{"title":"The Space of Traces in Symmetric Monoidal Infinity Categories","authors":"Jan Steinebrunner","doi":"10.1093/qmath/haab013","DOIUrl":null,"url":null,"abstract":"We define a tracelike transformation to be a natural family of conjugation invariant maps \n<tex>$T_{x,\\mathtt{C}}:\\hom_\\mathtt{C}(x, x) \\to \\hom_\\mathtt{C}(\\unicode{x1D7D9},\\unicode{x1D7D9})$</tex>\n for all dualizable objects x in any symmetric monoidal \n<tex>$\\infty$</tex>\n-category \n<tex>$\\mathtt{C}$</tex>\n. This generalizes the trace from linear algebra that assigns a scalar \n<tex>$\\operatorname{Tr}(\\,f\\,) \\in k$</tex>\n 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 \n<tex>$\\operatorname{Tr}$</tex>\n 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 \n<tex>$\\infty$</tex>\n-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 \n<tex>$f \\mapsto \\operatorname{Tr}(\\,f^{\\,n})$</tex>\n. Unlike for \n<tex>$\\operatorname{Tr}$</tex>\n, the relevant connected component of the moduli space is not contractible, but rather equivalent to \n<tex>$B\\mathbb{Z}/n\\mathbb{Z}$</tex>\n or BS\n<sup>1</sup>\n for n = 0. As a result, we obtain a \n<tex>$\\mathbb{Z}/n\\mathbb{Z}$</tex>\n-action on \n<tex>$\\operatorname{Tr}(\\,f^{\\,n})$</tex>\n as well as a circle action on \n<tex>$\\operatorname{Tr}(\\operatorname{id}_x)$</tex>\n.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab013","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quarterly Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://ieeexplore.ieee.org/document/9690934/","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
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 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
$\operatorname{Tr}(\operatorname{id}_x)$
.
期刊介绍:
The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.