A mor­phism is the ab­stract rep­re­sen­ta­tion of a re­la­tion be­tween math­e­mat­i­cal ob­jects.

Usu­ally, it is used to re­fer to func­tions map­ping el­e­ment of one set to an­other, but it may rep­re­sent a more gen­eral no­tion of a re­la­tion in cat­e­gory the­ory.


To un­der­stand a mor­phism, it is eas­ier to first un­der­stand the con­cept of an iso­mor­phism. Two math­e­mat­i­cal struc­tures (say two groups) are called iso­mor­phic if they are in­dis­t­in­guish­able us­ing the in­for­ma­tion of the lan­guage and the­ory un­der con­sid­er­a­tion.

Imag­ine you are the Count von Count. You care only about count­ing things. You don’t care what it is you count, you just care how many there are. You de­cide that you want to col­lect ob­jects you count into boxes, and you con­sider two boxes equal if there are the same num­ber of el­e­ments in both boxes. How do you know if two boxes have the same num­ber of el­e­ments? You pair them up and see if there are any left over in ei­ther box. If there aren’t any left over, then the boxes are “bi­jec­tive” and the way that you paired them up is a bi­jec­tion. A bi­jec­tion is a sim­ple form of an iso­mor­phism and the boxes are said to be iso­mor­phic.

For ex­am­ple, the the­ory of groups only talks about the way that el­e­ments are com­bined via group op­er­a­tion (and whether they are the iden­tity or in­verses, but that in­for­ma­tion is already given by the in­for­ma­tion of how el­e­ments are com­bined un­der the group op­er­a­tion (here­after called mul­ti­pli­ca­tion). The the­ory does not care in what or­der el­e­ments are put, or what they are la­bel­led or even what they are. Hence, if you are us­ing the lan­guage and the­ory of groups, you want to say two groups are es­sen­tially in­dis­t­in­guish­able if their mul­ti­pli­ca­tion acts the same way.


  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.