▁is ▁that ▁of ▁a ▁system ▁of ▁data ▁types ▁closed ▁under ▁a ▁type ▁constructor ▁that ▁takes ▁two ▁types ▁and ▁builds ▁an ▁aggregate ▁type ; ▁the ▁types ▁are ▁the ▁objects ▁and ▁ ▁is ▁the ▁aggregate ▁constructor . ▁The ▁associ ativity ▁up ▁to ▁isomorphism ▁is ▁then ▁a ▁way ▁of ▁express ing ▁that ▁different ▁ways ▁of ▁aggreg ating ▁the ▁same ▁data — such ▁as ▁ ▁and ▁— store ▁the ▁same ▁information ▁even ▁though ▁the ▁aggregate ▁values ▁need ▁not ▁be ▁the ▁same ; ▁similarly ▁the ▁identity ▁object ▁is ▁the ▁void ▁type , ▁which ▁stores ▁no ▁information . ▁The ▁concept ▁of ▁mon oid al ▁category ▁does ▁not ▁pres ume ▁that ▁values ▁of ▁such ▁aggregate ▁types ▁can ▁be ▁taken ▁apart ; ▁on ▁the ▁contrary , ▁it ▁provides ▁a ▁framework ▁that ▁un ifies ▁classical ▁and ▁quantum ▁information ▁theory . ▁ ▁In ▁category ▁theory , ▁mon oid al ▁categories ▁can ▁be ▁used ▁to ▁define ▁the ▁concept ▁of ▁a ▁mon oid ▁object ▁and ▁an ▁associated ▁action ▁on ▁the ▁objects ▁of ▁the ▁category . ▁They ▁are ▁also ▁used ▁in ▁the ▁definition ▁of ▁an ▁en rich ed ▁category . ▁ ▁Mon oid al ▁categories ▁have ▁numerous ▁applications ▁outside ▁of ▁category ▁theory ▁proper . ▁They ▁are ▁used ▁to ▁define ▁models ▁for ▁the ▁multip lic ative ▁fragment ▁of ▁intu ition istic ▁linear ▁logic . ▁They ▁also ▁form ▁the ▁mathematical ▁foundation ▁for ▁the ▁topological ▁order ▁in ▁cond ensed ▁matter . ▁Bra ided ▁mon oid al ▁categories ▁have ▁applications ▁in ▁quantum ▁information , ▁quantum ▁field ▁theory , ▁and ▁string ▁theory . ▁ ▁F ormal ▁definition ▁ ▁A ▁ ▁mon