Categorical trace


In 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. An object X in such a category C is called dualizable if there is another object playing the rôle of a dual object of X. In this situation, the trace of a morphism is defined as the compositition of the following morphisms:
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
which is the multiplication by the trace of the endomorphism f in the usual sense of linear algebra. In this sense, the categorical trace generalizes the linear-algebraic trace.
If C is the ∞-category of chain complexes of modules, 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:

Further applications

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.