Inclusion map

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, if 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a <a href="/facts/Subset/SJGERmbA">subset</a> of 
 
 
 
 B
 ,
 
 
 {\displaystyle B,}
 
 then the inclusion map is the <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> <a href="/facts/%CE%99/qwHge41z">
 
 
 
 ι
 
 
 {\displaystyle \iota }
 
</a> that sends each element 
 
 
 
 x
 
 
 {\displaystyle x}
 
 of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 to 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 treated as an element of 
 
 
 
 B
 :
 
 
 {\displaystyle B:}

ι
        :
        A
        →
        B
        ,
        
        ι
        (
        x
        )
        =
        x
        .
      
    
    {\displaystyle \iota :A\rightarrow B,\qquad \iota (x)=x.}

An inclusion map may also be referred to as an inclusion function, an insertion, or a canonical injection.
A "hooked arrow" (<a href="/facts/Unicode/DDu3MfPV">U+</a>21AA ↪ RIGHTWARDS ARROW WITH HOOK) is sometimes used in place of the function arrow above to denote an inclusion map; thus:

ι
        :
        A
        ↪
        B
        .
      
    
    {\displaystyle \iota :A\hookrightarrow B.}

(However, some authors use this hooked arrow for any <a href="/facts/Embedding/HKeVKc42">embedding</a>.)
This and other analogous <a href="/facts/Injective/5ES6tqf5">injective</a> functions from <a href="/facts/Substructure_(mathematics)/HqHnHKo1">substructures</a> are sometimes called natural injections.
Given any <a href="/facts/Morphism/Kdpuyz3S">morphism</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 between <a href="/facts/Object_(category_theory)/7WzxUThf">objects</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, if there is an inclusion map 
 
 
 
 ι
 :
 A
 →
 X
 
 
 {\displaystyle \iota :A\to X}
 
 into the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
, then one can form the <a href="/facts/Restriction_(mathematics)/JV59DhKy">restriction</a> 
 
 
 
 f
 ∘
 ι
 
 
 {\displaystyle f\circ \iota }
 
 of 
 
 
 
 f
 .
 
 
 {\displaystyle f.}
 
 In many instances, one can also construct a canonical inclusion into the <a href="/facts/Codomain/MGc43nwQ">codomain</a> 
 
 
 
 R
 →
 Y
 
 
 {\displaystyle R\to Y}
 
 known as the <a href="/facts/Range_of_a_function/66CeTV6u">range</a> of 
 
 
 
 f
 .
 
 
 {\displaystyle f.}

Inclusion map open-in-new

Inclusion map