The property of continuity (and hence smoothness) seems weaker than the properties of other morphisms, in the sense that a homeomorphism is a "continuous bijection whose inverse is continuous". In every other morphism type, the marking quality of the morphism is guaranteed for the inverse.
An isomorphism of vector spaces is "a bijective linear map", I don't need to verify that the inverse is linear.
An isomorphism of groups is "a bijective map that preserves group structure", I don't need to verify that the inverse preserves group structure.
An isomorphism of rings is a "bijective map that preserves ring structure", I don't need to verify that the inverse preserves ring structure.
There seems to be a trend that the bijective morphisms of "algebraic" categories seem to be guaranteed an inverse which is also a morphism, while in "topological" categories, that's not the case.
Is there an interesting explanation for this?
Thank you
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