A function from to is injective if whenever . Equivalently, a function is injective if all its fibers are subsingletons: for all elements and for all elements and , if and , then . An injective function is also called one-to-one or an injection; it is the same as a monomorphism in the category of sets.
A bijection is a function that is both injective and surjective.
In constructive mathematics, a strongly extensional function between sets equipped with tight apartness relations is called strongly injective if whenever (which implies that the function is injective). This is the same as a regular monomorphism in the category of such sets and strongly extensional functions (while any merely injective function, if strongly extensional, is still a monomorphism). Some authors use ‘one-to-one’ for an injective function as defined above and reserve ‘injective’ for the stronger notion.
Since an element in a set in the category of sets is just a global element , one could define injections in any category with a terminal object :
A morphism in is an injection or a one-to-one morphism if, given any two global elements , if .
The term injective morphism is already used in category theory in a different context to mean a morphism with a right lifting property.
In a category with a terminal object , every monomorphism is an injection.
This follows from the definition of a monomorphism.
In a category with a terminal object , every global element is an injection.
By definition of terminal object , the unique global element is the identity morphism of the terminal object. Thus for every global element , for any two global elements , is always true, making an injection.
If the category has a strict initial object , then every morphism is vacuously an injection, since there are no global elements .
Last revised on December 9, 2023 at 07:29:34. See the history of this page for a list of all contributions to it.