By Patrick Iglesias-zemmour

**Diffeology** is the 1st textbook at the topic. it truly is aimed to graduate scholars and researchers who paintings in differential geometry or in mathematical physics

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**Example text**

Let X and X be two diﬀeological spaces, whose diﬀeologies are denoted by D and D . The notion of pushforward of diﬀeologies gives a new interpretation of the notion of diﬀerentiability: a map SUBDUCTIONS 25 f : X → X is smooth if and only if f∗ (D) ⊂ D . In other words, C∞ (X, X ) = {f ∈ Maps(X, X ) | f∗ (D) ⊂ D }. Proof. Let us assume ﬁrst that f is smooth. Said diﬀerently, D is a diﬀeology of X such that f is smooth. But f∗ (D) is the intersection of all the diﬀeologies of X for which f is smooth (art.

Dxk dxk ˆ : f → f(x), from C∞ (R) to R, is 2) Show that, for every real number x, the map x smooth. Deduce that, for every real number x, for every integer k, the following map, called the k-jet or the jet of order k, is smooth: Dkx : C∞ (R) → Rk+1 deﬁned by Dkx (f) = (f(x), f (x), . . , f(k) (x)). 3) Show that, for any pair of real numbers a and b, the following map Ia,b is smooth, where the sign denotes the Riemann integral: Ia,b : C∞ (R) → R b with Ia,b (f) = f(t) dt . a 4) Let C∞ 0 (R) be the space of smooth real maps f such that f(0) = 0.

3) Generalize this construction to C∞ (Rn , Rn ). What can you say about the injection of GL(n, R) into C∞ (Rn )? SUMS OF DIFFEOLOGICAL SPACES 21 Sums of Diﬀeological Spaces The category {Diﬀeology} is closed for coproducts (also called sums or disjoint unions) of diﬀeological spaces. There exists a distinguished diﬀeology, called the sum diﬀeology, on the sum of any family of diﬀeological spaces. Reminder. Let us recall the formal construction of the sum (also called coproduct) of any family of sets [Bou72].