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ABSTRACT. Manifold approximate fibrations arise in the geometric topology of manifolds and group actions on topological manifolds. The primary purpose of this paper is to classify manifold approximate fibrations in terms of the lifting problem for a certain bundle. Our classification meshes well with the classical classifications of fibrations and bundles and, hence, we are able to attack questions such as the following. When is a fibration controlled homotopy equivalent to a manifold approximate fibration? When is a manifold approximate fibration controlled homeomorphic to a bundle?Let Bi be a topological manifold. Recall that a manifold approximate fibration over B is a proper map q: M -+ B such that M is a manifold (topological or Hilbert cube) and such that q satisfies an approximate lifting condition (see [8] or § I.D). This "bundle" theory plays an important role in the study of topological manifolds. Consider the following examples. Let Bi be a topological manifold. Recall that a manifold approximate fibration over B is a proper map q: M -+ B such that M is a manifold (topological or Hilbert cube) and such that q satisfies an approximate lifting condition (see [8] or § I.D). This "bundle" theory plays an important role in the study of topological manifolds. Consider the following examples.

Therapies to prevent diabetes in particular the progressive loss of β-cell mass and function and/or to improve the dysregulated metabolism associated with diabetes are highly sought. The incretin-based therapy comprising GLP-1R agonists and DPP-4 inhibitors have represented a major focus of pharmaceutical R&D over the last decade. The incretin hormone GLP-1 has powerful antihyperglycemic effect through direct stimulation of insulin biosynthesis and secretion within the β-cells; it normalizes β-cell sensitivity to glucose, has an antiapoptotic role, stimulates β-cell proliferation and differentiation, and inhibits glucagon secretion. However, native GLP-1 therapy is inappropriate due to the rapid post-secretory inactivation by DPP-4. Therefore, incretin mimetics developed on the backbone of the GLP-1 or exendin-4 molecule have been developed to behave as GLP-1R agonists but to display improved stability and clinical efficacy. New formulations of incretins and their analogs based on micro- and nanomaterials (i.e., PEG, PLGA, chitosan, liposomes and silica) and innovative encapsulation strategies have emerged to achieve a better stability of the incretin, to improve its pharmacokinetic profile, to lower the administration frequency or to allow another administration route and to display fewer adverse effects. An important advantage of these formulations is that they can also be used at the targeted non-invasive imaging of the beta-cell mass. This review therefore focuses on the current state of these efforts as the next step in the therapeutic evolution of this class of antidiabetic drugs.

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