# Monos, epis, and duality

Something I didn’t mention last time is duality! That was silly of me. Suppose we have a monomorphism $f:X\to Y$ in a category $\mathcal C$. I claim that $f^{\textrm{op}}:Y\to X$ is an epimorphism in $\mathcal C^{\textrm{op}}$. That is, monos and epis are dual!

#### Proof:

Just unravel the definitions. $f$ being monic means that whenever we have $g$ and $h$ with $f\circ g=f\circ h$, then $g=h$. Taking this to the opposite category, whenever we have $g^{\textrm{op}}$ andĀ $h^{\textrm{op}}$ as morphisms in $\mathcal C^{\textrm{op}}$, and $(f\circ g)^{\textrm{op}}=(f\circ h)^{\textrm{op}}$, then $g^{\textrm{op}}=h^{\textrm{op}}$. But we know how to “distribute” $-^{\textrm{op}}$. So whenever

$g^{\textrm{op}}\circ f^{\textrm{op}}=h^{\textrm{op}}\circ f^{\textrm{op}}$

we know that $g^{\textrm{op}}=h^{\textrm{op}}$. This is precisely the condition for $f^{\textrm{op}}$ to be an epimorphism in $\mathcal C^{\textrm{op}}$.

$\square$

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