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
D
{\mathcal {D}}
of diffeological spaces and develop the smooth homotopy theory of diffeological spaces via a homotopical algebraic study of the model category
D
{\mathcal {D}}
and the model category
C
0
{\mathcal {C}^0}
of arc-generated spaces, also known as
Δ
\Delta
-generated spaces. Then, the hereditary
C
∞
C^\infty
-paracompactness and semiclassicality conditions on
M
M
imply that
M
M
has the smooth homotopy type of a cofibrant object in
D
{\mathcal {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
C
W
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.