nLab higher trace

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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)) \,.

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).

Last revised on April 24, 2021 at 16:03:02. See the history of this page for a list of all contributions to it.