Categorical trace

Short description: Generalization of matrix trace

In category theory, a branch of mathematics, the categorical trace is a generalization of the trace of a matrix.

Definition

The trace is defined in the context of a symmetric monoidal category C, i.e., a category equipped with a suitable notion of a product $\otimes$. (The notation reflects that the product is, in many cases, a kind of a tensor product.) An object X in such a category C is called dualizable if there is another object $X^{\vee }$ playing the role of a dual object of X. In this situation, the trace of a morphism $f:X\to X$ is defined as the composition of the following morphisms: $\mathrm{t}\mathrm{r}(f):1\stackrel{coev}{⟶}X\otimes X^{\vee }\stackrel{f\otimes \mathrm{id}}{⟶}X\otimes X^{\vee }\stackrel{twist}{⟶}{X}^{\vee }\otimes X\stackrel{eval}{⟶}1$ where 1 is the monoidal unit and the extremal morphisms are the coevaluation and evaluation, which are part of the definition of dualizable objects.

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 k, the dualizable objects are precisely the finite-dimensional vector spaces, and the trace in the sense above is the morphism

$k\to k$

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 projective modules. The dual of such a module M is $M^{*}=\mathrm{Hom}(M,R)$, and the evaluation map $M^{*}{\otimes }_{R}M\to R$, $\varphi \otimes x\mapsto \varphi (x)$ (extended linearly), allows the identification $M^{*}{\otimes }_{R}M={\mathrm{End}}_R(M)$, under which the trace of an endomorphism is, again, given by multiplication with the trace, the value of the map $M^{*}{\otimes }_{R}M\to R$ above. Similarly, one can define a trace for endomorphisms of locally free sheaves of finite rank on a ringed space, see Sheaf of modules § Operations.
• If C is the ∞-category of chain complexes of modules (over a fixed commutative ring R), dualizable objects V in C are precisely the perfect complexes. The trace in this setting captures, for example, the Euler characteristic, which is the alternating sum of the ranks of its terms:

$\mathrm{t}\mathrm{r}({\mathrm{id}}_V)=\sum _i(-1{)}^i\mathrm{rank}{V}_i.$

Further applications

(Kondyrev Prikhodko) have used categorical trace methods to prove an algebro-geometric version of the Atiyah–Bott fixed point formula, an extension of the Lefschetz fixed point formula.

See also

• Pivotal category, a not necessarily symmetric category where traces can be defined
• Traced monoidal category

Further reading

• Kondyrev, Grigory; Prikhodko, Artem (2018), "Categorical Proof of Holomorphic Atiyah–Bott Formula", J. Inst. Math. Jussieu 19 (5): 1–25, doi:10.1017/S1474748018000543 
• Ponto, Kate; Shulman, Michael (2014), "Traces in symmetric monoidal categories", Expositiones Mathematicae 32 (3): 248–273, doi:10.1016/j.exmath.2013.12.003, Bibcode: 2011arXiv1107.6032P