Cosingularity Categories: Duals of Singularity Categories

• Cosingularity categories are triangulated invariants defined as Verdier quotients capturing complementary, dual, or relative singularity information of algebraic and geometric objects.
• They measure co-smoothness or coregularity by quotienting out objects generated from the residue field or analogous subcategories in derived and noncommutative contexts.
• Their applications span noncommutative resolutions, representation theory, and categorification of cluster algebras, elucidating duality, Koszul phenomena, and AR structures.

A cosingularity category is a Verdier quotient or triangulated invariant that captures the complement, dual, or relative structure to the singularity category of a ring, algebra, scheme, dg category, or more general geometric object. Its formalization, computation, and applications have developed in parallel to singularity theory and triangulated categories, notably in relative, derived, and noncommutative contexts. Cosingularity categories encode information about the "coregularity" or "co-smoothness" (often measured by quotienting out objects built from the residue field or from geometric data), and they achieve prominence in contexts such as noncommutative resolutions, derived algebraic geometry, representation theory, and the categorification of cluster algebras.

1. Formal Definitions and Foundational Constructions

Cosingularity categories arise as Verdier quotients in derived or triangulated settings. A typical construction, for a dg algebra, k-linear category, or ring $A$, is

$\mathrm{Cosg}(A) = D^b(A)/\mathrm{Thick}(k)$

where $\mathrm{Thick}(k)$ is the thick subcategory generated by the residue field $k$ as an $A$-module. This construction complements the classical singularity category

$\mathrm{Sg}(A) = D^b(A)/\mathrm{Perf}(A)$

where $\mathrm{Perf}(A)$ is the subcategory of perfect complexes (i.e., those finitely built from $A$ itself). In general, objects in the cosingularity category represent large or infinite-dimensional behavior not detected by the smooth/regular part of the derived category (Greenlees et al., 2017, Benson et al., 2021, Benson, 2022).

A relative version arises by replacing $k$ with an arbitrary subcategory $\mathcal{C}$ (e.g., Gorenstein projectives, or band objects) and forming

$$D_{\mathcal{C}\text{-sg}}(\mathcal{A}) = D_\mathcal{C}^{b}(\mathcal{A}) / K^{b}(\mathcal{C})$$

encoding singular or cosingular phenomena with respect to $\mathcal{C}$ (Li et al., 2015, Hafezi, 2020).

2. Duality, Koszul Phenomena, and Relationship to Singularity Categories

Cosingularity categories are often dual or Koszul dual to singularity categories. In many symmetric Gorenstein contexts, such as for augmented dg algebras with Noether normalizations, there is a deep duality: $D_\mathrm{cosg}(R) \simeq D_\mathrm{sg}(E)$ where $E$ is the Koszul dual of $R$. Under this duality, the cosingularity category for $R$ (measuring coregularity, i.e., how $R$ fails to be finitely built from $k$) corresponds to the singularity category of $E$ (measuring regularity), and vice versa (Greenlees et al., 2017, Benson et al., 2021, Benson, 2022).

In representation theory, as for block cohomology rings $H^*BG$ or their $A_\infty$/dg models, the cosingularity category $\mathrm{Cosg}(H^*BG)$ is equivalent to the stable category of maximal Cohen–Macaulay modules for $H^*BG$ if it is a complete intersection (Benson, 2022). Koszul duality then links this with a singularity category for $H_*(\Omega BG^{\wedge}_p)$, reflecting stable module categories for the group algebra $kG$ (Benson et al., 2021).

3. Structural and Classification Results

The structure of cosingularity categories often mirrors, but complements, that of singularity categories:

• Decomposition and Semisimplicity: In settings such as gentle algebras or categories built from quivers with radical square zero, singularity and cosingularity categories decompose into products of cluster or orbit categories, often exhibiting semisimple abelian or finite type behavior (Kalck, 2012, Bouhada, 2019, Kravets, 2019).
• Classification via Support or Cosupport: In tensor-triangular geometry, the (co)support theory classifies categorical substructures. For instance, cosupport assigns to an object $A$ the set of primes $p$ with $\mathrm{Hom}_R(g_p,A)\neq 0$, and costratification identifies (co)localizing subcategories of the singularity category $S(R)$ with subsets of the singular locus $\operatorname{Sing} R$ (Verasdanis, 2023, Stevenson, 2011).
• Auslander–Reiten (AR) Structure: Finite or tame cosingularity categories (e.g., for group algebras with tame representation type) display AR quivers of cylindrical form, with explicit combinatorial parameters indexed by homological invariants or derived autoequivalences (Benson et al., 2021).

4. Methodologies and Model Constructions

Cosingularity categories are accessed using several methodologies:

• Relative and DG Model Structures: Localized abelian or model-categorical structures (for instance, cotorsion pair localization, localization of injective or projective model structures) provide explicit dg or homotopy-theoretic models for cosingularity and singularity categories, including in the context of matrix factorizations, curved or mixed complexes, and derived noncommutative resolutions (Becker, 2012, Kalck et al., 2018, Christensen et al., 2020).
• Perverse Schober and Fukaya Category Techniques: For higher Teichmüller theory and cluster Fequations, cosingularity categories arise as categorical invariants constructed via perverse schobers, glued from local data of Calabi–Yau completions associated to triangulations of surfaces, and are identified with topological Fukaya categories valued in cluster categories (Christ, 7 Oct 2025).
• Amalgamation and Quiver Mutations: Amalgamation of ice quivers with potential and the gluing along perverse schobers correspond to the composition or mutation of seeds in the additive categorification of cluster algebras, yielding explicit models for cosingularity categories with cluster tilting objects (Liu, 1 Apr 2024, Christ, 7 Oct 2025).

5. Applications in Representation Theory, Geometry, and Topology

The deployment of cosingularity categories has concrete utility in multiple domains:

• Modular and Block Representation Theory: Cosingularity categories for group cohomology, blocks of group rings, and Brauer tree algebras directly encode the stable representation type, with their AR quivers and indecomposable objects delineating the representation-theoretic complexity (Benson et al., 2021, Benson, 2022).
• Algebraic Geometry and Calabi–Yau Categories: For Gorenstein schemes, toric varieties, or quotient singularities, the cosingularity category measures deviations from being coregular or of finite Cohen–Macaulay type. Equivalences with Higgs or cluster categories enable the translation of geometric information into categorical and representation-theoretic invariants (Liu, 1 Apr 2024, Christ, 7 Oct 2025).
• Cluster Algebras and Teichmüller Theory: Cosingularity categories serve as the categorical backbone for the additive categorification of cluster algebras attached to higher Teichmüller spaces; they arise as invariants of topological Fukaya categories of surfaces, with canonical clusters and tilting subcategories reflecting the combinatorial seed data of cluster varieties (Christ, 7 Oct 2025).

6. Generalizations, Dualizations, and Open Directions

Research continues to expand cosingularity category theory:

• Beyond Isolated, Gorenstein, or Hypersurface Singularities: Extending structural results and equivalences for cosingularity categories to more general singular, noncommutative, and higher-dimensional settings remains an active pursuit (Burban et al., 2011, Kalck, 2017, Liu, 1 Apr 2024).
• Theoretical Dualizations: Dual or relative versions, such as those replacing projectives with injectives or considering the quotient by band or geometric subcategories, offer alternative perspectives and finer invariants (e.g., stable categories of Gorenstein injectives) (Kalck, 2012, Li et al., 2015, Hafezi, 2020).
• Interaction with Tensor-Triangular Geometry and Homotopy Theory: The costratification and support/cosupport frameworks continue to provide a powerful language for classifying categorical substructures and their geometric avatars, with implications for stratification conjectures and telescope-type properties (Stevenson, 2011, Verasdanis, 2023).
• Categorical and Homological Mirror Symmetry: As cosingularity categories appear in Landau–Ginzburg mirror symmetry and higher Teichmüller theory, their paper is intertwined with the ongoing effort to categorify and dualize geometric and physical invariants (Kravets, 2019, Christ, 7 Oct 2025).

7. Summary Table: Cosingularity Category Constructions

Setting	Singularity Category	Cosingularity Category
Ring/dg algebra $A$	$D^b(A)/\mathrm{Perf}(A)$	$D^b(A)/\mathrm{Thick}(k)$
Local complete intersection $R$	Stable category of MCM modules	Stable Gorenstein injective modules
$A_\infty$-algebra $a$ (e.g. $C^*BG$)	$\mathrm{Sg}(a) = D^b(a)/\mathrm{Thick}(a)$	$\mathrm{Cosg}(a) = D^b(a)/\mathrm{Thick}(k)$
Relative context (subcategory $\mathcal{C}$)	$D_\mathcal{C}^{b}(\mathcal{A}) / K^b(\mathcal{C})$	$D_\mathcal{C}^{b}(\mathcal{A}) / K^b(\mathcal{S})$, $\mathcal{S} \subset \mathcal{C}$

Cosingularity categories provide a flexible, dual, and often categorically robust framework for understanding the "co-smooth," "coregular," and complementarily singular features of derived categories arising in algebra, geometry, topology, and categorified representation theory. Their structure, equivalences (often via Koszul duality), and computational models continue to inform advances in both theoretical and applied aspects of modern mathematics.