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Hopfian object
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<p>In the branch of mathematics called <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a hopfian object is an object <i>A</i> such that any <a href="/facts/Epimorphism/IrzonI01">epimorphism</a> of <i>A</i> onto <i>A</i> is necessarily an <a href="/facts/Automorphism/WnyxNFkQ">automorphism</a>. The <a href="/facts/Duality_(mathematics)/8jGuyQIU">dual notion</a> is that of a cohopfian object, which is an object <i>B</i> such that every <a href="/facts/Monomorphism/Q9BrtDTs">monomorphism</a> from <i>B</i> into <i>B</i> is necessarily an automorphism. The two conditions have been studied in the categories of <a href="/facts/Group_(mathematics)/IQTYqINt">groups</a>, <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a>, <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a>, and <a href="/facts/Topological_spaces/v2ut7CbK">topological spaces</a>. </p><p>The terms "hopfian" and "cohopfian" have arisen since the 1960s, and are said to be in honor of <a href="/facts/Heinz_Hopf/PMks10NV">Heinz Hopf</a> and his use of the concept of the hopfian group in his work on <a href="/facts/Fundamental_group/VyYtLqSP">fundamental groups</a> of surfaces. (Hazewinkel 2001, p. 63) </p>