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

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.
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 Δ-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.

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