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}} and 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?

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