▁a ▁strict ▁ 2 - mon ad ▁on ▁Cat . ▁ ▁Special izations ▁▁ ▁If , ▁in ▁a ▁mon oid al ▁category , ▁ ▁and ▁ ▁are ▁naturally ▁is omorphic ▁in ▁a ▁manner ▁compatible ▁with ▁the ▁coh er ence ▁conditions , ▁we ▁speak ▁of ▁a ▁bra ided ▁mon oid al ▁category . ▁If , ▁moreover , ▁this ▁natural ▁isomorphism ▁is ▁its ▁own ▁inverse , ▁we ▁have ▁a ▁symmetric ▁mon oid al ▁category . ▁ ▁A ▁closed ▁mon oid al ▁category ▁is ▁a ▁mon oid al ▁category ▁where ▁the ▁fun ctor ▁ ▁has ▁a ▁right ▁ad joint , ▁which ▁is ▁called ▁the ▁" internal ▁Hom - fun ctor " ▁. ▁ ▁Ex amples ▁include ▁cart esian ▁closed ▁categories ▁such ▁as ▁Set , ▁the ▁category ▁of ▁sets , ▁and ▁compact ▁closed ▁categories ▁such ▁as ▁F d V ect , ▁the ▁category ▁of ▁finite - dimensional ▁vector ▁spaces . ▁ ▁Aut onom ous ▁categories ▁( or ▁compact ▁closed ▁categories ▁or ▁rig id ▁categories ) ▁are ▁mon oid al ▁categories ▁in ▁which ▁du als ▁with ▁nice ▁properties ▁exist ; ▁they ▁abstract ▁the ▁idea ▁of ▁F d V ect . ▁ ▁D agger ▁symmetric ▁mon oid al ▁categories , ▁equ ipped ▁with ▁an ▁extra ▁d agger ▁fun ctor , ▁abstract ing ▁the ▁idea ▁of ▁F d H il b , ▁finite - dimensional ▁Hil bert ▁spaces . ▁ ▁These ▁include ▁the ▁d agger ▁compact ▁categories . ▁ ▁T ann ak ian ▁categories ▁are ▁mon oid al ▁categories ▁en rich ed ▁over ▁a ▁field , ▁which ▁are ▁very ▁similar