Euc\mathbf{Euc}Eucthe category of euclidean spaces and smooth maps between themobjects: R\mathbb{R}R for all n∈Nn \in \mathbb{N}n∈Nmorphisms: f:Rn→Rmf: \mathbb{R}^{n} \to \mathbb{R}^{m}f:Rn→Rm, smooth mapIt’s a cartesian category with Rn×Rm=Rn+m\mathbb{R}^{n} \times \mathbb{R}^{m} = \mathbb{R}^{n + m}Rn×Rm=Rn+m and 1=R01 = \mathbb{R}^{0}1=R0