Smooth Homotopy of Infinite-Dimensional C∞-Manifolds

In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional C infinity-manifolds in convenient calculus. More precisely, we discuss the smoothing of maps, sections, principal bundles, and gauge transformations.We first introduce the notion of hereditary C infinity-paracompactness along with the semiclassicality condition on a C infinity-manifold, which enables us to use local convexity in local arguments. Then, we prove that for C infinity-manifolds M and N, the smooth singular complex of C infinity(M,N) is weakly equivalent to the ordinary singular complex of C0(M,N) under the hereditary C infinity-paracompactness and semiclassicality conditions on M. We next generalize this result to sections of fiber bundles over a C infinity-manifold M under the same conditions on M. Further, we establish the Dwyer-Kan equivalence between the simplicial groupoid of smooth principal G-bundles over M and that of continuous principal G-bundles over M for a Lie group G and a C infinity-manifold M under the same conditions on M, encoding the smoothing results for principal bundles and gauge transformations.For the proofs, we fully faithfully embed the category C infinity of C infinity-manifolds into the category D of diffeological spaces and develop the smooth homotopy theory of diffeological spaces via a homotopical algebraic study of the model category D and the model category C0 of arc-generated spaces. Then, the hereditary C infinity-paracompactness and semiclassicality conditions on M imply that M has the smooth homotopy type of a cofibrant object in D. This result can be regarded as a smooth refinement of the results of Milnor, Palais, and Heisey on the homotopy type of infinite-dimensional topological manifolds