Assertion: isomorphism | /r/HasProperty | bijective
Facets:
value: on the underlying sets
type: LOCATION
Source sentences:
text: Although it can be true that an isomorphism is a bijective (homo)morphism, this does not hold in general.
text: In a concrete category (that is, 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 groups, rings, and modules, an isomorphism must be bijective on the underlying sets.
text: An isomorphism is a map (which is bijective and has various nice properties but we'll leave those aside for a minute) which basically keeps everything the same.
text: In a concrete category (that is, roughly speaking, a category whose objects are sets and morphisms are mappings between sets), such as the category of topological spaces or categories of algebraic objects like groups, rings, and modules, an isomorphism must be bijective on the underlying sets.