Calabi-Yau object


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See at cobordism hypothesis – For non-compact cobordisms.



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

This is (Lurie, def. 4.2.6).


Calabi-Yau algebras


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

Then a Calabi-Yau object in Alg(S)Alg(\mathbf{S}) is an algebra object AA equipped with an SO(2)SO(2)-equivariant morphism

tr: S 1A1 tr : \int_{S^1} A \to 1

satisfying the condition that the composite morphism

AA S 0A S 1Atr1 A \otimes A \simeq \int_{S^0} A \to \int_{S^1} A \stackrel{tr}{\to} 1

exhibits AA as its own dual A A^\vee.

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

This is (Lurie, example 4.2.8).


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

Z:Bord 2 nc𝒞 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\mathcal{C}.

This is Lurie, 4.2.11.

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


Section 4.2 of

Revised on July 6, 2014 06:39:45 by Urs Schreiber (