A category of local models is a category whose objects play the role of particularly well-controled test spaces in the sense of space and quantity. The major notions of spaces, such as topological spaces, algebraic spaces, smooth manifolds, are spaces modeled on a category of local models in the sense of structured generalized spaces.
A category of local models is
a small site ;
a morphism of sites Top;
a set of diagrams in
an object of
The objects of are usually called affine spaces. In particular the object is the affine line.
is the category of local models for algebraic spaces; here ;
section 1.1 of