Smooth Homotopy of Infinite-Dimensional 𝐶^{∞}-Manifolds

Abstract

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 M and N N , the smooth singular complex of the diffeological space C ∞ ( M , N ) C^\infty (M,N) is weakly equivalent to the ordinary singular complex of the topological space C 0 ( M , N ) {\mathcal {C}^0}(M,N) under the hereditary C ∞ C^\infty -paracompactness and semiclassicality conditions on M M . We next generalize this result to sections of fiber bundles over a C ∞ C^\infty -manifold M M under the same conditions on M M . Further, we establish the Dwyer-Kan equivalence between the simplicial groupoid of smooth principal G G -bundles over M M and that of continuous principal G G -bundles over M M for a Lie group G G and a C ∞ C^\infty -manifold M M under the same conditions on M 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