nLab category of local models

Idea

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.

Definition

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

  • $R$ is closed under limits of shape in $L$;

  • $U$ is a basis for its image in that…

  • $A$ generates $R$ under $L$-limits and gluing (?).

The objects of $R$ are usually called affine spaces. In particular the object $A$ is the affine line.

Applications

• For every category of local models there is the corresponding notion of locally modeled monoids. See the examples below.

Examples

• $R = Rings^{op}$ 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 $A = \mathbb{R}$.

References

section 1.1 of

• David Spivak, Quasi-smooth derived manifolds (pdf)