= Category of topological manifolds
The <category> of topological manifolds has <topological manifolds> without boundary as its <objects of a category> and all <continuous maps> between them as its <categorical morphisms>. Allowing every <dimension of a manifold> lets maps connect spaces of different dimensions. <Identity functions> are <continuous>, and <function composition> preserves <continuity>, so the <category> laws are inherited from <functions>. Its <isomorphisms> are exactly <homeomorphisms>.
In <coordinate charts> $(U,\varphi)$ on $M$ and $(V,\psi)$ on $N$, a <continuous map> $f:M\to N$ is represented locally by $\psi\circ f\circ\varphi^{-1}$ on $\varphi(U\cap f^{-1}(V))$. Changing <coordinate charts> changes this expression through <topological transition maps>, while $f$ itself stays fixed. This viewpoint separates the underlying <continuous map> from the coordinates used to describe it; restricting to <smooth manifolds> and <smooth maps> gives the corresponding smooth <category>.
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