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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 $$\mathrm{End_C}(A)$$.

### 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!

p29