Weak Relative Calabi–Yau Structures

• Weak relative Calabi–Yau structures are relative duality data that establish a quasi-isomorphism between relative Hochschild chains and the shifted duals of cochains.
• They are defined in various settings (dg, A∞, and stable ∞-categories) and underpin constructions like relative completions and gluing of duality data.
• Applications include enhanced mirror symmetry, Legendrian contact homology, and cluster algebra formulations via Ginzburg dg algebras over Gorenstein rings.

Searching arXiv for recent and foundational papers on weak relative Calabi–Yau structures. Weak relative Calabi–Yau structures are relative duality data attached to a functor between homologically enriched categories. Depending on the framework, they are formulated for dg functors, strictly unital $A_\infty$-functors, or $R$-linear stable $\infty$-functors, but the common content is a Poincaré–Lefschetz type identification between a relative diagonal or relative Hochschild object and an appropriate shifted dual. In the dg setting of relative Calabi–Yau completions, the datum is a class in relative negative cyclic homology whose Hochschild image induces a quasi-isomorphism between relative Hochschild chains and the shifted dual of relative Hochschild cochains (Yeung, 2016). In later formulations, especially over commutative Gorenstein rings and in stable $\infty$-categories, the adjective “weak” often refers to a nondegenerate class in Hochschild homology or dual Hochschild homology before any lift to cyclic or negative cyclic homology has been chosen (Hanihara et al., 3 Dec 2025, Christ, 2023).

1. Terminology and conceptual scope

The phrase “weak relative Calabi–Yau structure” is not completely uniform across the literature. In Yeung’s dg-categorical treatment, a weak relative $n$-Calabi–Yau structure on a dg functor $F:A\to B$ is the datum of a class

$[\omega]\in HN_n(B,A)$

whose image in $HH_n(B,A)$ induces a quasi-isomorphism of bimodules

$$C_{\bullet}(B,A)\xrightarrow{\simeq} C^\bullet(A,B)^\vee[-n].$$

Thus the structure already lives in relative negative cyclic homology, and “weak” refers to the relative, homotopy-level duality statement rather than to the absence of cyclic data (Yeung, 2016).

By contrast, in the formulation for dg categories over a commutative Gorenstein ring $R$, a weak structure is explicitly distinguished from a genuine one. There, “weak” means that one has only a Hochschild class $R$0, or a dual Hochschild class $R$1, satisfying nondegeneracy, but not necessarily a lift to $R$2 or to $R$3. A promotion to a genuine Calabi–Yau structure is the choice of such a lift (Hanihara et al., 3 Dec 2025).

An analogous distinction appears in the $R$4-linear stable $R$5-categorical setting. For a functor $R$6, a weak left or right $R$7-Calabi–Yau structure is a Hochschild or dual-Hochschild class satisfying the relevant nondegeneracy condition encoded by a fiber–cofiber diagram, while an unqualified left or right $R$8-Calabi–Yau structure is an $R$9-invariant lift to negative cyclic or dual-cyclic homology (Christ, 2023).

The relative/absolute distinction is stable across these settings. In the dg and $\infty$0-categorical formulations, setting one side equal to $\infty$1 recovers the ordinary absolute Calabi–Yau notion for a single category. A recurrent misconception is therefore that “weak” always means “degenerate” or “insufficient for duality.” The cited papers use “weak” instead to mark a precise stage in the homological or cyclic hierarchy, and in every case nondegeneracy remains essential.

2. Relative Hochschild models in the dg setting

For a dg functor

$\infty$2

between smooth, $\infty$3-flat small dg categories, the relative theory is built from cone constructions. Yeung defines the relative Hochschild chain complex by

$\infty$4

and similarly the relative negative cyclic complex by

$\infty$5

On the cochain side one has

$\infty$6

where $\infty$7 denotes the bimodule dual (Yeung, 2016).

Within this framework, a weak relative $\infty$8-Calabi–Yau structure on $\infty$9 is a class

$\infty$0

whose image in $\infty$1 induces the quasi-isomorphism

$\infty$2

This states that, up to homotopy, relative Hochschild chains are identified with the shifted dual of relative Hochschild cochains. In the absolute case $\infty$3, the construction recovers Keller–Van den Bergh’s $\infty$4-Calabi–Yau condition (Yeung, 2016).

The same paper packages the relative chain-level duality through short resolutions. Starting from the bar–Cuntz–Quillen resolutions $\infty$5 and $\infty$6, one forms

$\infty$7

The proof of nondegeneracy then proceeds by constructing a relative Casimir element in a cone of resolution bimodules and proving that the induced map

$\infty$8

is a quasi-isomorphism. The significance of this step is that the Calabi–Yau property is verified at the bimodule level, not only at the level of numerical pairings.

3. Relative Calabi–Yau completions

The principal construction in Yeung’s paper generalizes Keller’s deformed $\infty$9-Calabi–Yau completion to the relative setting. Given smooth, $n$0-flat small dg categories $n$1 and $n$2, and a dg functor $n$3, one first chooses cofibrant replacements

$n$4

and then forms the tensor algebras

$n$5

The original functor induces a bimodule map

$n$6

hence a dg-algebra map

$n$7

by sending generators to generators (Yeung, 2016).

Deformation data are supplied by a relative negative cyclic class

$n$8

represented by a cocycle

$n$9

Its lowest term $F:A\to B$0 produces degree $F:A\to B$1 bimodule maps $F:A\to B$2, which deform the differentials on both tensor algebras. One obtains deformed dg categories

$F:A\to B$3

together with an induced dg functor

$F:A\to B$4

The main theorem states that for any such $F:A\to B$5, the associated deformed relative tensor-algebra functor

$F:A\to B$6

carries a canonical weak relative $F:A\to B$7-Calabi–Yau structure. Equivalently, one constructs a class

$F:A\to B$8

whose Hochschild image induces

$F:A\to B$9

If $[\omega]\in HN_n(B,A)$0 is exact, in the sense of lying in the image of the Connes map $[\omega]\in HN_n(B,A)$1, then the resulting relative Calabi–Yau structure is exact (Yeung, 2016).

This construction is significant because it supplies a systematic mechanism for manufacturing relative duality from functorial data and a relative negative cyclic class. It also organizes the absolute deformed completion as the special case $[\omega]\in HN_n(B,A)$2.

4. Weak, genuine, left, and right structures over Gorenstein rings

Over a commutative Gorenstein ring $[\omega]\in HN_n(B,A)$3, the literature separates left and right Calabi–Yau structures and makes the weak/genuine distinction explicit. For a smooth small dg $[\omega]\in HN_n(B,A)$4-category $[\omega]\in HN_n(B,A)$5, a left $[\omega]\in HN_n(B,A)$6-Calabi–Yau structure is a class

$[\omega]\in HN_n(B,A)$7

whose image in $[\omega]\in HN_n(B,A)$8 is nondegenerate, meaning that the induced bimodule map

$[\omega]\in HN_n(B,A)$9

is an isomorphism in $HH_n(B,A)$0. For an arbitrary dg $HH_n(B,A)$1-category $HH_n(B,A)$2, a right $HH_n(B,A)$3-Calabi–Yau structure is a class

$HH_n(B,A)$4

whose image in $HH_n(B,A)$5 is nondegenerate, equivalently producing an isomorphism

$HH_n(B,A)$6

in $HH_n(B,A)$7. In this setting, “weak” means that one has the nondegenerate Hochschild or dual-Hochschild class without the cyclic lift (Hanihara et al., 3 Dec 2025).

The same paper analyzes exact sequences of small flat dg $HH_n(B,A)$8-categories

$HH_n(B,A)$9

via Keller’s distinguished triangle of Hochschild complexes and its dual. The connecting morphism

$$C_{\bullet}(B,A)\xrightarrow{\simeq} C^\bullet(A,B)^\vee[-n].$$0

is identified with the dg-level lift of Amiot’s connecting construction. Under the additional hypotheses that the canonical maps $$C_{\bullet}(B,A)\xrightarrow{\simeq} C^\bullet(A,B)^\vee[-n].$$1 and $$C_{\bullet}(B,A)\xrightarrow{\simeq} C^\bullet(A,B)^\vee[-n].$$2 are isomorphisms of $$C_{\bullet}(B,A)\xrightarrow{\simeq} C^\bullet(A,B)^\vee[-n].$$3-bimodules, $$C_{\bullet}(B,A)\xrightarrow{\simeq} C^\bullet(A,B)^\vee[-n].$$4 preserves nondegeneracy: a right $$C_{\bullet}(B,A)\xrightarrow{\simeq} C^\bullet(A,B)^\vee[-n].$$5-Calabi–Yau class on $$C_{\bullet}(B,A)\xrightarrow{\simeq} C^\bullet(A,B)^\vee[-n].$$6 is sent to a right $$C_{\bullet}(B,A)\xrightarrow{\simeq} C^\bullet(A,B)^\vee[-n].$$7-Calabi–Yau class on $$C_{\bullet}(B,A)\xrightarrow{\simeq} C^\bullet(A,B)^\vee[-n].$$8 (Hanihara et al., 3 Dec 2025).

A central application concerns a symmetric $$C_{\bullet}(B,A)\xrightarrow{\simeq} C^\bullet(A,B)^\vee[-n].$$9-order $R$0. Let

$R$1

Auslander–Reiten duality gives a weak right $R$2-Calabi–Yau class on $R$3, the connecting morphism carries it to a weak right $R$4-Calabi–Yau class on $R$5, and a base-change isomorphism for Gorenstein $R$6-algebras shifts this to a genuine right $R$7-Calabi–Yau structure over the ground field $R$8. Under the listed hypotheses, one then deduces that the singularity category is triangle equivalent to a generalized cluster category of a deformed $R$9-preprojective dg algebra (Hanihara et al., 3 Dec 2025).

5. Stable $R$00-categorical reformulation and gluing

The $R$01-categorical treatment replaces dg categories by dualizable $R$02-linear stable $R$03-categories, with $R$04 a commutative $R$05-ring spectrum. For a functor $R$06, the left and right relative theories are formulated through endofunctor-valued duality objects. On the smooth side, a weak left $R$07-Calabi–Yau structure is a map

$R$08

such that the associated diagram built from $R$09, $R$10, and the functor

$R$11

has all three vertical maps equivalences. A left $R$12-Calabi–Yau structure is a lift of $R$13 to

$R$14

On the proper side, a weak right $R$15-Calabi–Yau structure is a map

$R$16

satisfying the analogous nondegeneracy condition, and a right $R$17-Calabi–Yau structure is a lift to $R$18 (Christ, 2023).

One of the main structural results is that relative Calabi–Yau structures glue. For homotopy pushouts of smooth $R$19-linear categories, left $R$20-Calabi–Yau structures that agree on the intersection induce a left $R$21-Calabi–Yau structure on the composite functor. Dually, for homotopy pullbacks of proper $R$22-linear categories, compatible right $R$23-Calabi–Yau structures glue to a right $R$24-Calabi–Yau structure on the induced functor. The proofs analyze unit and counit diagrams of adjunctions and show that the relative Hochschild classes force the relevant squares to be bi-Cartesian (Christ, 2023).

These gluing theorems are applied to perverse schobers on surfaces with boundary. For periodic topological Fukaya categories, the natural cup-functor to boundary values admits a relative left $R$25-Calabi–Yau structure, and its proper subcategory admits a right $R$26-Calabi–Yau structure. For spherical objects, a weak right $R$27-Calabi–Yau structure on a category induces a weak right $R$28-Calabi–Yau structure on the associated spherical functor. For relative Ginzburg algebras, if either $R$29 is odd or the spanning graph is orientable, the derived category of the relative Ginzburg algebra admits a left $R$30-Calabi–Yau structure relative to the boundary evaluation functor; a parallel integral statement gives weak right $R$31-Calabi–Yau structures (Christ, 2023).

A further point is the role of monodromy. The local-transport functors of a perverse schober assemble to a local system only after passage to the frame bundle or projectivized tangent bundle, and the global gluing theorem requires compatibility between these local transports and the underlying cyclic classes. This makes the relative Calabi–Yau datum sensitive not only to local duality but also to the framing-dependent global topology of the surface.

6. $R$32-bimodules, Legendrian contact homology, and concrete examples

In the $R$33 framework, a weak right relative Calabi–Yau structure of dimension $R$34 on a strictly unital $R$35-functor

$R$36

over $R$37 consists of an $R$38-morphism

$R$39

and an $R$40-pre-morphism

$R$41

such that $R$42 is a null-homotopy of

$R$43

equivalently

$R$44

and such that the associated exact triangles fit into a commutative Poincaré–Lefschetz diagram in the derived category, with $R$45 and the induced maps on cone and cocone quasi-isomorphisms (Ma et al., 2 Sep 2025).

The main example is Legendrian contact homology. For a Legendrian knot $R$46, the positive augmentation $R$47-category $R$48, the circle category $R$49, and the projection functor

$R$50

are constructed using holomorphic disk counts in the $R$51-copy of $R$52. If $R$53 is defined by a “simply perturbed” Morse function, then $R$54 carries a weak right relative Calabi–Yau structure of dimension $R$55 (Ma et al., 2 Sep 2025).

The proof uses three bimodules from the separated $R$56-copy of $R$57: the positive bimodule $R$58, a negative submodule $R$59 quasi-isomorphic to the Serre bimodule, and the circle bimodule $R$60. The short exact sequence

$R$61

produces a bimodule morphism $R$62, and direct disk counts show that its adjoint $R$63, together with the Poincaré map $R$64, satisfies a very weak Calabi–Yau condition. An algebraic lemma then upgrades this to the full weak relative structure on $R$65 (Ma et al., 2 Sep 2025).

The derived exact triangles recover the classical long exact sequence for linearized Legendrian contact homology: $R$66 For the Legendrian figure-eight knot, the paper computes the Chekanov–Eliashberg DGA, its two augmentations, the relevant $R$67-operations, an explicit homotopy $R$68, and the exact sequence

$R$69

thereby verifying the classical duality statement in a fully relative $R$70-categorical form (Ma et al., 2 Sep 2025).

A more algebraic example already appears in the completion formalism: when $R$71 is the path algebra of a finite quiver $R$72 and $R$73, the absolute completion $R$74 is precisely the Ginzburg dg algebra of $R$75, while a frozen-vertex subquiver $R$76 yields an inclusion $R$77 whose relative completion recovers the ice Ginzburg algebra (Yeung, 2016). This example shows that weak relative Calabi–Yau structures are not confined to abstract duality theory; they are built into the standard algebraic models used in cluster theory and related parts of representation theory.