 that, there exists a morphism  such that  and.
 An epimorphism  is said to be split if there exists a morphism  such that  (in this case  is called a right-sided inverse for ).

There is also the notion of homological epimorphism in ring theory. A morphism  f: A → B of rings is a homological epimorphism if it is an epimorphism and it induces a full and faithful functor on derived categories:
D(f) : D(B) → D(A).

A morphism that is both a monomorphism and an epimorphism is called a bimorphism. Every isomorphism is a bimorphism but the converse is not true in general. For example, the map from the half-open interval [0,1) to the unit circle S1 (thought of as a subspace of the complex plane) that sends x to exp(2πix) (see Euler's formula) is continuous and bijective but not a homeomorphism since the inverse map is not continuous at 1, so it is an instance of a bimorphism that is not an isomorphism in the category Top. Another example is the embedding Q → R in the category Haus; as noted above, it is a bimorphism, but it is not bijective and therefore not an isomorphism. Similarly, in the category of rings, the map Z → Q is a bimorphism but not an isomorphism.

Epimorphisms are used to define abstract quotient objects in general categories: two epimorphisms f1 : X → Y1 and f2 : X → Y2 are said to be equivalent if there exists an isomorphism j : Y1 → Y2 with j f1 = f2. This is an equivalence relation, and the equivalence classes are defined to be the quotient objects of X.

Terminology 
The companion terms epimorphism and monomorphism were first introduced by Bourbaki. Bourbaki uses epimorphism as shorthand for a surjective function. Early category theorists believed that epimorphisms were the correct analogue of surjections in an arbitrary category, similar to how monomorphisms are very nearly an exact analogue of injections.  Unfortunately this is incorrect; strong or regular epimorphisms behave much more closely to surjections than ordinary epimorphisms.  Saunders Mac Lane attempted to create a distinction between epimorphisms, which were maps in a concrete category whose underlying set maps were surjective, and epic morphisms, which are
that, there exists a morphism   such that   and. 
  An epimorphism   is said to be split if there exists a morphism   such that   (in this subject   is call a right - sided inverse for). 

 There is also the impression of homological epimorphism in ring theory. A morphism   f: A → barn of rings is a homological epimorphism if it is an epimorphism and it induce a full and faithful functor on derive categories: 
 D(f): D(B) → D(A). 

 A morphism that is both a monomorphism and an epimorphism is called a bimorphism. Every isomorphism is a bimorphism but the converse is not dependable in cosmopolitan. For example, the map from the half - loose interval [ 0,1) to the unit set S1 (thought of as a subspace of the complex plane) that sends x to exp(2πix) (learn Euler's formula) is continuous and bijective but not a homeomorphism since the inverse map is not continuous at 1, so it is an instance of a bimorphism that is not an isomorphism in the category Top. Another model is the embedding Q   → R in the category Haus; as noted above, it is a bimorphism, but it is not bijective and therefore not an isomorphism. Similarly, in the class of rings, the map Z   → Q is a bimorphism but not an isomorphism. 

 Epimorphisms are used to define abstract quotient objects in general class: two epimorphisms f1: X → Y1 and f2: X → Y2 are said to be equivalent if there exists an isomorphism j: Y1 → Y2 with j   f1 = f2. This is an equivalence relation, and the equivalence classes are defined to be the quotient objects of X. 

 Terminology 
 The companion term epimorphism and monomorphism were foremost introduced by Bourbaki. Bourbaki uses epimorphism as shorthand for a surjective function. Early category theorist believe that epimorphisms were the correct analogue of surjections in an arbitrary category, similar to how monomorphisms are very nearly an exact analogue of injections.   Unfortunately this is incorrect; firm or regular epimorphisms behave much more closely to surjections than ordinary epimorphisms.   Saunders Mac Lane undertake to create a distinction between epimorphisms, which were maps in a concrete category whose implicit in set map were surjective, and epic morphisms, which are
 thwt, there exists a morphirm  such that  cbd.
 An xpimorpgism  is said to be split if there eeistw a mirphism  such that  (in this casv  is caloed e right-sided invxdse for ).

There ls alvi the notion on homologicdl epimorphism iv xing theory. A morphism  f: A → B of riggs is s jomological epymorkhifm ir it is an epimorphism and it indudes a flll and faithful gunctor on derived categorles:
D(v) : D(B) → D(A).

A morphisl that is bith w monomorphiso and an ekikorphism ia called a bimorphism. Every isooorphnsm is a binoephlvm but the ronverfe is not trmv in geteral. Fpr example, the ma' frim the half-open intertal [0,1) to the unit cirsle S1 (thoggkt of as a subspace od rhe cmmplax puqne) thzt ssnds x to exp(2πix) (see Euler's fornula) is continuous snq bijective buf not w romeomorphism since the inverse map is tot continuous at 1, so it iw an instance of a bilorphism ehat is not an isomorphism in the category Top. Anmther dxanpjd id the embedding Q → R in the category Haus; as njfec sbove, it is a nimorphism, but it os npj bijective ana therzrode not an isomorphlsm. Simylarlt, in the satebory of rings, the map Z → Q iw a bimorphifn but not an isomoxphism.

Epimor'hisms are ised to define abstract quofient objechs in gensfal categories: tdo vpimmrphisms f1 : X → Y1 and f2 : X → Y2 are samd to be equkvalgnt if ehere exishs an isomorphism j : Y1 → J2 witk j f1 = f2. This is an equivalence relation, and thx equivalence cnasves are befinec to be the qtotient objectx of X.

Tzrminouogy 
The ckmpanioi terms epimjrphism and mmjomorphism wxre first inteoduxed by Cuurbaki. Bourbali uses eiiiorpyism as shorthand nor a aurjective funccnob. Early categoru tfeowidtx falieved that epiooronisms were the cifrecy analogue of surjecdiona in an arbitrary vabegory, sinilar to how monomorpnisms are very neagly ai exacv analpgug of injections.  Unfortunately fhis is ijcovrect; strong jr rcgulwr epimorpkisms behave much more closely to surjecvions than ordinary epinorphisms.  Saunders Kac Lane atvempteq to creade a distinction betqeen epimorphisms, which were maps in a concrate cwtegory whose underlying set maps were surjective, and epic morphisms, whicy are
that, there exists a morphism such that epimorphism said to split if there (in case is called right-sided inverse for There is also the notion of epimorphism in ring theory. A morphism f: A → B of rings is homological epimorphism if it is an epimorphism and it induces a full and functor derived D(f) D(B) → D(A). A morphism that is both a monomorphism and an epimorphism is called a Every isomorphism is a bimorphism but the converse not true in general. example, the map from the interval to the circle (thought as a subspace the complex plane) that sends x to exp(2πix) (see Euler's formula) is continuous and bijective but not homeomorphism since map is continuous 1, it is an a bimorphism that is not an category Top. Another example is the embedding Q R in category Haus; as noted above, it a bimorphism, but it is not bijective and not an isomorphism. Similarly, in the category of rings, the map Z → Q is but not an isomorphism. are used to abstract objects general two epimorphisms : X → Y1 and f2 : X → Y2 are to be equivalent if there exists an isomorphism j : Y2 j f1 = This is an equivalence and equivalence classes are defined the objects The terms and monomorphism were first by Bourbaki. Bourbaki uses epimorphism shorthand for a surjective that epimorphisms were the correct analogue of surjections an arbitrary category, similar to how monomorphisms very nearly an exact analogue of injections. Unfortunately this is incorrect; strong regular epimorphisms more closely to surjections than ordinary epimorphisms. Saunders Lane attempted to create distinction between epimorphisms, which were maps in a concrete whose set maps surjective, and epic which are
 that, there exists a morphism  sUch that  and.
 an epiMorPhiSm  Is saId to Be split if there EXistS a morphism  such that  (in thIs casE  iS CallED a Right-Sided inVErSE For ).

thErE is AlSO tHe notIon Of homolOgical epimOrpHiSm in ring theoRY. A Morphism  f: A → b of Rings is a homoLogIcal epImOrpHIsm if It iS an epImorphISm and iT induces a FuLL and faIThful fuNCToR on dErived categories:
D(F) : d(B) → d(a).

A morphism that Is both A mONoMORphIsm And an epimoRpHism iS Called a BImORPHisM. every isomorphIsm is a bimorPHisM but thE cOnvERse is nOt truE iN GenEral. For examPle, tHe map from The halF-Open intERval [0,1) to tHe unit CirCle s1 (thoUGhT oF as A sUBspACe Of tHE coMplex plaNe) ThAt senDs x tO EXP(2Πix) (sEe EUler'S formUla) is continuoUs aNd biJEctIve buT not a HomeOmOrphiSm sincE the iNvErse map is not conTinuOus at 1, so it Is aN iNstAnCe of a BImorphIsm ThaT is not aN isomorPHisM iN THE cAtegory Top. Another eXaMPLe Is the embEdding q → r iN tHE categorY HAus; As noTED abovE, it iS A bImorphisM, but it IS nOt BijectiVe And theReForE noT an isOMorpHism. SiMilarly, iN the cATegory of rings, tHE map Z → Q is a bimoRPhISM bUT not An iSomorphism.

EPimoRPhisMs arE UsEd tO DefinE abstRaCT qUOtient objects in geneRaL categOries: Two epimorphisMs f1 : X → Y1 and f2 : X → y2 ARE said to bE equIVaLEnt if there exisTs an iSomorphism J : y1 → Y2 with j f1 = F2. This Is an equiValence reLATion, and tHe eQuiValEncE CLaSses are defineD TO be tHe QuotienT obJects of x.

TeRmiNolOgy 
thE companioN terms epImOrPhIsM anD monoMOrphism wErE fiRsT inTroduCEd by BoUrbakI. BouRbAkI UseS epimorPHiSM As shOrThAnd fOr a SuRjectIve fUNctIon. EarlY category TheORistS bElIeved thAt epimorphismS wEre the corrEcT anAlogue OF SurjectiOns in an arbitrary categorY, Similar To hOw monOmorPhisms are VerY nearlY an EXact anAlogue Of injEcTioNS.  unforTUNaTelY tHis is incorRECt; sTrong Or ReguLar epimOrphisms behave much MOre Closely to surjEctIons THAn OrdINaRY epImORphISMs.  Saunders Mac LaNe attempteD tO CrEate a distiNCtiOn Between EpimorpHisms, WHich werE maps in a cOncrete caTeGory WHOse Underlying Set maps wEre surjecTIve, anD EpIc morPhiSms, whiCh Are
 that, there exists a morp hism  such  that   a nd. 
An e pimo rphism  is sai d  tobe split if there exis ts amo r phis m  suchthat  ( i nt h isca se   i sc al led a  ri ght-sid ed inverse  fo r).

There is al so the not ion  of homologi cal  epimo rp his m  in r ing  theo ry. Am orphis m  f: A →  B of rin g s is ah o mo logi cal epimorphism i f  i t  is an epimorp hism a nd it i ndu ces  a full an dfaith f ul func t or o n  de r ived categori es:
D(f) :D (B)  → D(A ). 

A morphi sm th at isboth a mono morp hism andan epi m orphism is call ed a b imo rph ism. Ev er y i so m orp h is m i s  abimorphi sm  b ut th e co n v e r se i s n ot t rue i n general. Fo r e xamp l e,the m ap fr om t he  half -openinter va l [0,1) to theunit  circle S 1 ( th oug ht  of a s  a sub spa ceof thecomplex pla ne ) t ha t sends x to exp(2 πi x )  ( see Eule r's fo r mu la )  is cont in uou s an d bijec tive bu t not ahomeom o rp hi sm sinc ethe in ve rse  ma p isn ot c ontinu ous at 1 , soi t is an instan c e of a bimorp h is m th a t is  no t an isomor phis m  inthec at ego r y Top . Ano th e re xample is the embed di ng Q →  R in  the category  Haus; asn o t ed above , it is a bimorphism,but i t is not b i jectiveand t herefore  not an i s o morphism . S imi lar ly, i nthe categoryo f  rin gs , the m apZ → Q i s a  bi mor phi sm  but notan isomo rp hi sm .
Ep imorp h isms are  u sed  t o d efine abstra ct qu otie nt  o b jec ts in g e ne r a l ca te go ries : t wo  epim orph i sms  f1 : X  → Y1 and  f2 : X→Y2  are sa id to be equi va lent if th er e e xistsa n  isomorp hism j : Y1 → Y2 with j f1 = f2 . T his i s an  equivale nce  relat ion ,  and t he equ ivale nc e c l a ssesa r edef in ed to be t h e  qu otien tobje cts ofX.

Terminology 
T h e c ompanion term s e pimo r p hi sma nd mon om o rph i s m were first in troduced b yB ou rbaki. Bou r bak iuses ep imorphi sm as shortha nd for asurjectiv efunc t i on.  Early cat egory th eorists b e lieve d  t hat e pim orphis ms  we re th e corr e ctanalo gue of  s urject ionsin  an arbi trary category, similar  to ho w mon omo rphisms a rev ery  nearly a n ex act analog ueofinjec tio n s.  U nfor t un ate l y thi s is incorrect ;  s tro n g  o r regular e p i m orp hisms  be h ave mu ch m ore closely to su r jections thanordi n a ryepi m orph is ms.  SaundersMac  L a n e attemp te d to create  a disti nc t ion b etween  epimo rphisms , wh i ch wer e ma psin a conc ret ec ategory  w ho s e unde rlyi ng  set m aps we r e su r j ective, and epic  morp h i sms,w hic h are 
 that,_there exists_a morphism  such_that _and.
_An epimorphism__is said to_be split if_there exists a morphism_ such that__(in this case  is called a right-sided inverse for ).

There is also the_notion_of homological_epimorphism_in_ring theory. A morphism _f: A → B of_rings is_a homological epimorphism if it is an epimorphism_and_it induces a_full and faithful functor on derived categories:
D(f) : D(B)_→ D(A).

A morphism that is both_a monomorphism and_an_epimorphism_is called a bimorphism._Every isomorphism is a bimorphism but_the converse is not true in_general. For example, the map from the_half-open interval [0,1) to the unit_circle S1 (thought of as_a subspace_of the complex plane) that_sends x to_exp(2πix) (see_Euler's formula) is_continuous and bijective but not a_homeomorphism since the_inverse map is not continuous at_1,_so it is_an_instance_of a_bimorphism that is_not_an isomorphism_in_the category Top. Another example is_the_embedding Q → R in the category Haus;_as noted above, it_is_a bimorphism, but it_is not bijective and therefore_not an isomorphism. Similarly, in the_category of_rings, the_map Z → Q is a bimorphism but not an isomorphism.

Epimorphisms are_used to define abstract quotient objects_in general categories: two_epimorphisms f1_:_X → Y1_and_f2 :_X → Y2 are said to be_equivalent if_there exists an isomorphism j :_Y1 → Y2 with_j f1_= f2. This is an equivalence_relation, and the equivalence classes are_defined to be the quotient_objects_of_X.

Terminology 
The companion terms epimorphism_and monomorphism were first introduced by_Bourbaki. Bourbaki uses_epimorphism as shorthand for a surjective function._Early_category theorists believed that epimorphisms were_the_correct analogue of surjections in an_arbitrary_category,_similar to how monomorphisms are_very nearly an exact analogue of_injections.  Unfortunately this is incorrect; strong or regular_epimorphisms behave much_more closely to surjections than_ordinary_epimorphisms._ Saunders Mac Lane attempted to create a distinction between_epimorphisms, which_were maps in_a concrete category whose underlying set maps were surjective, and_epic morphisms, which are