Homological algebra for diffeological vector spaces

Abstract

Diffeological spaces are natural generalizations of smooth manifolds, introduced by J.M. Souriau and his mathematical group in the 1980's. Diffeological vector spaces (especially fine diffeological vector spaces) were first used by P. Iglesias-Zemmour to model some infinite dimensional spaces in [I1, I2]. K. Costello and O. Gwilliam developed homological algebra for differentiable diffeological vector spaces in Appendix A of their book [CG]. In this paper, we present homological algebra of general diffeological vector spaces via the projective objects with respect to all linear subductions, together with some applications in analysis.

Diffeological vector pseudo-bundles

“…Applying now the lattice property of the family of all diffeologies on a given set (this is mentioned in Section 1, see [6], Section 1.25 for the detailed treatment), we conclude that the diffeology D * is well-defined. In addition, while an a priori question could be, whether the subset diffeology relative to D * on each fibre is indeed its functional diffeology (which is the standard diffeology of a diffeological dual, see [13], [14]), 37 part (3) of the proof of Proposition 5.3.2 in [13] asserts that the answer is positive (and so the fibres are indeed diffeologial duals).…”

“…In particular, every fibre has coarse diffeology. 14 The second example we provide, stems from the fact that the condition of the local triviality is absent from the definition of a diffeological vector space over a given X; it shows that, in addition to a large diffeology on the fibres, the definition as given allows for topologically complicated (in the sense of the usual topology) total spaces. Example 1.6.…”

“…The tensor product The definition of the diffeological tensor product was given first in [13] and then in [14] (see Section 3); we recall the latter version. Let V 1 , ..., V n be diffeological vector spaces, let…”

“…Consider an arbitrary point y ∈ Y ; the restriction off to its preimage is a smooth linear map π −1 1 (y) → π −1 2 (f (y)) between two diffeological vector spaces. Then (see [14]; some details can also be found in [8]) there is a natural (smooth and linear) dual mapf * : (π −1 2 (f (y))) * → (π −1 1 (y)) * , which is defined in the usual way (that is, by the rulef * (v * 2 )(w 1 ) = v * 2 (f (w 1 ))). However, since it goes in the opposite (with respect tof ) direction, it obviously cannot be a lift of the existing map f : Y → X 2 .…”

“…An example of a fine vector space is R n with the standard diffeology, i.e., one that consists of all the usual smooth maps with values in R n . 12 The dual of a diffeological vector space The definition of the diffeological dual was first given in [13] and then in [14]; this concept is a very natural (and obvious) one: Definition 1.3. Let V be a diffeological vector space.…”