In Fascieaux pervers, Beilinson, Bernstein and Deligne define a recollement situation as a triple of triangulated categories $\mathcal{D}_U,\mathcal{D}_F$ and $\mathcal{D}$, together with functors $$ i_*\colon \mathcal{D}_F\to \mathcal{D},\qquad j_*\colon \mathcal{D}\to \mathcal{D}_U $$ with suitable properties. In particular, $i_*$ has both a left adjoint $i^*$ and a right adjoint $i^!$. Beilinson, Bernstein and Deligne show that, in this situation, given $t$-structures $(\mathcal{D}^{\leq0}_U,\mathcal{D}^{\geq 0}_U)$ on $\mathcal{D}_U$ and $(\mathcal{D}^{\leq0}_F,\mathcal{D}^{\geq 0}_F)$ on $\mathcal{D}_F$ one obtains a $t$-structure $(\mathcal{D}^{\leq0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$ by setting $$ \mathcal{D}^{\leq0}=\{K\in \mathcal{D}\,|\, j^*K\in\mathcal{D}_U^{\leq 0} \text{ and } i^*K\in \mathcal{D}^{\leq0}_F\} $$ $$ \mathcal{D}^{\geq0}=\{K\in \mathcal{D}\,|\, j^*K\in\mathcal{D}_U^{\geq 0} \text{ and } i^!K\in \mathcal{D}^{\geq0}_F\} $$ It is not hard to check that this is indeed a $t$-structure on $\mathcal{D}$. One may wonder what happens by switching the role of the left and right adjoint of $i_*$ in the above definition, i.e., if one tries to define a $t$-structure by setting $$ \tilde{\mathcal{D}}^{\leq0}=\{K\in \mathcal{D}\,|\, j^*K\in\mathcal{D}_U^{\leq 0} \text{ and } i^!K\in \mathcal{D}^{\leq0}_F\} $$ $$ \tilde{\mathcal{D}}^{\geq0}=\{K\in \mathcal{D}\,|\, j^*K\in\mathcal{D}_U^{\geq 0} \text{ and } i^*K\in \mathcal{D}^{\geq0}_F\} $$ The argument that shows that $(\mathcal{D}^{\leq0},\mathcal{D}^{\geq 0})$ is a $t$-structure breaks down if one tries to adapt it to $(\tilde{\mathcal{D}}^{\leq0},\tilde{\mathcal{D}}^{\geq 0})$, precisely since the adjoints are not in the right place. However this is not a proof that $(\tilde{\mathcal{D}}^{\leq0},\tilde{\mathcal{D}}^{\geq 0})$ is not a $t$-structure, but only that it is not obviously so. And indeed, if one tries to set up a recollement situation in the much more symmetric setting of stable $\infty$-categories (where all constructions show a very nice symmetric behaviour due to the fact that every pullback is a pushout and vice versa), then it seems that not only also $(\tilde{\mathcal{D}}^{\leq0},\tilde{\mathcal{D}}^{\geq 0})$ is a $t$-structure, but that this actually coincides with $(\mathcal{D}^{\leq0},\mathcal{D}^{\geq 0})$ (this however does not imply that $i^*=i^!$). Or at least, this is what I believe we have shown with Fosco Loregian in http://arxiv.org/abs/1507.03913 Assuming our result is correct (which may well not be the case), the natural questions are:
is this actually true also for triangulated categories? (I have not been able to locate a statement like this in the literature)
it is this maybe false for a general triangulated category but true for a triangulated category which is the homotopy category of a stable $\infty$-category?
is this always manifestly false for triangulated categories? (something that would make me suspect of some mistake in my argument for stable $\infty$-categories, or a hint of the fact that giving a recollement situation in the stable setting is such a strong requirement that there are actually no natural examples of stable recollements)
