nLab higher trace

Idea

For morphisms between dualizable objects in a symmetric monoidal category there is a notion of trace. More generally, for a fully dualizable object $V$ in a symmetric monoidal (∞,n)-category $\mathcal{C}^\otimes$ there is a notion of $k$-dimensional trace for each $k \leq n$ and indeed for each diffeomorphism type of a closed manifold.

The cobordism theorem says that a monoidal (∞,n)-functor $Z \colon Bord_n \to \mathcal{C}^\otimes$ from the (∞,n)-category of cobordisms picks a fully dualizable object $V \coloneqq Z(\ast) \in \mathcal{C}$ and then sends the $k$-sphere $S^k$ to the $k$-dimensional higher trace of the identity on $X$:

$$tr_{S^k}(id_{id_{ \cdots id_V}}) \in \Omega^k \mathcal{C} \,.$$

This are the “round traces”. More generally the cobordism theorem gives a higher dimensional trace of the “shape” of any closed manifold $\Sigma$ of dimension $k$ on any fully dualizable object $X$

$$tr_{\Sigma}(id_V) \in \Omega^k \mathcal{C}$$

and for every $k$-manifold with boundary $\partial \Sigma$ the relative trace is a morphism

$$(1 \stackrel{tr_{\Sigma}(V)}{\longrightarrow} tr_{\partial \Sigma}(id_V)) \in Hom_{\Omega^{k-1}\mathcal{C}}(1,tr_{\partial \Sigma}(id_V)) \,.$$

Examples

Higher dimensional traces in a an (∞,n)-category of correspondences are give by higher span traces. Those of the shape of Riemann surfaces are spelled out for instance in (lpqft).

Related concepts

• trace

• bicategorical trace

• span trace

• opetopic type theory

References

• David Ben-Zvi, David Nadler, Secondary Traces (arXiv:1305.7177)

• David Ben-Zvi, David Nadler, Nonlinear traces (arXiv:1305.7175)

• Local prequantum field theory