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 formula_1. (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 formula_2 playing the rôle of a dual object of "X". In this situation, the trace of a morphism formula_3 is defined as the composition of the following morphisms:
formula_4
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 (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:
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.