Monomorphism

A monomorphism is a left-cancellable morphism (denoted with $↣$ ). A morphism $𝑚 : 𝑌 \to 𝑍$ is monic iff for any $𝑋 \in 𝖢$ and $𝑓, 𝑔 : 𝑋 \to 𝑌$ #m/def/cat

𝑚 \circ 𝑓 = 𝑚 \circ 𝑔 ⟹ 𝑓 = 𝑔

In $𝖲 𝖾 𝗍$ a function is a monic iff it is injective iff it is left-invertible (i.e. split monic), but these are not equivalent in every concrete category, rather:

graph LR;
  left-invertible ==>|implies| injective ==>|implies| monic

Properties

See the dual properties.

If $𝑓 𝑔$ is monic then $𝑔$ is monic.

Proof of 1

Note $𝑔 𝑎 = 𝑔 𝑏$ implies $𝑓 𝑔 𝑎 = 𝑓 𝑔 𝑏$ which holds iff $𝑎 = 𝑏$ , proving ^P1.

#state/tidy | #lang/en | #SemBr