The Space of Traces in Symmetric Monoidal Infinity Categories

We define a tracelike transformation to be a natural family of conjugation invariant maps
$T_{x,\mathtt{C}}:\hom_\mathtt{C}(x, x) \to \hom_\mathtt{C}(\unicode{x1D7D9},\unicode{x1D7D9})$
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 BS
¹
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
$\operatorname{Tr}(\operatorname{id}_x)$
.