Skip to main content

hom-Sets

Let C\mathbb{C} be a fixed category.

Consider a morphism f:ABf:A\to B in C\mathbb{C} then, for any object XX in C\mathbb{C}, hom(X,1A):hom(X,A)hom(X,A)\operatorname{hom}(X,1_A):\operatorname{hom}(X,A)\to\operatorname{hom}(X,A) is the identity map of hom(X,A)\operatorname{hom}(X,A) since for any aAa\in A,

hom(X,1A)(a)=1Aa=a.\operatorname{hom}(X,1_A)(a)=1_Aa=a.

Given g:BCg:B\to C a morphism in C\mathbb{C}, we have that

hom(X,g)hom(X,f)(a)=hom(X,g)(fa)=gfa=hom(X,gf)(a)\operatorname{hom}(X,g)\operatorname{hom}(X,f)(a) = \operatorname{hom}(X,g)(fa) = gfa = \operatorname{hom}(X,gf)(a)

and so hom(X,g)hom(X,f)=hom(X,gf)\operatorname{hom}(X,g)\operatorname{hom}(X,f) = \operatorname{hom}(X,gf). Similarly, for a morphism chom(C,X)c\in\operatorname{hom}(C,X) we have that,

hom(f,X)hom(g,X)(c)=hom(f,X)(cg)=cgf=hom(gf,X)(c)\operatorname{hom}(f,X)\operatorname{hom}(g,X)(c) = \operatorname{hom}(f,X)(cg) = cgf = \operatorname{hom}(gf,X)(c)

and so hom(f,X)hom(g,X)=hom(gf,X)\operatorname{hom}(f,X)\operatorname{hom}(g,X) = \operatorname{hom}(gf,X).


Consider morphisms f:ABf:A\to B and g:CDg:C\to D in C\mathbb{C}, we can define the induced map hom(g,f):hom(D,A)hom(C,B)\operatorname{hom}(g,f):\operatorname{hom}(D,A)\to\operatorname{hom}(C,B) for dhom(D,A)d\in\operatorname{hom}(D,A) by

hom(g,f)(d)=fdg\operatorname{hom}(g,f)(d)=fdg

and we have that

hom(g,f)=hom(C,f)hom(g,A).\operatorname{hom}(g,f)=\operatorname{hom}(C,f)\operatorname{hom}(g,A).

For a morphism ff in Sets\textbf{Sets} we have that ff is a monomorphism if and only if hom(X,f)\operatorname{hom}(X,f) is injective for every object XX in Sets\textbf{Sets}. Suppose f:ABf:A\to B is a monomorphism in Sets\textbf{Sets}, then for any object XX in Sets\textbf{Sets} and any pair of parallel morphisms a,ahom(X,A)a,a'\in\operatorname{hom}(X,A), we have that

hom(X,f)(a)=hom(X,f)(a)    fa=fa    a=a\operatorname{hom}(X,f)(a)=\operatorname{hom}(X,f)(a')\implies fa=fa' \implies a = a'

since ff is a monomorphism. So, hom(X,f)\operatorname{hom}(X,f) is injective for any object XX in Sets\textbf{Sets}. Conversely, suppose hom(X,f)\operatorname{hom}(X,f) is injective for any object XX, then for any pair of parallel morphisms a,ahom(X,A)a,a'\in\operatorname{hom}(X,A),

hom(X,f)(a)=hom(X,f)(a)    a=a\operatorname{hom}(X,f)(a)=\operatorname{hom}(X,f)(a')\implies a = a'

and so

fa=fa    a=afa=fa'\implies a=a'

hence, f is a monomorphism in Sets\textbf{Sets}. Similarly, ff is an epimorphism in Sets\textbf{Sets} if and only if for every object XX the map hom(f,X)\operatorname{hom}(f,X) is injective.