George Ciprian Modoi
Babeş-Bolyai University, Cluj-Napoca, Romania

Consider two triangulated categories $\mathcal{D}_1$ and $\mathcal{D}_2$ and a triangle functor $\varphi_*:\mathcal{D}_2\to\mathcal{D}_1$ having both left adjoint $\varphi^*$ and a right adjoint $\varphi^!$. We say that a property of an object in $\mathcal{D}_1$ a (co)ascends, respectively (co)descends via these functors provided that it is preserved, respectively reflected by $\varphi^*$ ($\varphi^!$). We present several results describing how the property of being (co)silting (co)ascends and/or (co)descends. The motivating example is the study of the conditions in which (co)silting property (co)ascends/descends via the induction, restriction and coinduction functors induced by a morphism of commutative algebras $\varphi:A\to B$. We generalize this example by imposing condition that the categories $\mathcal{D}_1$ and $\mathcal{D}_2$ have an internal tensor (that is, they are actually tensor triangulated categories), such that the unit object for the tensor functor is silting in both cases. We want to advertise this setting, by explaining how natural this supplementary condition is.