nLab injection

Definition

A function $f$ from $A$ to $B$ is injective if $x = y$ whenever $f(x) = f(y)$. Equivalently, a function is injective if all its fibers are subsingletons: for all elements $b \in B$ and for all elements $x \in A$ and $y \in A$, if $f(x) = b$ and $f(y) = b$, then $x = y$. 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 $x:1\rightarrow\emptyset$.

Related concepts

• surjection

• monomorphism

• embedding

• injective-on-objects functor

• monomorphism in an (infinity,1)-category

• monotone function

• equivariant function

• decidable injection