[[Differential geometry MOC]]
# Category of manifolds

Let $\alpha \in \mathbb{N}_{0}$, $\alpha = \infty$, or $\alpha=\omega$.
The **category of manifolds** $\Man^\alpha$ is a [[category]] where
an object is a $C^\alpha$-[[differentiable manifold|manifold]]
and a morphism is a $C^\alpha$-[[Differentiability|map]]. #m/def/geo/diff 
Often one also sees the terms **smooth category** for $\Man^\infty$
and **analytic category** for $\Man^\omega$.

## See also

- [[Holomorphic category]]

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