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Monomorphisms and epimorphisms

The classes of monomorphisms, epimorphisms, bimorphisms, split monomorphisms, and split epimorphisms are closed under composition and contain all isomorphisms. To show these classes are closed under composition, consider two morphisms f:ABf:A\to B and g:BCg:B\to C. If ff and gg are monomorphisms then for a,ahom(X,A)a,a'\in\operatorname{hom}(X,A),

(gf)a=(gf)a    fa=fa    a=a(gf)a=(gf)a'\implies fa=fa'\implies a = a'

and so gfgf is a monomorphism. If ff and gg are epimorphisms then for c,chom(C,X)c,c'\in\operatorname{hom}(C,X),

c(gf)=c(gf)    cg=cg    c=cc(gf)=c'(gf)\implies cg=cg'\implies c=c'

and so gfgf is an epimorphism. If ff and gg are split monomorphisms then there exists F:BAF:B\to A and G:CBG:C\to B such that Ff=1AFf=1_A and Gg=1BGg=1_B. So gfgf is a split monomorphism since (FG)(gf)=1A(FG)(gf)=1_A. Similarly, if gg and ff are split epimorphisms then (gf)(FG)=1C(gf)(FG)=1_C and so gfgf is a split epimorphism.

Let f:ABf:A\to B be an isomorphism then, ff has inverse g:BAg:B\to A and so for every pair of morphisms a,ahom(X,A)a,a'\in\operatorname{hom}(X,A) and b,bhom(B,X)b,b'\in\operatorname{hom}(B,X) we have that

fa=fa    gfa=gfa    a=afa=fa'\implies gfa=gfa'\implies a=a'

and

bf=bf    bfg=bfg    b=b.bf=b'f\implies bfg=b'fg\implies b=b'.

Hence, every isomorphism is a monomorphism and an epimorphism and so it is a bimorphism. Also, every isomorphism is a split monomorphism and a split epimorphism since it has a unique inverse.

Let f:ABf:A\to B and g:BCg:B\to C be two morphisms:

  • If the composition gfgf is a monomorphism then for a,ahom(X,A)a,a'\in\operatorname{hom}(X,A), fa=fa    gfa=gfa    a=afa=fa'\implies gfa=gfa'\implies a=a' and so ff is a monomorphism.
  • If the composition gfgf is an epimorphism then for c,chom(C,X)c,c'\in\operatorname{hom}(C,X), cg=cg    cgf=cgf    c=ccg=c'g\implies cgf=c'gf\implies c=c' and so gg is an epimorphism.
  • If the composition gfgf is a split monomorphism then there exists a morphism h:CAh:C\to A such that h(gf)=1Ah(gf)=1_A, so (hg)f=1A(hg)f=1_A and hence, ff is a split monomorphism.
  • If the composition gfgf is a split epimorphism then there exists a morphism h:CAh:C\to A such that (gf)h=1C(gf)h=1_C, so g(fh)=1Cg(fh)=1_C and hence, gg is a split epimorphism.
  • If f:ABf:A\to B is a split monomorphism then there is a morphism g:BAg:B\to A with gf=1Agf=1_A and so (fg)f=f(gf)=f1A=f=1Bf(fg)f=f(gf)=f1_A=f=1_Bf and since ff is an epimorphism, we have that fg=1Bfg=1_B. Hence, ff is an isomorphism.

The axiom of choice can be stated as follows.

For every surjective map of sets f:ABf:A\to B, there exists a map g:BAg:B\to A with fg=1Bfg=1_B 1.

Using this, and the fact that every epimorphism in textbfSets\\textbf{Sets} is a surjective map of sets, it is easy to see that the axiom of choice can be formulated as

Every epimorphism in textbfSets\\textbf{Sets} is split 1.


The inclusion map from the additive monoid of natural numbers to the additive group of integers (considered as a monoid) is a bimorphism in textbfMon\\textbf{Mon}. Let N=(N,+,0)\mathbb{N}=(\mathbb{N},+,0) denote the additive monoid of natural numbers and Z=(Z,+,0)\mathbb{Z}=(\mathbb{Z},+,0) denote the additive group of integers considered as a monoid. Then for every object XX in textbfMon\\textbf{Mon} and for every pair of morphisms a,ahom(X,N)a,a'\in\operatorname{hom}(X,\mathbb{N}) we have that if ia=iaia=ia' then for every xXx\in X, ia(x)=ia(x)ia(x)=ia'(x) and so i(a(x))=i(a(x))    a(x)=a(x)i(a(x))=i(a'(x))\implies a(x)=a'(x). Hence, ii is a monomorphism. Additionally, for every pair of morphisms b,bhom(Z,X)b,b'\in\operatorname{hom}(\mathbb{Z},X) and nNn\in\mathbb{N}, bi(n)=bi(n)    b(α)=b(α)bi(n)=b'i(n)\implies b(\alpha)=b'(\alpha) for all αZ+\alpha\in\mathbb{Z}^+ with α=i(n)\alpha = i(n). But this means b(0)=b(0)b(0)=b'(0) and so b(α)+b1(α)=b(αα)=b(0)=b(0)=b(αα)=b(α)+(b)1(α)b(\alpha)+b^{-1}(\alpha)=b(\alpha -\alpha)=b(0)=b'(0)=b'(\alpha -\alpha) = b'(\alpha)+(b')^{-1}(\alpha) since b(0)=b(0)=0b(0)=b'(0)=0. Thus, we have that b(α)=b1(α)=(b)1(α)=b(α)b(-\alpha) = b^{-1}(\alpha)=(b')^{-1}(\alpha) = b'(-\alpha) and so b=bb=b'. Hence, ii is an epimorphism and so it is a bimorphism.


  1. George Janelidze. Category theory: A first course. Lecture Notes, 2020. ↩︎ ↩︎