Forming the total space of a principal U(1)-bundle can be seen as a functor from topological spaces over the classifying space BU(1) to topological spaces. This functor has an adjoint given by the cyclification; namely taking the (homotopy) quotient of the free loop space by the loop rotation action. This implies that if two spaces have isomorphic cyclifications, then there is an isomorphism between cocycles over the total spaces of the corresponding principal U(1)-bundles. Moreover, if the isomorphism between the cyclifications happens to cover the topological T-duality automorphism of the classifying space of the so-called topological T-duality 2-group, then the isomorphism between cocycles is given by a push-tensor-pull formula and so it is a topological T-duality isomorphism. While at the global geometric level this may sound mysterious, at the infinitesimal/rational homotopy theory level it reduces to some very simple (differential graded) polynomial algebra computation. Remarkably, someone with an expertise in string theory (i.e., not me) can read in this computation the topological T-duality between K^0-cocycles in type IIA string theory and K^1-cocycles in type IIB string theory as given by the Buscher rules for Ramond-Ramond fields and Hori's formula. Joint work with Hisham Sati and Urs Schreiber; arXiv:1611.06536.