In mathematics, if is a subset of then the inclusion map is the function that sends each element of to treated as an element of
An inclusion map may also referred to as an inclusion function, an insertion,[1] or a canonical injection.
A "hooked arrow" (U+21AA ↪ RIGHTWARDS ARROW WITH HOOK)[2] is sometimes used in place of the function arrow above to denote an inclusion map; thus:
(However, some authors use this hooked arrow for any embedding.)
This and other analogous injective functions[3] from substructures are sometimes called natural injections.
Given any morphism between objects and , if there is an inclusion map into the domain , then one can form the restriction of In many instances, one can also construct a canonical inclusion into the codomain known as the range of
Note that "insertion" is a function S → U and "inclusion" a relation S ⊂ U; every inclusion relation gives rise to an insertion function.