Editing Isomorphism (section)

===Isomorphism vs. bijective morphism===
In a [[concrete category]] (roughly, a category whose objects are sets (perhaps with extra structure) and whose morphisms are structure-preserving functions), such as the [[category of topological spaces]] or categories of algebraic objects (like the [[category of groups]], the [[category of rings]], and the [[category of modules]]), an isomorphism must be bijective on the [[underlying set]]s. In algebraic categories (specifically, categories of [[variety (universal algebra)|varieties in the sense of universal algebra]]), an isomorphism is the same as a homomorphism which is bijective on underlying sets. However, there are concrete categories in which bijective morphisms are not necessarily isomorphisms (such as the category of topological spaces).