Here are some key points of the definition of tensot triangulated categories.

A tensor triangulated category is a category with both a symmetric monoidal structure (denoted ⊗, 1 etc.) and a triangulated structure. In addition one has natural in X and Y isomorphisms of the form :

r_{X,Y} : X ⊗ Y [1] → (X ⊗ Y)[1]
l_{X,Y} : X[1] ⊗ Y → (X ⊗ Y)[1]

which commute in the obvious sense with associativity, commutativity and unity isomorphisms. (In particular r can be recovered from l and vice versa. In addition r_{X,Y} can be recovered from r_{X} := r_{X,1}:

X ⊗ 1[1] → X[1])

One also assumes that for a distinguished triangle

X_{0} → X_{1} → X_{2} → X_{0}[1]

and any Y the triangles

X_{0} ⊗ Y → X_{1} ⊗ Y → X_{2} ⊗ Y → (X_{0} ⊗ Y)[1]  (1)

and

Y ⊗ X_{0} → Y ⊗ X_{1} → Y ⊗ X_{2} → (Y ⊗ X_{0})[1]  (2)

where the last morphisms are, respectively,

X_{2} ⊗ Y → X_{0}[1] ⊗ Y →^{l} (X_{0} ⊗ Y)[1]

and

Y ⊗ X_{2} → Y ⊗ X_{0}[1] →^{r} (Y ⊗ X_{0})[1]

are distinguished (it is sufficient for one of (1), (2) to be distinguished, then the other one is as well).

These are all “obvious” axioms. The only non obvious one says that for any X and Y the following diagram

X[1] ⊗ Y [1] ⟶ (X[1] ⊗ Y)[1]
↓  ↓
(X ⊗ Y [1])[1] ⟶ (X ⊗ Y)[2]

anticommutes i.e. commutes up to multiplication by −1. The following is the only general result about t.t.c.’s:

Proposition 0.1 Let A be a symmetric monoidal additive category such that the symmetric monoidal structure commutes with the direct sums. Then the homotopy category of complexes over A with the standard monoidal and triangulated structures and the standard choice of r and l is a tensor triangulated category.