1.6Morphisms

Updated 11 May 2026Chapter (PDF)

We now redefine certain concepts about morphisms categorically.

[1.6.1]Definition(Isomorphism)#

Let $\sf{C}$ be a category and let $f \in \Hom_{\sf{C}}(A, B)$ be a morphism with $A, B \in \Obj(\sf{C})$ . $f$ is an isomorphism if there exists some $g \in \Hom_{\sf{C}}(b, a)$ such that $gf = 1_A$ and $fg = 1_B$ .

[1.6.2]Example(Bijections in

\Set

are Isomorphisms)#

Note that morphisms in $\Set$ are precisely set-functions, and bijective set-functions are precisely the ones that have two-sided inverses, so they are isomorphisms in the $\Set$ category.

[1.6.3]Theorem(Properties of Isomorphisms)#

Isomorphisms have unique inverses.
Each identity $1_A$ is an isomorphism and is involutory.
If $f$ is an isomorphism, then so is $f^{-1}$ . Furthermore, $(f^{-1})^{-1} = f$ .
If $f \in \Hom_{\sf{C}}(A, B)$ and $g \in \Hom_{\sf{C}}(B, C)$ are isomorphisms, then $gf \in \Hom_{\sf{C}}(A, C)$ is an isomorphism whose inverse is given by $(gf)^{-1} = f^{-1}g^{-1}$ .

Proof.

These are straight forward arguments whose proofs generalize cleanly from naive set theory, so we omit the proof.

[1.6.4]Definition(Endomorphism, Automorphism)#

Let $\sf{C}$ be a category. A morphism $f$ in $\sf{C}$ is an endomorphism if it maps an object $A$ to itself. The set of all endomorphisms is denoted by $\End_{\sf{C}}(A)$ . $f$ is an automorphism if it is an endomorphism that is an isomorphism. The set of all automorphisms is denoted by $\Aut_{\sf{C}}(A) \subseteq \End_{\sf{C}}(A)$ .

[1.6.5]Definition(Groupoid)#

A category $\sf{C}$ is a groupoid if every morphism it comprises is an isomorphism.

Monomorphisms and epimorphisms were already defined on $\Set$ , but note their equivalence to injectivity and surjectivity—as established in $\Set$ —does not necessarily hold in other categories.