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Let $K^b(\mathrm{proj}\, A)$ be the bounded homotopy category of chain complexes over $\mathrm{proj}\, A$. In Rickard's paper 'Derived categories and stable equivalence', he defines a tilting complex as an object $T$ such that $\mathrm{Hom}_{K^b(\mathrm{proj}\, A)}(T,T[n])=0$ for all $n \neq 0$ and such that the full subcategory $\mathrm{add}(T)$ of $K^b(\mathrm{proj}\, A)$ generates $K^b(\mathrm{proj}\, A)$ as a triangulated category.

What exactly does it mean for $\mathrm{add}(T)$ to generate $K^b(\mathrm{proj}\, A)$ as a triangulated category? Under what operation or process does this generation occur?

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It means that $K^b(\text{proj }A)$ is the smallest full triangulated subcategory containing $\text{add}(T)$. This is spelled out more explicitly in Section 5 of the earlier paper "Morita theory for derived categories", where the definition of a tilting complex first appears.

More concretely, starting with the objects of $\text{add}(T)$, you can construct new objects by shifting or completing triangles: i.e., if there is a distinguished triangle where you have two of the objects, then you get the third object.

This isn't quite the same as the definition in the link given by @hic in comments, which is "generating as a thick subcategory" (i.e., additionally you can take direct summands of objects you've already constructed). However, in the presence of the other parts of the definition of a tilting complex, it turns out that it doesn't make any difference.

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