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 $\otimes$. (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: $\mathrm {tr} (f):1\ {\stackrel {coev}{\longrightarrow }}\ X\otimes X^{\vee }\ {\stackrel {f\otimes \operatorname {id} }{\longrightarrow }}\ X\otimes X^{\vee }\ {\stackrel {twist}{\longrightarrow }}\ X^{\vee }\otimes X\ {\stackrel {eval}{\longrightarrow }}\ 1$ 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

$k\to k$

which is the multiplication by the trace of the endomorphism f in the usual sense of linear algebra.

• 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:

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

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.

Further reading

• Kondyrev, Grigory; Prikhodko, Artem (2018), "Categorical Proof of Holomorphic Atiyah–Bott Formula", J. Inst. Math. Jussieu, 19 (5): 1–25, arXiv:1607.06345, doi:10.1017/S1474748018000543
• Ponto, Kate; Shulman, Michael (2014), "Traces in symmetric monoidal categories", Expositiones Mathematicae, 32 (3): 248–273, arXiv:1107.6032, Bibcode:2011arXiv1107.6032P, doi:10.1016/j.exmath.2013.12.003, S2CID 119129371