Smooth classifying spaces

Sheaves, principal bundles, and Čech cohomology for diffeological spaces

“…As an immediate corollary, we combine the above theorem with the main results of [10,4]. A diffeological space X is smoothly paracompact if any open cover of X in the D-topology admits a smooth partition of unity subordinate to it.…”

“…fiber products, coproducts, quotients, etc.) to more specific adaptations of higher level concepts (including de Rham forms and their cohomology, principal/fiber bundles, and classifying spaces; see [4] and [10] for two independent developments of classifying spaces). The purpose of this note is to build a theory of sheaves over diffeological spaces and their Čech cohomology.…”

“…Given a smoothly paracompact diffeological space X and an abelian diffeological group G, there is a natural bijection from isomorphism classes of principal G-bundles on X to smooth homotopy classes of smooth maps from X to the classifying space BG of G; denote the set of homotopy classes by [X, BG]. (Note: the Hausdorff and second-countable requirements appearing in [10] are unnecessary; see [4,Lemmas 4.2,4.3].) See the cited papers for definitions and full details.…”

Smooth singular complexes and diffeological principal bundles

“…(2) Christensen and Wu constructed a nontrivial locally trivial principal R >0 -bundle : −→ with ≃ D * (see [6,Example 3.12]). By [6, Theorem 5.10], the locally trivial principal R >0 -bundle is not D-numerable.…”

Pushforward and smooth vector pseudo-bundles

“…On the other hand, the theory of diffeological vector spaces and their homological algebra is intimately related to analysis and geometry; see [W,CW16,CWa]. The projective objects there deserve special attention.…”