{{Short description|Generalization of matrix trace}} In [[category theory]], a branch of [[mathematics]], the '''categorical trace''' is a generalization of the [[trace (linear algebra)|trace]] of a [[matrix (mathematics)|matrix]].
==Definition== The trace is defined in the context of a [[symmetric monoidal category]] ''C'', i.e., a [[category (mathematics)|category]] equipped with a suitable notion of a product <math>\otimes</math>. (The notation reflects that the product is, in many cases, a kind of a [[tensor product]].) An [[object (category theory)|object]] ''X'' in such a category ''C'' is called [[dualizable object|dualizable]] if there is another object <math>X^\vee</math> playing the role of a dual object of ''X''. In this situation, the trace of a [[morphism]] <math>f: X \to X</math> is defined as the composition of the following morphisms: <math>\mathrm{tr}(f) : 1 \ \stackrel{coev}{\longrightarrow}\ X \otimes X^\vee \ \stackrel{f \otimes \operatorname{id}}{\longrightarrow}\ X \otimes X^\vee \ \stackrel{twist}{\longrightarrow}\ X^\vee \otimes X \ \stackrel{eval}{\longrightarrow}\ 1</math> where 1 is the monoidal unit and the extremal morphisms are the coevaluation and evaluation, which are part of the definition of dualizable objects.<ref>{{harvtxt|Ponto|Shulman|2014|loc=Def. 2.2}}</ref>
The same definition applies, to great effect, also when ''C'' is a symmetric monoidal ∞-category.
==Examples== * If ''C'' is the [[category of vector spaces]] over a fixed [[field (mathematics)|field]] ''k'', the dualizable objects are precisely the [[dimension (vector space)|finite-dimensional]] [[vector space]]s, and the trace in the sense above is the morphism ::<math>k \to k</math> :which is the multiplication by the trace of the [[endomorphism]] ''f'' in the usual sense of [[linear algebra]]. * More generally, in the [[category of modules]] over a ring ''R'', the dualizable objects are the [[finitely generated module|finitely generated]] [[projective module]]s. The dual of such a module ''M'' is <math>M^*=\operatorname{Hom}(M,R)</math>, and the evaluation map <math>M^*\otimes_RM\to R</math>, <math>\phi\otimes x\mapsto\phi(x)</math> (extended linearly), allows the identification <math>M^*\otimes_RM=\operatorname{End}_R(M)</math>, under which the trace of an endomorphism is, again, given by multiplication with the [[Tensor_product_of_modules#Trace|trace]], the value of the map <math>M^*\otimes_RM\to R</math> above.<ref>{{Citation|last1=Dold|first1=Albrecht|author1link = Albrecht Dold|author2link = Dieter Puppe|last2=Puppe|first2=Dieter|chapter=Duality, trace, and transfer|title=Proceedings of the International Conference on Geometric Topology (Warsaw, 1978) |page=88 |publisher=PWN-Polish Scientific Publishers |year=1980|mr=656721 |isbn=9788301017873 |oclc=681088710}}</ref> Similarly, one can define a trace for endomorphisms of locally free sheaves of finite rank on a [[ringed space]], see {{section link|Sheaf of modules|Operations}}. * If ''C'' is the [[∞-category]] of [[chain complex]]es of [[module (mathematics)|modules]] (over a fixed [[commutative ring]] ''R''), dualizable objects ''V'' in ''C'' are precisely the [[perfect complex]]es. The trace in this setting captures, for example, the [[Euler characteristic]], which is the alternating sum of the ranks of its terms: ::<math>\mathrm{tr}(\operatorname{id}_V) = \sum_i (-1)^i \operatorname {rank} V_i.</math><ref>{{harvtxt|Ponto|Shulman|2014|loc=Ex. 3.3}}</ref>
==Further applications==
{{harvtxt|Kondyrev|Prikhodko|2018}} have used categorical trace methods to [[mathematical proof|prove]] an [[algebraic geometry|algebro-geometric]] version of the [[Atiyah–Bott fixed point formula]], an extension of the [[Lefschetz fixed point formula]].
==References==
<references/>
==Further reading== {{refbegin}} *{{Citation|title=Categorical Proof of Holomorphic Atiyah–Bott Formula|last1=Kondyrev|first1=Grigory|last2=Prikhodko|first2=Artem|journal=J. Inst. Math. Jussieu|year=2018|volume=19|issue=5|pages=1–25|doi=10.1017/S1474748018000543|arxiv=1607.06345}} * {{Citation| first1 = Kate | last1 = Ponto | first2 = Michael | last2 = Shulman|title = Traces in symmetric monoidal categories | journal = Expositiones Mathematicae | volume = 32 | issue = 3 | year = 2014 | pages = 248–273 | doi = 10.1016/j.exmath.2013.12.003 | arxiv = 1107.6032| bibcode = 2011arXiv1107.6032P | s2cid = 119129371 }} {{refend}}
[[Category:Category theory]] [[Category:Fixed-point theorems]] [[Category:Geometry]] [[Category:Trace theory]]