The Space of Traces in Symmetric Monoidal Infinity Categories
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The Space of Traces in Symmetric Monoidal Infinity Categories. / Steinebrunner, Jan Paul.
In: The Quarterly Journal of Mathematics, Vol. 72, No. 4, 09.12.2021, p. 1461–1493.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - The Space of Traces in Symmetric Monoidal Infinity Categories
AU - Steinebrunner, Jan Paul
PY - 2021/12/9
Y1 - 2021/12/9
N2 - Abstract We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,\mathtt{C}}:\hom_\mathtt{C}(x, x) \to \hom_\mathtt{C}(\unicode{x1D7D9},\unicode{x1D7D9})$ 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 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 $\operatorname{Tr}(\operatorname{id}_x)$.
AB - Abstract We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,\mathtt{C}}:\hom_\mathtt{C}(x, x) \to \hom_\mathtt{C}(\unicode{x1D7D9},\unicode{x1D7D9})$ 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 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 $\operatorname{Tr}(\operatorname{id}_x)$.
UR - http://dx.doi.org/10.1093/qmath/haab013
U2 - 10.1093/qmath/haab013
DO - 10.1093/qmath/haab013
M3 - Journal article
VL - 72
SP - 1461
EP - 1493
JO - Quarterly Journal of Mathematics
JF - Quarterly Journal of Mathematics
SN - 0033-5606
IS - 4
ER -
ID: 318208123