nLab
locally modeled monoid

Idea

A locally modeled monoid or, $R$-Ring, is a generalized quantity in the sense of space and quantity which is modeled on a category of local models $R$.

Definition

Let $(R,U,L,A)$ be a category of local models. Then an $R$-ring or monoid locally modeled on $R$ is a co-presheaf

$$R\to \mathrm{Set}$$R \to Set

which preserves the limits of shape in $L$.

A morphism of such locally modeled monoids is a natural transformation.

Examples

• for $R={\mathrm{Ring}}^{\mathrm{op}}$, $R$-rings are just ordinary rings.
• for $R=$ CartSp, $R$-rings are generalized smooth algebras.

References

This is definition 1.1.6 of

• David Spivak, Quasi-smooth derived manifolds, PhD thesis, Berkeley (2007) (pdf)

where it appears as part of the discussion of derived smooth manifolds.