# Categorical trace

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Generalization of matrix trace

In [category theory](/source/Category_theory), a branch of [mathematics](/source/Mathematics), the **categorical trace** is a generalization of the [trace](/source/Trace_(linear_algebra)) of a [matrix](/source/Matrix_(mathematics)).

## Definition

The trace is defined in the context of a [symmetric monoidal category](/source/Symmetric_monoidal_category) *C*, i.e., a [category](/source/Category_(mathematics)) equipped with a suitable notion of a product ⊗ {\displaystyle \otimes } . (The notation reflects that the product is, in many cases, a kind of a [tensor product](/source/Tensor_product).) An [object](/source/Object_(category_theory)) *X* in such a category *C* is called [dualizable](/source/Dualizable_object) if there is another object X ∨ {\displaystyle X^{\vee }} playing the role of a dual object of *X*. In this situation, the trace of a [morphism](/source/Morphism) f : X → X {\displaystyle f:X\to X} is defined as the composition of the following morphisms: t r ( f ) : 1 ⟶ c o e v X ⊗ X ∨ ⟶ f ⊗ id X ⊗ X ∨ ⟶ t w i s t X ∨ ⊗ X ⟶ e v a l 1 {\displaystyle \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} where 1 is the monoidal unit and the extremal morphisms are the coevaluation and evaluation, which are part of the definition of dualizable objects.[1]

The same definition applies, to great effect, also when *C* is a symmetric monoidal ∞-category.

## Examples

- If *C* is the [category of vector spaces](/source/Category_of_vector_spaces) over a fixed [field](/source/Field_(mathematics)) *k*, the dualizable objects are precisely the [finite-dimensional](/source/Dimension_(vector_space)) [vector spaces](/source/Vector_space), and the trace in the sense above is the morphism

- - k → k {\displaystyle k\to k}

- which is the multiplication by the trace of the [endomorphism](/source/Endomorphism) *f* in the usual sense of [linear algebra](/source/Linear_algebra).

- More generally, in the [category of modules](/source/Category_of_modules) over a ring *R*, the dualizable objects are the [finitely generated](/source/Finitely_generated_module) [projective modules](/source/Projective_module). The dual of such a module *M* is M ∗ = Hom ⁡ ( M , R ) {\displaystyle M^{*}=\operatorname {Hom} (M,R)} , and the evaluation map M ∗ ⊗ R M → R {\displaystyle M^{*}\otimes _{R}M\to R} , ϕ ⊗ x ↦ ϕ ( x ) {\displaystyle \phi \otimes x\mapsto \phi (x)} (extended linearly), allows the identification M ∗ ⊗ R M = End R ⁡ ( M ) {\displaystyle M^{*}\otimes _{R}M=\operatorname {End} _{R}(M)} , under which the trace of an endomorphism is, again, given by multiplication with the [trace](/source/Tensor_product_of_modules#Trace), the value of the map M ∗ ⊗ R M → R {\displaystyle M^{*}\otimes _{R}M\to R} above.[2] Similarly, one can define a trace for endomorphisms of locally free sheaves of finite rank on a [ringed space](/source/Ringed_space), see [Sheaf of modules § Operations](/source/Sheaf_of_modules#Operations).

- If *C* is the [∞-category](/source/%E2%88%9E-category) of [chain complexes](/source/Chain_complex) of [modules](/source/Module_(mathematics)) (over a fixed [commutative ring](/source/Commutative_ring) *R*), dualizable objects *V* in *C* are precisely the [perfect complexes](/source/Perfect_complex). The trace in this setting captures, for example, the [Euler characteristic](/source/Euler_characteristic), which is the alternating sum of the ranks of its terms:

- - t r ( id V ) = ∑ i ( − 1 ) i rank ⁡ V i . {\displaystyle \mathrm {tr} (\operatorname {id} _{V})=\sum _{i}(-1)^{i}\operatorname {rank} V_{i}.} [3]

## Further applications

[Kondyrev & Prikhodko (2018)](#CITEREFKondyrevPrikhodko2018) have used categorical trace methods to [prove](/source/Mathematical_proof) an [algebro-geometric](/source/Algebraic_geometry) version of the [Atiyah–Bott fixed point formula](/source/Atiyah%E2%80%93Bott_fixed_point_formula), an extension of the [Lefschetz fixed point formula](/source/Lefschetz_fixed_point_formula).

## References

1. **[^](#cite_ref-1)** [Ponto & Shulman (2014](#CITEREFPontoShulman2014), Def. 2.2)

1. **[^](#cite_ref-2)** [Dold, Albrecht](/source/Albrecht_Dold); [Puppe, Dieter](/source/Dieter_Puppe) (1980), "Duality, trace, and transfer", *Proceedings of the International Conference on Geometric Topology (Warsaw, 1978)*, PWN-Polish Scientific Publishers, p. 88, [ISBN](/source/ISBN_(identifier)) [9788301017873](https://en.wikipedia.org/wiki/Special:BookSources/9788301017873), [MR](/source/MR_(identifier)) [0656721](https://mathscinet.ams.org/mathscinet-getitem?mr=0656721), [OCLC](/source/OCLC_(identifier)) [681088710](https://search.worldcat.org/oclc/681088710)

1. **[^](#cite_ref-3)** [Ponto & Shulman (2014](#CITEREFPontoShulman2014), Ex. 3.3)

## Further reading

- Kondyrev, Grigory; Prikhodko, Artem (2018), "Categorical Proof of Holomorphic Atiyah–Bott Formula", *J. Inst. Math. Jussieu*, **19** (5): 1–25, [arXiv](/source/ArXiv_(identifier)):[1607.06345](https://arxiv.org/abs/1607.06345), [doi](/source/Doi_(identifier)):[10.1017/S1474748018000543](https://doi.org/10.1017%2FS1474748018000543)

- Ponto, Kate; Shulman, Michael (2014), "Traces in symmetric monoidal categories", *Expositiones Mathematicae*, **32** (3): 248–273, [arXiv](/source/ArXiv_(identifier)):[1107.6032](https://arxiv.org/abs/1107.6032), [Bibcode](/source/Bibcode_(identifier)):[2011arXiv1107.6032P](https://ui.adsabs.harvard.edu/abs/2011arXiv1107.6032P), [doi](/source/Doi_(identifier)):[10.1016/j.exmath.2013.12.003](https://doi.org/10.1016%2Fj.exmath.2013.12.003), [S2CID](/source/S2CID_(identifier)) [119129371](https://api.semanticscholar.org/CorpusID:119129371)

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