  In our previous paper entitled "Axiomatic differential geometry -towards
model categories of differential geometry-, we have given a category-theoretic
framework of differential geometry. As the first part of our series of papers
concerned with differential-geometric developments within the above axiomatic
scheme, this paper is devoted to vector fields. The principal result is that
the totality of vector fields on a microlinear and Weil exponential object
forms a Lie algebra.
