nLab
category of local models

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 RR;

  • a morphism of sites U:RU : R \to Top;

  • a set LL of diagrams I LRI_L \to R in RR

  • an object AA of RR

  • such that

    • RR is closed under limits of shape in LL;

    • UU is a basis for its image in that

    • AA generates RR under LL-limits and gluing (?).

The objects of RR are usually called affine spaces. In particular the object AA 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 opR = Rings^{op} is the category of local models for algebraic spaces; here A=[x]A = \mathbb{Z}[x];

  • R=R = CartSp is the category of local models for smooth manifolds and generalized smooth algebras; here A=A = \mathbb{R}.

References

section 1.1 of

  • David Spivak, Quasi-smooth derived manifolds (pdf)
Revised on August 15, 2010 20:09:59 by Toby Bartels (75.88.65.54)