nLab Calabi-Yau category



Category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



The notion of Calabi-Yau category is a horizontal categorification of that of Frobenius algebra – a Frobenius algebroid. Their name derives from the fact that the definition of Calabi-Yau categories have been originally studied as an abstract version of the derived category of coherent sheaves on a Calabi-Yau manifold.



A Calabi-Yau category is a Vect-enriched category CC equipped for each object cCc \in C with a trace-like map

Tr C:C(c,c)k Tr_C : C(c,c) \to k

to the ground field, such that for all objects dCd \in C the induced pairing

, c,d:C(c,d)C(d,c)k \langle -,-\rangle_{c,d} : C(c,d) \otimes C(d,c) \to k

given by

f,g=Tr(gf) \langle f,g \rangle = Tr(g \circ f)

is symmetric and non-degenerate.

[Question: this 1-categorical definition seems to allow for different Frobenius structures on the endomorphism algebras of isomorphic objects. Would it be better to define it as a dinatural transformation from the Hom-functor to the constant functor with value the ground field ?]

A Calabi-Yau category with a single object is the same (or rather: is equivalently) the pointed monoid delooping of a Frobenius algebra.


A Calabi-Yau A A_\infty-category of dimension dd \in \mathbb{N} is an A-∞ category CC equipped with, for each pair a,ba,b of objects, a morphism of chain complexes

, a,b:C(a,b)C(b,a)k[d] \langle -,-\rangle_{a,b} : C(a,b) \otimes C(b,a) \to k[d]

such that

  1. this is non-degenerate and is symmetric in that

    , a,b=, b,aσ a,b \langle - , - \rangle_{a,b} = \langle - , - \rangle_{b,a} \circ \sigma_{a,b}

    for σ a,b:C(a,b)C(b,a)C(b,a)C(a,b)\sigma_{a,b} : C(a,b)\otimes C(b,a) \to C(b,a) \otimes C(a,b) the symmetry isomorphism of the symmetric monoidal category of chain complexes;

  2. this is cyclically invariant in that for all elements (α i)(\alpha_i) is the respective hom-complexes we have

    m n1(α 0α n2),α n1=(1) (n+1)+|α 0| i=1 n1|α i|m n1(α 1α n2),α 0. \langle m_{n-1}(\alpha_0 \otimes \cdots \otimes \alpha_{n-2}), \alpha_{n-1} \rangle = (-1)^{(n+1)+ |\alpha_0| \sum_{i = 1}^{n-1}|\alpha_i|} \langle m_{n-1}(\alpha_1 \otimes \cdots \otimes \alpha_{n-2}), \alpha_0 \rangle \,.


From Calabi-Yau varieties

  • Let XX be a smooth projective Calabi-Yau variety of dimension dd. Write D b(X)D^b(X) for the bounded derived category of that of coherent sheaves on XX.

    Then D b(X)D^b(X) is a CY A A_\infty-category in a naive way:

    • the non-binary composition maps are all trivial;

    • the pairing is given by Serre duality (one needs also a choice of trivialization of the canonical bundle of XX)

This is however not the morally correct CY A A_\infty-structure associated with a Calabi-Yau. A correct choice is, for example, the Dolbeault dg-enhancement of the derived category

(Costello 04, 7.2, Costello 05, 2.2)

From symplectic manifolds

The Fukaya category associated with a symplectic manifold XX. But see this MO discussion for more.

From string topology

string topology: for XX a compact simply connected oriented manifold, its cohomology H (X)H^{\bullet}(X) is naturally a Calabi-Yau A A_\infty-category with a single object. The A A_\infty structure comes from the homological perturbation lemma. One could also use the dg algebra of cochains C (X)C^\bullet(X).


Classification of 2d TQFT

Calabi-Yau A A_\infty-categories classify TCFTs. This remarkable result is what actually one should expect. Indeed, TCFTs originally arose as an abstract version of the CFTs constructed from sigma-models whose targets are Calabi-Yau spaces.

2d TQFT (“TCFT”)coefficientsalgebra structure on space of quantum states
open topological stringVect k{}_kFrobenius algebra AAfolklore+(Abrams 96)
open topological string with closed string bulk theoryVect k{}_kFrobenius algebra AA with trace map BZ(A)B \to Z(A) and Cardy condition(Lazaroiu 00, Moore-Segal 02)
non-compact open topological stringCh(Vect)Calabi-Yau A-∞ algebra(Kontsevich 95, Costello 04)
non-compact open topological string with various D-branesCh(Vect)Calabi-Yau A-∞ category
non-compact open topological string with various D-branes and with closed string bulk sectorCh(Vect)Calabi-Yau A-∞ category with Hochschild cohomology
local closed topological string2Mod(Vect k{}_k) over field kkseparable symmetric Frobenius algebras(SchommerPries 11)
non-compact local closed topological string2Mod(Ch(Vect))Calabi-Yau A-∞ algebra(Lurie 09, section 4.2)
non-compact local closed topological string2Mod(S)(\mathbf{S}) for a symmetric monoidal (∞,1)-category S\mathbf{S}Calabi-Yau object in S\mathbf{S}(Lurie 09, section 4.2)


A relative version is defined for functors instead of categories,

  • Christopher Brav , Tobias Dyckerhoff, Relative Calabi–Yau structures, Compositio Math. 155 (2019) 372–412 doi; Relative Calabi–Yau structures II: shifted Lagrangians in the moduli of objects, Sel. Math. New Ser. 27, 63 (2021). doi
  • Bernhard Keller, Yu Wang, An introduction to relative Calabi-Yau structures, arXiv:2111.10771

Last revised on October 18, 2022 at 10:32:19. See the history of this page for a list of all contributions to it.