nLab injection set



Given sets AA and BB, the injection set Inj(A,B)\mathrm{Inj}(A, B) is the set of injection between the set AA and BB. 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.


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

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

Last revised on January 13, 2023 at 05:50:24. See the history of this page for a list of all contributions to it.