category Euc

\(\mathbf{Euc}\)

• the category of euclidean spaces and smooth maps between them
• objects: \(\mathbb{R}\) for all \(n \in \mathbb{N}\)
• morphisms: \(f: \mathbb{R}^{n} \to \mathbb{R}^{m}\), smooth map
• It's a cartesian category with \(\mathbb{R}^{n} \times \mathbb{R}^{m} = \mathbb{R}^{n + m}\) and \(1 = \mathbb{R}^{0}\)