Suppose that we have:


1) triangulated categories $C,D$, each equipped with a $t$-structure.
2) triangulated functor $F: C \to D$ which is $t$-exact.
3) $F$ reflects isomorphisms, i.e. is conservative.


Point 3) is equivalent to $F$ reflecting zero objects.


It seems that the restriction of $F$ to the hearts is faithful, as a conservative exact functor between abelian categories.


Is there a nice example when $F$ itself is not faithful?



A conservative, non faithful functor between triangulated categories
Here is also a simple example where the restriction of $F$ to the heart is faithful but $F$ itself is not. Let $Vec$ be the abelian category of complex vector spaces and let $Rep(\mathbb{Z})$ be the abelian category of complex representations of the free cyclic group $(\mathbb{Z},+)$. The forgetful functor $U: Rep(\mathbb{Z}) \longrightarrow Vec$ is faithful, conservative and exact. Let $C$ be the derived category of $Rep(\mathbb{Z})$ and $D$ be the derived category of $Vec$, both equipped with their natural $t$-structures. Then $U$ induces a triangulated, $t$-exact, conservative functor
$$ F: C \longrightarrow D $$
whose restriction to the corresponding hearts coincides with $U$. However, $F$ is not faithful. To see this, let $T$ be the $1$-dimensional representation of $\mathbb{Z}$ with trivial action and let $V$ be the $2$-dimensional representation of $\mathbb{Z}$ where the generator of $\mathbb{Z}$ acts via the matrix $\left(\begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix}\right)$. There is a non-split short exact sequence in $Rep(\mathbb{Z})$ of the form
$$ 0 \longrightarrow T \longrightarrow V \longrightarrow T \longrightarrow 0 $$
classified by a non-trivial map $T \longrightarrow T\left<1\right>$ in $C$. But the corresponding map $F(T) \longrightarrow F(T\left<1\right>) = F(T)\left<1\right>$ is trivial because $Ext^1\_{Vec}(\mathbb{C},\mathbb{C}) = 0$.


