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 $R$;
a morphism of sites $U : R \to$ Top;
a set $L$ of diagrams $I_L \to R$ in $R$
an object $A$ of $R$
such that
The objects of $R$ are usually called affine spaces. In particular the object $A$ is the affine line.
$R = Rings^{op}$ is the category of local models for algebraic spaces; here $A = \mathbb{Z}[x]$;
$R =$ CartSp is the category of local models for smooth manifolds and generalized smooth algebras; here $A = \mathbb{R}$.
section 1.1 of