nLab
CartSp

Definition

By $\mathrm{CartSp}$ we denote the category

• whose objects are the cartesian spaces ${ℝ}^n$, $n\in \mathbb{N}$, equipped with their standard smooth structure;

• whose morphisms are all smooth (infinitely differentiable) maps between these spaces.

So $\mathrm{CartSp}$ is the full subcategory of Diff consisting of the spaces for ${ℝ}^n$ for $n\in \mathbb{N}$.

Remarks

• Let ${\mathrm{CartSp}}_{\mathrm{pr}}\hookrightarrow \mathrm{CartSp}$ be the subcategory of $\mathrm{CartSp}$ whose morphisms are restricted to be the standard injections and projections on $\mathrm{CartSp}$. If we write FinSet for the skeletal version of the category of finite sets, with objects identified with the natural numbers, then ${\mathrm{CartSp}}_{\mathrm{pr}}\simeq \mathrm{FinSet}$.

• $\mathrm{CartSp}$ appear as test objects in the context of generalized smooth spaces and generalized smooth algebras.

• One can define $\mathrm{Cart}\mathrm{Sp}$ with other choices of morphism; see cartesian space for some idea.

category: category