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See at cobordism hypothesis – For non-compact cobordisms.
For $C$ a symmetric monoidal (infinity,2)-category, a Calabi-Yau object in $C$ is
a morphism $\eta : dim(X) = ev_X \circ coev_X \to Id_x$ in $\Omega_x C$ which is equivariant with respect to the canonical ∞-action of the circle group $SO(2)$ on $dim(X)$ and which is the counit for an adjunction between the evaluation map $ev_X$ and coevaluation map $coev_X$.
This is (Lurie 09, def. 4.2.6).
Let $\mathbf{S}$ be a good symmetric monoidal (∞,1)-category. Write $Alg(\mathbf{S})$ for the symmetric monoidal (∞,2)-category whose objects are algebra objects in $\mathbf{S}$ and whose morphisms are bimodule objects.
Then a Calabi-Yau object in $Alg(\mathbf{S})$ is an algebra object $A$ equipped with an $SO(2)$-equivariant morphism
from the Hochschild homology $\int_{S^1} A \simeq A \otimes_{A \otimes A} A$, satisfying the condition that the composite morphism
exhibits $A$ as its own dual object $A^\vee$.
Such an algebra object is called a Calabi-Yau algebra object.
This is (Lurie 09, example 4.2.8).
A version of the cobordism hypothesis says that symmetric monoidal $(\infty,2)$-functors
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 $\mathcal{C}$.
This is (Lurie 09, theorem 4.2.11).
Here the trace condition translates to the cobordism which is the “disappearance of a circle”.
Its would-be adjoint, the “appearance of a circle” is not included in $Bord_2^{nc}$.
This is closely related to the description of 2d TQFT as TCFTs (Lurie 09, theorem 4.2.13).
Last revised on January 3, 2015 at 17:08:05. See the history of this page for a list of all contributions to it.