulated Categories

• Silting subcategory is an additive subcategory characterized by self-orthogonality (vanishing positive degree morphisms) and the generation of its triangulated category.
• It enables mutation processes via approximations that yield presilting subcategories and organizes a partial order through irreducible mutations.
• Applications include establishing connections with t-structures, torsion theories, and categorification in higher homological algebra.

A silting subcategory is a full additive subcategory of a triangulated (or extriangulated) category that both generates the ambient category and satisfies a vanishing condition on positive degree morphisms. Silting subcategories form the core connective tissue between tilting theory, t-structures, torsion/cotorsion theory, and categorical mutation, appearing in both algebraic and categorical representation theory. The following sections provide a technical overview, emphasizing definitions, structural properties, reduction and mutation techniques, correspondences with t-structures and cluster-tilting theory, and salient applications.

1. Definitions and Characteristic Properties

Let $\mathcal{T}$ be a triangulated category (or, more generally, an extriangulated category). A full additive subcategory $\mathcal{M} \subseteq \mathcal{T}$ is called a silting subcategory if:

1. Self-orthogonality:

$$\operatorname{Hom}_{\mathcal{T}}(M, M'[i]) = 0 \quad \text{for all } M, M' \in \mathcal{M}, \, i > 0.$$

This is often referred to as semi-selforthogonality, and such an $\mathcal{M}$ is called presilting.

1. Generating property:

$\operatorname{thick}(\mathcal{M}) = \mathcal{T},$

where $\operatorname{thick}(\mathcal{M})$ is the smallest thick subcategory containing $\mathcal{M}$ (i.e., closed under shifts, cones, extensions, and direct summands).

In the presence of (co)products, one may alternatively require closure under all (co)products in the “large” setting. For a single object $M$, the silting subcategory is typically $\operatorname{add}(M)$. These definitions extend coherently to extriangulated categories by replacing $\operatorname{Hom}$ with the relevant extension bifunctor $\mathbb{E}$ and using closure under appropriate s-conflations (Adachi et al., 2023).

Silting subcategories generalize tilting subcategories: a tilting subcategory requires vanishing in all nonzero degrees, while silting only requires vanishing in positive degrees.

2. Silting Mutation and Partial Orders

A haLLMark property is the existence of silting mutations (Aihara et al., 2010, Adachi et al., 2023). Given a silting subcategory $\mathcal{M}$ and a covariantly finite subcategory $\mathcal{D} \subseteq \mathcal{M}$, for each $M \in \mathcal{M}$ one chooses a left $\mathcal{D}$-approximation $f: M \to D$ and then constructs a triangle: $M \to D \to N_M \to M[1].$ The left mutation is: $$\mu^+(\mathcal{M}; \mathcal{D}) = \operatorname{add}\left( \mathcal{D} \cup \{ N_M \mid M \in \mathcal{M} \} \right).$$ This process always yields a presilting (often silting) subcategory, circumventing obstructions present in classical tilting mutation (Aihara et al., 2010).

Silting subcategories are naturally partially ordered: for silting subcategories $\mathcal{M}, \mathcal{N}$,

$$\mathcal{M} \geq \mathcal{N} \iff \operatorname{Hom}_{\mathcal{T}}(\mathcal{M}, \mathcal{N}[i]) = 0 \quad \forall\, i > 0.$$

The Hasse quiver of this poset encodes irreducible silting mutations, which organize the “mutation graph” of silting objects and track their combinatorics (Aihara et al., 2010, Adachi et al., 2023).

3. Reduction Procedures and Silting Intervals

Silting reduction removes a (pre)silting subcategory $\mathcal{P} \subset \mathcal{T}$ by forming the Verdier quotient $U = \mathcal{T}/\operatorname{thick}(\mathcal{P})$ (Iyama et al., 2014, Børve, 1 May 2024). Under mild hypotheses, there are bijections: $\text{(pre)silting subcategories in %%%%17%%%% containing %%%%18%%%%} \longleftrightarrow \text{(pre)silting subcategories in %%%%19%%%%}.$ This reduction is compatible with partial orders and mutation (Iyama et al., 2014), and is generalized in extriangulated categories using cotorsion pairs and ideal quotients (Børve, 1 May 2024).

A refinement is silting interval reduction (Pan et al., 24 Jan 2024): given silting subcategories $\mathcal{M} \leq \mathcal{N}$, the set $[ \mathcal{M}, \mathcal{N} ]$ of silting subcategories intermediate between them is naturally identified with the silting subcategories of $\mathcal{M} \cap \mathcal{N}$. This enables a “local-to-global” classification by reducing to smaller subcategories.

4. Correspondences with t-Structures and Torsion Theories

There is a deep correspondence between silting subcategories and t-structures (and their dual co-t-structures) (Liu et al., 2012, Hügel, 2018, Hügel et al., 2016). Given a silting subcategory $\mathcal{M}$ in $\mathcal{T}$, define: $$\mathcal{T}^{\leq 0}_{\mathcal{M}} = \{ X \in \mathcal{T} \mid \operatorname{Hom}_{\mathcal{T}}(\mathcal{M}, X[i]) = 0 \;\forall i > 0 \},$$

$$\mathcal{T}^{\geq 0}_{\mathcal{M}} = \{ X \in \mathcal{T} \mid \operatorname{Hom}_{\mathcal{T}}(\mathcal{M}, X[i]) = 0 \;\forall i < 0 \}.$$

Then $$(\mathcal{T}^{\leq 0}_{\mathcal{M}}, \mathcal{T}^{\geq 0}_{\mathcal{M}})$$ is a t-structure whose heart, in many cases, contains (or is generated by) $\mathcal{M}$ (Aihara et al., 2010).

Silting subcategories also induce co-t–structures and are in bijection with bounded co-t–structures whose cohearts are $\operatorname{add}(M)$ (Liu et al., 2012). Via TTF (torsion-torsionfree) triples, silting subcategories play a universal role in classifying the ambient category’s “torsion-theoretic” decompositions (Hügel, 2018, Hügel et al., 2019).

Over finite-dimensional algebras, the poset of (two-term) silting objects in $K^b(\operatorname{proj} \Lambda)$ is isomorphic to the poset of functorially finite torsion (and cotorsion) classes in $\operatorname{mod} \Lambda$ and in suitable truncated categories (Zhou et al., 2018, Gupta, 15 Jul 2024).

5. Gluing, Complements, and Structural Decomposition

Gluing techniques enable the explicit construction of silting objects in categories realized as recollements, or via ring epimorphisms and universal localizations (Liu et al., 2012, Bonometti, 2020). If $D$ has a recollement built from categories $X$ and $Y$ with silting objects $X$ and $Y$, then the glued silting object is

$Z = i_* Y \oplus K_X,$

where $K_X$ is defined by a distinguished triangle involving truncations in the co-t–structures of $X$ and $Y$. Explicit criteria are given for when the glued silting is tilting (orthogonality conditions for functor images of the summands) (Liu et al., 2012).

Complements and completions are addressed via the averaging of coaisles/co-t–structures: a presilting object $X$ can be completed to a silting object $X \oplus V$ if and only if $X$ is intermediate with respect to some silting $M$ and the intersection of associated coaisles is itself a coaisle. For hereditary and silting-discrete algebras, every presilting admits a complement (Hügel et al., 20 Feb 2024).

6. Mutation Graphs, Discreteness, and Classification

Silting mutation produces a quiver on the set of silting subcategories—the silting quiver. In many settings (e.g., derived-discrete/silting-discrete algebras, local, hereditary, or canonical algebras) this quiver is connected, meaning that any silting object can be reached from any other by iterated mutation (Aihara et al., 2010, Hügel, 2018). For silting-discrete algebras, every large silting object is equivalent to a compact/classically silting object (Hügel et al., 20 Feb 2024).

Classification results for silting subcategories in derived categories of commutative noetherian rings are tightly linked to spectral topological data (Thomason filtrations), while in finite-dimensional settings, combinatorial tools such as poset lattices, cluster combinatorics, and Hasse quivers (and, for extriangulated/0-Auslander categories, picture categories) mediate parametrization and mutation (Hügel, 2018, Børve, 1 May 2024).

7. Applications in Higher and Cluster Categories

Recent work extends silting theory to $d$-silting objects associated with higher homological algebra and Calabi–Yau categories. For a dg algebra $A$ and its $(d+1)$-Calabi–Yau completion $\Pi$, the induction functor

$$- \otimes^{\mathrm{L}}_A \Pi : \operatorname{per} A \to \operatorname{per} \Pi$$

embeds $d$-silting objects into the silting objects of $\Pi$, with tight connections to $d$-cluster tilting subcategories in the associated cluster category $C(\Pi)$ (Hanihara et al., 18 Aug 2025). For “$F$-liftable” Calabi–Yau dg algebras, every $d$-cluster tilting in $C(\Pi)$ can be realized by a silting object in the fundamental domain—a property which is equivalent to $\Pi$ being a Calabi–Yau completion of a hereditary algebra in the class $H^0\Pi$ hereditary, (Hanihara et al., 18 Aug 2025).

Reduction and mutation techniques, cluster-tilting correspondences, and explicit geometric or dg models all further enable applications in the categorification of cluster algebras, representation-theoretic parametrization, and the description of singularity categories.

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Table: Characteristic Conditions for a Silting Subcategory in $\mathcal{T}$

Property	Formulation	Context
Presilting	$\operatorname{Hom}_{\mathcal{T}}(M,M'[i]) = 0$ for $i>0$	All $\mathcal{T}$
Silting	Presilting $+$ $\operatorname{thick}(M) = \mathcal{T}$	Generated subcategory
Mutation	$$\mu^+(\mathcal{M}; \mathcal{D}) = \operatorname{add}(\mathcal{D} \cup \{N_M\})$$	Cov. finite $\mathcal{D}\subset \mathcal{M}$
t-structure aisle	$$\mathcal{T}^{\leq 0}_M = \{ X : \operatorname{Hom}_{\mathcal{T}}(M,X[i])=0,\,i>0 \}$$	Silting $\mathcal{M}$

• (Aihara et al., 2010): "Silting mutation in triangulated categories"
• (Liu et al., 2012): "Glueing silting objects"
• (Iyama et al., 2014): "Silting reduction and Calabi--Yau reduction of triangulated categories"
• (Wei, 2015): "Relative singularity categories, Gorenstein objects and silting theory"
• (Marks et al., 2016): "Universal localisations via silting"
• (Hügel et al., 2016): "Torsion pairs in silting theory"
• (Hügel, 2018): "Silting objects"
• (Zhou et al., 2018): "Two-term relative cluster tilting subcategories, $τ$-tilting modules and silting subcategories"
• (Hügel et al., 2019): "Partial silting objects and smashing subcategories"
• (Bonometti, 2020): "Gluing silting objects along recollements of well generated triangulated categories"
• (Chang et al., 2020): "A geometric realization of silting theory for gentle algebras"
• (Liu et al., 2021): "Silting reduction in extriangulated categories"
• (Breaz et al., 2022): "Silting, cosilting, and extensions of commutative ring"
• (Gao et al., 2022): "(Gorenstein) silting modules in recollements"
• (Adachi et al., 2023): "An assortment of properties of silting subcategories of extriangulated categories"
• (Pan et al., 24 Jan 2024): "Silting interval reduction and 0-Auslander extriangulated categories"
• (Hügel et al., 20 Feb 2024): "Fishing for complements"
• (Børve, 1 May 2024): "Silting reduction and picture categories of 0-Auslander extriangulated categories"
• (Gupta, 15 Jul 2024): "$d$-term silting objects, torsion classes, and cotorsion classes"
• (Hanihara et al., 18 Aug 2025): "Silting correspondences and Calabi-Yau dg algebras"

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Silting subcategories are thus at the intersection of categorical, geometric, algebraic, and combinatoric approaches, providing a flexible and unifying structure underpinning modern developments in higher homological algebra and the theory of triangulated and extriangulated categories.