nLab
injection

Definition

A function $f$ from $A$ to $B$ is injective if $x = y$ whenever $f(x) = f(y)$. 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 $f(x) \ne f(y)$ whenever $x \ne y$ (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 $a$ in a set $A$ in the category of sets is just a global element $a:1\rightarrow A$, one could define injections in any category $\mathcal{C}$ with a terminal object $1$:

Definition

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

Remark

The term injective morphism is already used in category theory in a different context to mean a morphism with a right lifting property.

Proposition

In a category $\mathcal{C}$ with a terminal object $1$, every monomorphism is an injection.

This follows from the definition of a monomorphism.

Proposition

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

Proof

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

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

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.

Related concepts

• surjection

• monomorphism

• embedding

• monomorphism in an (infinity,1)-category