  In this paper, we endow the family of closed oriented genus $g$ surfaces,
starting with torus, with a structure of a (co)cyclic object in the category of
$3$-dimensional cobordisms. As a corollary, any $3$-dimensional TQFT induces a
(co)cyclic module, which we compute algebraically for the Reshetikhin-Turaev
TQFT.
