This paper is a much expanded version of the Appendix of the previously posted paper entitled "Picard groups, Grothendieck rings, and Burnside rings of categories". In it, we explain a fundamental additivity theorem for Euler characteristics and generalized trace maps in triangulated categories. The proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation. The refinement consists of several new axioms relating products and distinguished triangles. The axioms hold in the examples and shed light on generalized homology and cohomology theories.