Comments to tensor triangulated categories:

1. Prop. 8.8. (dcTTC) One formulate the proof a little different. Just say that D^{−} = K^{−}(P) where P is additive symmetric monoidal. Then apply 8A.4 (KTTC).

2. Right before (TTC1) in def. 8A.1 (TTC): “... unity isomorphisms” instead of “.... unity axioms”

3. In definition 8A.3 (asmc) - better not to consider infinite direct sums

4. Example 8A.5 - same comment as for Prop. 8.8

5. Example 8A.5 - “based” not “bases”

6. proof of Prop. 8A.7 (locTTC) - the reason for ⊗ to be defined on the localized category is the universal property of the localiztion: the functor C ↦ C ⊗ D takes W to W hence descends to the localization. This does not depend on how good W is.

7. Examples 8A.8 and 8A.9: First - I do not understand the second example at all. What D[T^{−1}] has to do with localization? The first example is correct but there is a subtle things which is skipped. In general for a symmetric monoidal category D the category D[T^{−1}] is *not* monoidal. It is monoidal if and only if the cyclic permutation on T^{⊗3} becomes the identity in D[T^{−1}]. If the condition of the first example holds then this is the case, but not in general.

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References

[1] Vladimir Voevodsky, Carlo Mazza, and Charles Weibel. Lectures on motivic cohomology. www.math.uiuc.edu/K-theory/486, 2000.