nLab injection set

Definition

Given sets $A$ and $B$, the injection set $\mathrm{Inj}(A, B)$ is the set of injections between the set $A$ and $B$. In the foundations of mathematics, the existence of such a set may be taken to follow from the existence of power sets, from the axiom of subset collection, from the existence of function sets, or as an axiom (the axiom of injection sets) in its own right.

Properties

If the set theory has function sets, then the injection set between $A$ and $B$ is a subset of the function set between $A$ and $B$: $\mathrm{Inj}(A, B) \subseteq B^A$.

The universal property of the bijection set states that given a function $X \to \mathrm{Inj}(A, B)$ from a set $X$ to the injection set between sets $A$ and $B$, there exists a injection $A \times X \hookrightarrow B \times X$ in the slice category $\mathrm{Set}/X$.

Related concepts

• embedding type

• object of monomorphisms

• function set, bijection set