Essentially surjective functor

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically in <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a <a href="/facts/Functor/l2u3rIBE">functor</a>

F
        :
        C
        →
        D
      
    
    {\displaystyle F:C\to D}

is essentially surjective if each object 
 
 
 
 d
 
 
 {\displaystyle d}
 
 of 
 
 
 
 D
 
 
 {\displaystyle D}
 
 is isomorphic to an object of the form 
 
 
 
 F
 c
 
 
 {\displaystyle Fc}
 
 for some object 
 
 
 
 c
 
 
 {\displaystyle c}
 
 of 
 
 
 
 C
 
 
 {\displaystyle C}
 
.
Any functor that is part of an <a href="/facts/Equivalence_of_categories/k6uzqAcy">equivalence of categories</a> is essentially surjective. As a partial converse, any <a href="/facts/Full_and_faithful_functor/bMM75v7e">full and faithful functor</a> that is essentially surjective is part of an equivalence of categories.

Essentially surjective functor open-in-new

Essentially surjective functor