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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., Comment: 48 pages, many figures. Minor changes done. Comments are still welcome!

In this paper, we first endow the set of ribbon string links (up to isotopy) with a structure of a cyclic and of a cocyclic set. Next, we relate these (co)cyclic sets with those associated with the coend of a ribbon category. The relationship is given by the universal quantum invariants \`a la Reshetikhin-Turaev., Comment: 23 pages, many figures

In $1998$, Connes and Moscovici defined the cyclic cohomology of Hopf algebras. In $2010$, Khalkhali and Pourkia proposed a braided generalization: to any Hopf algebra $H$ in a braided category $\mathcal B$, they associate a paracocyclic object in $\mathcal B$. In this paper we explicitly compute the powers of the paracocyclic operator of this paracocyclic object. Also, we introduce twisted modular pairs in involution for $H$ and derive (co)cyclic modules from them. Finally, we relate the paracocyclic object associated with $H$ to that associated with an $H$-module coalgebra via a categorical version of the Connes-Moscovici trace., Comment: 42 pages, many figures