# Contents

## Definition

###### Definition

For $C$ a symmetric monoidal (infinity,2)-category, a Calabi-Yau object in $C$ is

• a dualizable object;

• a morphism $\eta :\mathrm{dim}\left(X\right):{\mathrm{ev}}_{X}\circ {\mathrm{circoev}}_{X}\to {\mathrm{Id}}_{x}$ in ${\Omega }_{x}C$ which is equivariant with respect to the canonical action of the circle group on $\mathrm{dim}\left(X\right)$ and which is the counit for an adjunction between ${\mathrm{ev}}_{X}$ and ${\mathrm{coev}}_{X}$.

This is (Lurie, def. 4.2.6).

## Examples

### Calabi-Yau algebras

###### Example

Let $S$ be a good symmetric monoidal (∞,1)-category. Write $\mathrm{Alg}\left(S\right)$ for the symmetric monoidal (∞,2)-category whose objects are algebra objects in $S$ and whose morphisms are bimodule objects.

Then a Calabi-Yau object in $\mathrm{Alg}\left(S\right)$ is an algebra object $A$ equipped with an $\mathrm{SO}\left(2\right)$-equivariant morphism

$\mathrm{tr}:{\int }_{{S}^{1}}A\to 1$tr : \int_{S^1} A \to 1

satisfying the condition that the composite morphism

$A\otimes A\simeq {\int }_{{S}^{0}}A\to {\int }_{{S}^{1}}A\stackrel{\mathrm{tr}}{\to }1$A \otimes A \simeq \int_{S^0} A \to \int_{S^1} A \stackrel{tr}{\to} 1

exhibits $A$ as its own dual ${A}^{\vee }$.

Such an algebra object is called a Calabi-Yau algebra object.

This is (Lurie, example 4.2.8).

## Properties

A version of the cobordism hypothesis says that symmetric monoidal $\left(\infty ,2\right)$-functors

$Z:{\mathrm{Bord}}_{2}^{\mathrm{nc}}\to 𝒞$Z : Bord_2^{nc} \to \mathcal{C}

out of a version of the (infinity,2)-category of cobordisms where all 2-cobordisms have at least one outgoing (ingoing) boundary component, are equivalently given by their value on the point, which is a Calabi-Yau object in $𝒞$.

This is Lurie, 4.2.11.

This is closely related to the description of TCFTs (Lurie, theorem 4.2.13).

## References

Section 4.2 of

Revised on July 21, 2011 22:32:27 by Urs Schreiber (82.113.99.58)