Categorical 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 ${\displaystyle \otimes }$. (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 ${\displaystyle X^{\vee }}$ playing the role of a dual object of X. In this situation, the trace of a morphism ${\displaystyle f:X\to X}$ is defined as the composition of the following morphisms: ${\displaystyle \mathrm {tr} (f):1\ {\stackrel {coev}{\longrightarrow }}\ X\otimes X^{\vee }\ {\stackrel {f\otimes \operatorname {id} }{\longrightarrow }}\ X\otimes X^{\vee }\ {\stackrel {twist}{\longrightarrow }}\ X^{\vee }\otimes X\ {\stackrel {eval}{\longrightarrow }}\ 1}$ where 1 is the monoidal unit and the extremal morphisms are the coevaluation and evaluation, which are part of the definition of dualizable objects.

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

${\displaystyle k\to k}$

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 ${\displaystyle M^{*}=\operatorname {Hom} (M,R)}$, and the evaluation map ${\displaystyle M^{*}\otimes _{R}M\to R}$, ${\displaystyle \phi \otimes x\mapsto \phi (x)}$ (extended linearly), allows the identification ${\displaystyle M^{*}\otimes _{R}M=\operatorname {End} _{R}(M)}$, under which the trace of an endomorphism is, again, given by multiplication with the trace, the value of the map ${\displaystyle M^{*}\otimes _{R}M\to R}$ 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:

${\displaystyle \mathrm {tr} (\operatorname {id} _{V})=\sum _{i}(-1)^{i}\operatorname {rank} V_{i}.}$

Further applications

Kondyrev & Prikhodko (2018) 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

References

Further reading