  The central object of synthetic differential geometry is microlinear spaces.
In our previous paper [Microlinearity in Frolicher spaces -beyond the regnant
philosophy of manifolds-, International Journal of Pure and Applied
Mathematics, 60 (2010), 15-24] we have emancipated microlinearity from within
well-adapted models to Frolicher spaces. Therein we have shown that Frolicher
spaces which are microlinear as well as Weil exponentiable form a cartesian
closed category. To make sure that such Frolicher spaces are the central object
of infinite-dimensional differential geometry, we develop the theory of vector
fields on them in this paper. The central result is that all vector fields on
such a Frolicher space form a Lie algebra.
