Natural transformation
Opposite group

Statements such as

abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category Grp {\displaystyle {\textbf {Grp}}} of all groups with group homomorphisms as morphisms. If (G, ∗) {\displaystyle (G,*)} is a group, we define its opposite group (G op, ∗ op) {\displaystyle (G^{\text{op}},{*}^{\text{op}})} as follows: G op {\displaystyle G^{\text{op}}} is the same set as G {\displaystyle G}, and the operation ∗ op {\displaystyle *^{\text{op}}} is defined by a ∗ op b = b ∗ a {\displaystyle a*^{\text{op}}b=b*a}. All multiplications in G op {\displaystyle G^{\text{op}}} are thus "turned around". Forming the opposite group becomes a (covariant) functor from Grp {\displaystyle {\textbf {Grp}}} to Grp {\displaystyle {\textbf {Grp}}} if we define f op = f {\displaystyle f^{\text{op}}=f} for any group homomorphism f: G → H {\displaystyle f:G\to H}. Note that f op {\displaystyle f^{\text{op}}} is indeed a group homomorphism from G op {\displaystyle G^{\text{op}}} to H op {\displaystyle H^{\text{op}}}:

The content of the above statement is:

To prove this, we need to provide isomorphisms η G: G → G op {\displaystyle \eta _{G}:G\to G^{\text{op}}} for every group G {\displaystyle G}, such that the above diagram commutes. Set η G (a) = a − 1 {\displaystyle \eta _{G}(a)=a^{-1}}.The formulas (a ∗ b) − 1 = b − 1 ∗ a − 1 = a − 1 ∗ op b − 1 {\displaystyle (a*b)^{-1}=b^{-1}*a^{-1}=a^{-1}*^{\text{op}}b^{-1}} and (a − 1) − 1 = a {\displaystyle (a^{-1})^{-1}=a} show that η G {\displaystyle \eta _{G}} is a group homomorphism with inverse η G op {\displaystyle \eta _{G^{\text{op}}}}. To prove the naturality, we start with a group homomorphism f: G → H {\displaystyle f:G\to H} and show η H ∘ f = f op ∘ η G {\displaystyle \eta _{H}\circ f=f^{\text{op}}\circ \eta _{G}}, i.e. (f (a)) − 1 = f op (a − 1) {\displaystyle (f(a))^{-1}=f^{\text{op}}(a^{-1})} for all a {\displaystyle a} in G {\displaystyle G}. This is true since f op = f {\displaystyle f^{\text{op}}=f} and every group homomorphism has the property (f (a)) − 1 = f (a − 1) {\displaystyle (f(a))^{-1}=f(a^{-1})}.
punctuation:no_comma