  We construct the t-motivic mixed Carlitz zeta category over $\F_q(t)$ and
show that it contains all the (mixed) t-motives with Carlitz-Thakur multi-zeta
values as periods constructed by Anderson and Thakur. Our construction is
canonical and our category is Tannakian and neutral and every object is
equipped with a weight filtration whose graded pieces are Carlitz motives over
$\F_q(t)$. For any finite separable extension L/\F_q(t) we show that existence
of a similar category over $L$ is a consequence of a version of a conjecture of
L. Taelman. Along the way we also prove the existence of the category of mixed
$t$-motives and the category of mixed Carlitz motives over any $L$ (these two
existence results are independent of any conjectures).
