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Context: The ordinary Euler characteristic of a complex (satisfying appropriate finiteness conditions so that all cohomology groups are finite-dimensional over some field ''k'', say, and only finitely many are nonzero) is defined as $$\chi(X) = \sum (-1)^n \dim H^n(X)$$ In fact, this is an example of a categorical trace, namely $\chi(X)$ is the dimension of $X$, where dimension refers to the trace of the identity, and trace refers to the trace of an endomorphism of a dualizable object (see e.g. this survey by Ponto and Shulman. Indeed, a complex as above, i.e., a perfect complex, is precisely a dualizable object in the derived category of $k$-vector spaces: $$\chi(X) = \mathrm{tr} (X \stackrel{id} \to X).$$

The secondary (and similarly, higher) Euler characteristic is defined as $$\chi_2(X) := \sum (-1)^n n \dim H^n(X).$$ It appears in many places, surveyed e.g. by Ramachandran in Higher Euler Characteristics.

Question: Is the secondary (and higher) Euler characteristic also an example of a categorical trace?

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