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Math and science::Algebra::Aluffi

Endomorphisms and automorphisms

Endomorphism

An endomorphism is a morphism that has the same source and target object.

For an object \( A \) in category \( \cat{C} \) the set of endomorphisms of \( A \) are all morphisms in \( \cathom{C}(A, A) \), which is denoted as \( \mathrm{End_C}(A) \).

Automorphism

An automorphism is a morphism that has the same source and target object and is isomorphic (i.e. is an isomorphism).

For an object \( A \) in category \( \cat{C} \), the set of automorphisms for \( A \) is denoted as \( \mathrm{Aut_C}(A) \). \( \mathrm{Aut_C}(A) \) is a subset of \( \mathrm{End_C}(A) \).


Example

Groups and automorphisms

For an object \( A \) in category \( \cat{C} \), all morphisms in \( \mathrm{Aut_C}(A) \) 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 \( \mathrm{Aut_C}(A) \):

closed under composition
For any \( f, g \in \mathrm{Aut_C}(A) \), \( f \, g \in \mathrm{Aut_C}(A) \).
composition is associative
For any \( f, g, h \in \mathrm{Aut_C}(A) \), \( (f \, g) h = f (g \, h) \). This is a direct property of morphisms.
identity element under composition
\( 1_A \) is isomorphic, so it must be in \( \mathrm{Aut_C}(A) \).
inverse under composition
any \( f \in \mathrm{Aut_C}(A) \) has an inverse \( f^{-1} \in \mathrm{Aut_C}(A) \).

In other words, \( \mathrm{Aut_C}(A) \) is a group!

Source

p29