nLab higher trace



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

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

tr S k(id id id V)Ω k𝒞. 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 Σ\Sigma of dimension kk on any fully dualizable object XX

tr Σ(id V)Ω k𝒞 tr_{\Sigma}(id_V) \in \Omega^k \mathcal{C}

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

(1tr Σ(V)tr Σ(id V))Hom Ω k1𝒞(1,tr Σ(id V)). (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.