# Monos, epis, and duality

January 12, 2012 Leave a comment

Something I didn’t mention last time is duality! That was silly of me. Suppose we have a monomorphism in a category . I claim that is an epimorphism in . That is, monos and epis are dual!

#### Proof:

Just unravel the definitions. being monic means that whenever we have and with , then . Taking this to the opposite category, whenever we have and as morphisms in , and , then . But we know how to “distribute” . So whenever

we know that . This is precisely the condition for to be an epimorphism in .

So in some sense, injections and surjections are “the opposite of each other.” Isn’t that neat?