Recollements of Triangulated Categories

• Recollements of Triangulated Categories are precise structures that decompose a central triangulated category into two complementary subcategories via exact triangle functors, enabling effective localization and colocalization.
• They support the construction and gluing of t-structures, silting objects, and torsion theories, thereby impacting representation theory, algebraic geometry, and higher category theory.
• Iterated forms like ladders and polygons extend the framework, facilitating advanced analyses of homological dimensions and categorical enhancements across various mathematical contexts.

A recollement of triangulated categories is a precise structure encoding how a central triangulated category may be reconstructed from two complementary subcategories, capturing both localization and colocalization data. Originally formalized by Beĭlinson, Bernstein, and Deligne in the context of perverse sheaves, recollements have become central to the structure theory of triangulated and stable ∞-categories, impacting areas such as homological algebra, representation theory, algebraic geometry, and higher category theory.

1. Formal Definition and Key Axioms

Let $\mathcal{D}$, $\mathcal{X}$, and $\mathcal{Y}$ be triangulated categories. A recollement of $\mathcal{D}$ by $\mathcal{X}$ and $\mathcal{Y}$ consists of six exact triangle functors arranged as follows: $\begin{tikzcd} \mathcal{Y} \ar[r, "i_*"] & \mathcal{D} \ar[l, shift left=2, "i^!"] \ar[l, shift right=2, "i^*"'] \ar[r, "j^*"] & \mathcal{X} \ar[l, shift left=2, "j_*"] \ar[l, shift right=2, "j_!"'] \end{tikzcd}$ such that:

• $(i^*, i_*, i^!)$ and $(j_!, j^*, j_*)$ are adjoint triples: $i^*\dashv i_* \dashv i^!$, $j_! \dashv j^* \dashv j_*$.
• $i_*, j_!$, and $j_*$ are fully faithful.
• $i^* j_! = 0$ (equivalently, $j^* i_* = 0$, $i^! j_* = 0$).
• For every $D \in \mathcal{D}$, there are functorial triangles:

$$i_* i^! D \to D \to j_* j^* D \to i_* i^! D[1], \qquad j_! j^* D \to D \to i_* i^* D \to j_! j^* D[1].$$

This structure identifies $\mathcal{D}$ as “glued” from $\mathcal{Y}$ and $\mathcal{X}$ by means of these adjunctions and truncation sequences (Hügel et al., 2010, Sun et al., 2022, Fiorenza et al., 2015).

2. Gluing of Structures along Recollements

Recollements provide a canonical framework to construct and glue various categorical and homological structures:

• t-Structures and Silting Theory: Both t-structures and co-t-structures can be glued along recollements, yielding complex invariants and hearts with prescribed simple objects. Explicit gluing constructions for silting and tilting objects, as well as simple-minded collections (SMCs), have been established for (co)compact and non-compact settings (Bonometti, 2020, Sun et al., 2022, Saorín et al., 2018).
• Torsion Theories in Stable ∞-Categories: In the language of stable $\infty$-categories, recollement and the associated gluing operation on torsion pairs correspond to the formalism for higher factorization systems. The “gluing” of t-structures is associative up to canonical equivalence, allowing, for instance, the construction of perverse t-structures on stratified spaces independent of the parenthesization of glue order (Fiorenza et al., 2015).

3. Iterated Recollements, Ladders, and Polygons

Iterated recollements, also known as “ladders” or “stratifications,” generalize the basic recollement by introducing composable chains (or infinite/infinite periodic ladders) of adjunctions between individual or derived/singularity categories (Gao et al., 2016, Gao et al., 2020). This yields hierarchies of decompositions analogous to composition series in group theory.

• Ladders of Recollements: A ladder of height $n$ consists of functor chains in which each consecutive pair forms an adjunction, and under suitable conditions allow the lifting of abelian recollements to derived/singularity categories. Ladders control which homological properties, such as Gorensteinness, are preserved or lifted through these constructions.
• Polygons of Recollements: In certain Calabi–Yau triangulated categories, intricate cyclic collections—so-called $n$-gons of recollements—arise, decomposing the triangulated category into orthogonal subcategories via successively orthogonal complements and Serre functors (Iyama et al., 2016).

4. Model Structures, Extriangulated Generalizations, and Categorical Enhancements

Recollement formalisms extend beyond classical triangulated categories:

• Model Categories and Cotorsion Pairs: There is a tight relationship between cotorsion pairs and recollements in WIC exact categories. For instance, three compatible injective cotorsion pairs produce a recollement among their associated homotopy categories, including classical cases such as Verdier localization and Gorenstein recollements (Gillespie, 2013, Gillespie, 2012).
• Extriangulated and Right-Triangulated Generalizations: Extriangulated categories, encompassing abelian and triangulated categories, admit s-recollements, which become genuine recollements after localization. Passage between extriangulated and triangulated settings is enabled by precise biresolving and thick subcategory conditions (Hu et al., 2021, Ma et al., 2021).
• Comma Categories and Morphic Enhancements: Every recollement induces a canonical epivalence (full and dense on objects) between the central category and a comma category constructed from a triangle functor between the outer categories. This provides an enriched view on the role of recollements in categorical enhancement and module realization (Chen et al., 2019).

5. Gluing and Restriction of Recollements to Subcategories

Recent advances address the restriction and induction of recollements to distinguished or intrinsic subcategories:

• Preservation under Restriction: For compactly generated triangulated categories, a recollement restricts to compacts provided certain adjoint functors preserve compactness. Analogous results are established for bounded, bounded-finite, or singularity subcategories, relying on orthogonality and homological finiteness conditions (Kostas et al., 20 Dec 2025).
• Gluing Approximability: Approximable categories—those admitting suitable resolutions—retain approximability under recollement, thereby ensuring the robust behavior of approximation-theoretic tools under categorical glueing operations (Burke et al., 2018).
• Behavior of Homological Dimensions: Under recollement, properties such as global dimension, finitistic dimension, and regularity can be transferred or compared between components, provided boundedness and regularity conditions are met. Recent work formalizes these preservation and equivalence criteria via “far-away orthogonality” (Kostas et al., 20 Dec 2025).

6. Applications and Examples

• Stratifications and Jordan–Hölder Theory: Iterated recollements yield stratifications of derived module categories analogous to composition series. A Jordan–Hölder theorem holds for derived categories of hereditary Artin algebras: all stratifications have the same multiset of simple factors, up to derived equivalence and order (Hügel et al., 2010).
• Classification Problems: The structure and gluing theory for silting and tilting objects, SMCs, and hearts of t-structures under recollement are central to modern classification results in the representation theory of algebras (Bonometti, 2020, Sun et al., 2022, Saorín et al., 2018).
• Non-commutative and Algebraic Geometry: Recollements govern the behavior of derived categories of (possibly noncommutative) schemes under open-closed decompositions, perverse sheaf theory, and categorical blowing-up procedures (Burke et al., 2018, Fiorenza et al., 2015).
• Categorified Homological Algebra: The passage from abelian to derived, stable, or extriangulated settings, via ladders and model structures, underpins advances in Gorenstein/homological algebra, cotorsion pair theory, and spectral/∞-categorical methods (Gillespie, 2012, Gillespie, 2013, Kostas et al., 20 Dec 2025).

7. Higher and Future Perspectives

• Stable ∞-Categories and Coherence: The stable $\infty$-categorical framework captures higher coherence data omitted in triangulated settings, enabling canonical and associative gluing at all levels (e.g., Urizen compasses, Jacob’s ladders). This formalism is essential for the study of perverse sheaves, stratified spaces, and homological invariants in derived algebraic geometry (Fiorenza et al., 2015).
• Open Problems: The existence and uniqueness of recollement-based stratifications for general rings, the extension of Jordan–Hölder-type theorems, and the classification of derived simple rings remain active areas of research (Hügel et al., 2010). Recent work on intrinsic homological algebra aims to generalize these issues without reference to explicit compact or projective generators, using categorical orthogonality alone (Kostas et al., 20 Dec 2025).

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References:

• (Hügel et al., 2010) “On the uniqueness of stratifications of derived module categories”
• (Fiorenza et al., 2015) “Recollements in stable $\infty$-categories”
• (Iyama et al., 2016) “Polygon of recollements and $N$-complexes”
• (Gao et al., 2016) “Ladders of recollements, categories of monomorphisms and singularity categories”
• (Burke et al., 2018) “Gluing Approximable Triangulated Categories”
• (Saorín et al., 2018) “Lifting of recollements and gluing of partial silting sets”
• (Chen et al., 2019) “Recollements, comma categories and morphic enhancements”
• (Bonometti, 2020) “Gluing silting objects along recollements of well generated triangulated categories”
• (Gao et al., 2020) “Ladders of recollements of abelian categories”
• (Hu et al., 2021) “s-Recollements and its localization”
• (Ma et al., 2021) “Recollements induced by left Frobenius pairs”
• (Sun et al., 2022) “Gluing simple-minded collections in triangulated categories”
• (Gillespie, 2012) “Gorenstein complexes and recollements from cotorsion pairs”
• (Gillespie, 2013) “Exact model structures and recollements”
• (Kostas et al., 20 Dec 2025) “Intrinsic homological algebra for triangulated categories”