The Space of Traces in Symmetric Monoidal Infinity Categories

Abstract

Abstract We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,\mathtt{C}}:\hom_\mathtt{C}(x, x) \to \hom_\mathtt{C}(𝟙,𝟙)$ for all dualizable objects x in any symmetric monoidal $\infty$-category $\mathtt{C}$. This generalizes the trace from linear algebra that assigns a scalar $\operatorname{Tr}(\,f\,) \in k$ to any endomorphism f : V → V of a finite-dimensional k-vector space. Our main theorem computes the moduli space of tracelike transformations using the one-dimensional cobordism hypothesis with singularities. As a consequence, we show that the trace $\operatorname{Tr}$ can be uniquely extended to a tracelike transformation up to a contractible space of choices. This allows us to give several model-independent characterizations of the $\infty$-categorical trace. By restricting the aforementioned notion of tracelike transformations from endomorphisms to automorphisms one can in particular recover a theorem of Toën and Vezzosi. Other examples of tracelike transformations are for instance given by $f \mapsto \operatorname{Tr}(\,f^{\,n})$. Unlike for $\operatorname{Tr}$, the relevant connected component of the moduli space is not contractible, but rather equivalent to $B\mathbb{Z}/n\mathbb{Z}$ or BS1 for n = 0. As a result, we obtain a $\mathbb{Z}/n\mathbb{Z}$-action on $\operatorname{Tr}(\,f^{\,n})$ as well as a circle action on $\operatorname{Tr}(\operatorname{id}_x)$.