You said that you'd often like to talk about (small, or maybe infinitesimal) variations of a smooth map. There is a well known (in some circles at least) topology on the space of smooth maps. It's called the Whitney $C^{\infty}$-topology. 

First you define the Whitney $C^k$-topology using the natural projection $\pi : C^{\infty}(A,B) \twoheadrightarrow J^k(A,B),$ where the base space is the k-jet space; which is a finite dimensional, real vector space. As a basis for the $C^k$-topology on $C^{\infty}(A,B)$ we take the preimages of the open sets in $J^k(A,B).$

If $W^k$ denotes the set of open sets of $C^{\infty}(A,B)$ under the $C^k$-topology, then the Whitney $C^{\infty}$-topology on $C^{\infty}(A,B)$ is defined to be the topology whose basis is $W$, where

$W := \bigcup_{k=0}^{\infty} W^k.$

The Whitney $C^{\infty}$-topology makes $C^{\infty}(A,B)$ into a Baire space.

See pages 42 - 50 of Golubitsky & Guillemin, "Stable Mappings and their Singularities", (1974).