Are There Examples of Triangulated Categories that $\bigcap_n\mathcal{D}^{\geq n}\neq\{0\} $?

Question

In chapter 4 of Gelfand&Manin's book Methods in Homological Algebra they mentions the property that if a triangulated category $\mathsf{T} $ with the $t$-structure $(\mathcal{D}^{\leq 0} ,\mathcal{D}^{\geq 0} )$ satisfies $\bigcap_n\mathcal{D}^{\geq n} =\bigcap_n\mathcal{D}^{\leq n} =\{0\} $, then it will behave much like a derived category with the core as the underlying abelian category, and I'm curious if there is a natural example of triangulated category does not satisfy this assumption, or it can be derived from the axioms.

Answer 1

Well, there is always the trivial $t$-structure where the truncations are $t_{\leq n}=0$ and $t_{\geq n}=id$.

A bit less trivially, you can consider a category of couples of complexes $C(\mathcal{A})\times C(\mathcal{A})$. In other words, an object is the data of a first complex $...\to C^n\to C^{n+1}\to C^{n+2}\to...$ and another one where indices are denoted by $\omega+n$ (just a notation) : $...\to C^{\omega+n}\to C^{\omega+n+1}\to ...$. Shifts works as usual on both complexes. A quasi-isomorphism is a morphism inducing isomorphisms in all degree ($n$ and $\omega+n$). Invert quasi-isomorphism and you have a triangulated categories which is simply $D(\mathcal{A})\times D(\mathcal{A})$.

You now have truncation functors $t_{\leq n}$ which keeps only the degrees $\leq n$ and in particular kills everything in degree $\omega+n$. And the functor $t_{\geq n}$ which keeps only the degrees $\geq n$ and in particular keeps everything in degree $\omega+n$.

In this situation $\bigcap_{n}D^{\geq n}=\{0\}\times D(\mathcal{A})$ is the whole derived category of complexes in degree $\omega+n$.

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In this somewhat artificial example (which is simply the product of the canonical $t$-structure and the trivial one), I wanted to show that you can assume that in some triangulated categories, things might happen "at infinity". Of course this is not the case in the derived category of an abelian category.