Math and science::Algebra::Aluffi

Example

Groups and automorphisms

For an object $A$ in category $\mathrm{C}$, all morphisms in ${\mathrm{Aut}}_{\mathrm{C}}(A)$ enjoy the property of being isomorphic. This, along with the usual properties of morphisms allows us to collect the following set of propositions related to composing elements of ${\mathrm{Aut}}_{\mathrm{C}}(A)$:

closed under composition

For any $f,g\in {\mathrm{Aut}}_{\mathrm{C}}(A)$, $fg\in {\mathrm{Aut}}_{\mathrm{C}}(A)$.

composition is associative

For any $f,g,h\in {\mathrm{Aut}}_{\mathrm{C}}(A)$, $(fg)h=f(gh)$. This is a direct property of morphisms.

identity element under composition

$1_A$ is isomorphic, so it must be in ${\mathrm{Aut}}_{\mathrm{C}}(A)$.

inverse under composition

any $f\in {\mathrm{Aut}}_{\mathrm{C}}(A)$ has an inverse $f^{-1}\in {\mathrm{Aut}}_{\mathrm{C}}(A)$.

In other words, ${\mathrm{Aut}}_{\mathrm{C}}(A)$ is a group!

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Source

p29