# Monics and epics

Given a category we define a **monic** $m$ as:

and an **epic** $e$ as:

where $f$ and $g$ are parallel morphisms. Note that since I don't have fancier LaTeX at my disposal I'm separating the two morphisms $f$ and $g$ by a comma which should be interpreted as two separate morphisms (arrows) between the two objects.

Here's a picture:

To make the idea more *familiar* let's consider the category $Set$ which is the category of structured sets and total functions between those sets.

If $m$ is **injective** as a function then it is **monic** as a morphism (also called a **monomorphism**).

Similarly if $e$ is **surjective** as a function then it is **epic** as a morphism (also called an **epimorphism**).

There is a one-to-one correspondence between the two relations when considering the category $Set$ and the "real" set but in general this does not hold for arbitrary categories.