  We define the affinization of an arbitrary monoidal category $\mathcal{C}$,
corresponding to the category of $\mathcal{C}$-diagrams on the cylinder. We
also give an alternative characterization in terms of adjoining dot generators
to $\mathcal{C}$. The affinization formalizes and unifies many constructions
appearing in the literature. In particular, we describe a large number of
examples coming from Hecke-type algebras, braids, tangles, and knot invariants.
When $\mathcal{C}$ is rigid, its affinization is isomorphic to its horizontal
trace, although the two definitions look quite different. In general, the
affinization and the horizontal trace are not isomorphic.
