A function ff from AA to BB is injective if x=yx = y whenever f(x)=f(y)f(x) = f(y). 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 f(x)f(y)f(x) \ne f(y) whenever xyx \ne y (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.

In other categories

Since an element aa in a set AA in the category of sets is just a global element a:1Aa:1\rightarrow A, one could define injections in any category 𝒞\mathcal{C} with a terminal object 11:


A morphism f:ABf:A\rightarrow B in 𝒞\mathcal{C} is an injection or a one-to-one morphism if, given any two global elements x,y:1Ax, y:1\rightarrow A, x=yx = y if fx=fyf \circ x = f \circ y.


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 𝒞\mathcal{C} with a terminal object 11, every monomorphism is an injection.

This follows from the definition of a monomorphism.


In a category 𝒞\mathcal{C} with a terminal object 11, every global element e:1Ae:1\rightarrow A is an injection.


By definition of terminal object 11, the unique global element i:11i:1\rightarrow 1 is the identity morphism of the terminal object. Thus for every global element e:1Ae:1\rightarrow A, for any two global elements x,y:11x, y:1\rightarrow 1, x=yx = y is always true, making e:1Ae:1\rightarrow A an injection.

If the category has a strict initial object \emptyset, then every morphism f:Bf:\emptyset\rightarrow B is vacuously an injection, since there are no global elements x:1x:1\rightarrow\emptyset.

Anonymous: Under what conditions are all injections in a category monomorphisms? Obviously injections are monomorphisms in a well-pointed? topos or pretopos (those are models of particular types of set theories), but does that remain true in a (pre)topos without well-pointedness, a coherent category or an exact category?

Anonymous: There is this stackexchange post, but the answers only refer to concrete categories with a forgetful functor to Set and a free functor from Set, rather than arbitrary abstract categories.

Last revised on April 16, 2021 at 08:05:31. See the history of this page for a list of all contributions to it.