The space of traces in symmetric monoidal infinity categories

Abstract

We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,C}: hom_C(x,x) \to hom_C(1,1)$ for all dualisable objects $x$ in any symmetric monoidal infinity-category $C$. This generalises the trace from linear algebra that assigns a scalar $Tr(f) \in k$ to any endomorphism $f:V \to V$ 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 $Tr$ can be uniquely extended to a tracelike transformation up to a contractible space of choices. This allows us to give several model-independent characterisations of the infinity-categorical trace. Restricting our notion of tracelike transformations from endomorphisms to automorphisms we in particular recover a theorem of To\"en and Vezzosi. Other examples of tracelike transformations are for instance given by $f \mapsto Tr(f^n)$. Unlikefor $Tr$ the relevant connected component of the moduli space is not contractible, but ratherequivalent to $B\mathbb{Z}/n\mathbb{Z}$ or $BS^1$ for $n=0$. As a result we obtain a $\mathbb{Z}/n\mathbb{Z}$-action on $Tr(f^n)$ as well as a circle action on $Tr(id_x)$.