als . ▁They ▁were ▁first ▁defined ▁( follow ing ▁Alexandre ▁Gro th end ieck ) ▁by ▁Ne ant ro ▁Sa aved ra - R iv ano ▁in ▁his ▁th esis ▁on ▁T ann ak ian ▁categories . ▁ ▁Definition ▁ ▁There ▁are ▁at ▁least ▁two ▁equivalent ▁definitions ▁of ▁a ▁rig id ity . ▁ ▁An ▁object ▁X ▁of ▁a ▁mon oid al ▁category ▁is ▁called ▁left ▁rig id ▁if ▁there ▁is ▁an ▁object ▁Y ▁and ▁morph isms ▁ ▁and ▁ ▁such ▁that ▁both ▁compos itions ▁ ▁are ▁ident ities . ▁A ▁right ▁rig id ▁object ▁is ▁defined ▁similarly . ▁ ▁An ▁inverse ▁is ▁an ▁object ▁X − 1 ▁such ▁that ▁both ▁X ▁ ⊗ ▁X − 1 ▁and ▁X − 1 ▁ ⊗ ▁X ▁are ▁is omorphic ▁to ▁ 1 , ▁the ▁one ▁object ▁of ▁the ▁mon oid al ▁category . ▁If ▁an ▁object ▁X ▁has ▁a ▁left ▁( resp . ▁right ) ▁inverse ▁X − 1 ▁with ▁respect ▁to ▁the ▁tensor ▁product ▁then ▁it ▁is ▁left ▁( resp . ▁right ) ▁rig id , ▁and ▁X * ▁= ▁X − 1 . ▁ ▁The ▁operation ▁of ▁taking ▁du als ▁gives ▁a ▁cont rav ari ant ▁fun ctor ▁on ▁a ▁rig id ▁category . ▁ ▁Us es ▁ ▁One ▁important ▁application ▁of ▁rig id ity ▁is ▁in ▁the ▁definition ▁of ▁the ▁trace ▁of ▁an ▁end omorphism ▁of ▁a ▁rig id ▁object . ▁The ▁trace ▁can ▁be ▁defined ▁for ▁any ▁rig id ▁category ▁such ▁that ▁taking ▁the ▁(   ) ** , ▁the ▁fun