▁j arg on ▁which ▁refers ▁to ▁mapping ▁a ▁fun ctor ▁to ▁a ▁category ▁by ▁highlight ing ▁or ▁defining ▁remain ders . ▁ ▁Struct ures ▁The ▁term ▁" mod ulo " ▁can ▁be ▁used ▁differently — when ▁referring ▁to ▁different ▁mathematical ▁structures . ▁For ▁example : ▁ ▁Two ▁members ▁a ▁and ▁b ▁of ▁a ▁group ▁are ▁con gru ent ▁mod ulo ▁a ▁normal ▁subgroup , ▁if ▁and ▁only ▁if ▁ab − 1 ▁is ▁a ▁member ▁of ▁the ▁normal ▁subgroup ▁( see ▁quot ient ▁group ▁and ▁isomorphism ▁theorem ▁for ▁more ). ▁ ▁Two ▁members ▁of ▁a ▁ring ▁or ▁an ▁algebra ▁are ▁con gru ent ▁mod ulo ▁an ▁ideal , ▁if ▁the ▁difference ▁between ▁them ▁is ▁in ▁the ▁ideal . ▁ ▁U sed ▁as ▁a ▁verb , ▁the ▁act ▁of ▁factor ing ▁out ▁a ▁normal ▁subgroup ▁( or ▁an ▁ideal ) ▁from ▁a ▁group ▁( or ▁ring ) ▁is ▁often ▁called ▁" mod ding ▁out ▁the ..." ▁or ▁" we ▁now ▁mod ▁out ▁the ... ". ▁ ▁Two ▁subsets ▁of ▁an ▁infinite ▁set ▁are ▁equal ▁mod ulo ▁finite ▁sets ▁precisely ▁if ▁their ▁symmetric ▁difference ▁is ▁finite , ▁that ▁is , ▁you ▁can ▁remove ▁a ▁finite ▁piece ▁from ▁the ▁first ▁subset , ▁then ▁add ▁a ▁finite ▁piece ▁to ▁it , ▁and ▁get ▁the ▁second ▁subset ▁as ▁a ▁result . ▁ ▁A ▁short ▁exact ▁sequence ▁of ▁maps ▁leads ▁to ▁the ▁definition ▁of ▁a ▁quot ient ▁space ▁as ▁being ▁one ▁space ▁mod ulo ▁another ; ▁thus , ▁for ▁example , ▁that ▁a ▁coh om ology ▁is ▁the ▁space ▁of