Smooth Homotopy of Infinite-Dimensional $C^{\infty }$-Manifolds

In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional C$C^{\infty }$-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$C^{\infty }$-paracompactness along with the semiclassicality condition on a C$C^{\infty }$-manifold, which enables us to use local convexity in local arguments. Then, we prove that for C$C^{\infty }$-manifolds M and N, the smooth singular complex of the diffeological space C$C^{\infty }$(M,N) is weakly equivalent to the ordinary singular complex of the topological space C0(M,N) under the hereditary C$C^{\infty }$-paracompactness and semiclassicality conditions on M. We next generalize this result to sections of fiber bundles over a C$C^{\infty }$-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$C^{\infty }$-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$C^{\infty }$ of C$C^{\infty }$-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, also known as -generated spaces. Then, the hereditary C$C^{\infty }$-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, which give sufficient conditions under which an infinite-dimensional topological manifold has the homotopy type of a CW-complex. We also show that most of the important C$C^{\infty }$-manifolds introduced and studied by Kriegl, Michor, and their coauthors are hereditarily C$C^{\infty }$-paracompact and semiclassical, and hence, results can be applied to them.