Surjectivity, injectivity, and bijectivity

Surjective, injective, and bijective functions are epimorphisms, monomorphisms, and isomorphisms respectively in $𝖲 𝖾 𝗍$ . Thus the morphisms of any concrete category may be described as such, but these concepts may not align exactly (for example, there exist bijective continuous functions that are not homeomorphisms). Specifically, given a function $𝑓 : 𝐴 \to 𝐵$

$f$ is surjective iff for every $b \in B$ there exists $a \in A$ such that $f(a) = b$ #m/def/general
- In particular if there exists a right-inverse. These are equivalent assuming AC.
- A surjective function induces an Equivalence relation.
$f$ is injective iff $f(a_{1}) = f(a_{2}) \iff a_{1} = a_{2}$ . #m/def/general
- Equivalently, there exists a left-inverse.
$f$ is bijective iff it is surjective and injective. #m/def/general
- In particular, there exists a unique ambidextrous inverse. Again these are equivalent assuming AC.

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