Simplicial cochain algebras for diffeological spaces

Local systems in diffeology

“…For a simplicial set L, the cohomology algebra H * (A P L (L)) is isomorphic to the singular cohomology of L with coefficients in Q; see, for example, [10, Theorem 10.9]. Therefore, we see that for each diffeological space M , the cohomology [24,Theorem 2.4] and comments in the end of [24,Section 5]. It is possible to restrict R f to the category of simply-connected simplicial rational (real) spaces.…”

“…Remark 2.2. The functor S D ( ) here is different from the singular simplex functor S D ( ) ′ in [24]. In constructing the latter functor, we use the affine space A p instead of the standard simplex ∆ p mentioned above.…”

“…In constructing the latter functor, we use the affine space A p instead of the standard simplex ∆ p mentioned above. However, the singular homology for S D ( ) ′ is quasi-isomorphic to that for S D ( ); see the paragraph before [24,Theorem 5.4], the comments in the end of [24,Section 5] and also [21,Remark 3.9].…”

“…Thanks to the highly versatility, our results in this article are applicable to the polynomial de Rham complex (A * P L ) • and also the singular de Rham complex (A * DR ) • ; see Section 2 for the details. We observe that there is a quasi-isomorphism between (A * P L ) • (K) ⊗ R and (A * DR ) • (K) for each simplicial set K; see [24,Corollary 3.5]. In general, it seems to be difficult to construct explicitly a rational model and an R-local model for a given diffeological space; see Remark 2.11.…”

“…Therefore, adjunction spaces, mapping spaces and classifying spaces are defined naturally as diffeological spaces with smooth structures. Moreover, it is possible to make de Rham calculus [17,Chapters 6,7,8 and 9] expanding that on manifolds in Diff with the de Rham complex in the sense of Souriau [31] and the singular de Rham complex [24,25]. The calculus will be more accessible via algebraic models such as minimal models in rational homotopy theory.…”