# On explaining monomorphisms and epimorphisms

I almost always see explanations of monomorphisms such as: if $$\forall g_1,g_2, \ m\circ g_1 = m\circ g_2 \Rightarrow g_1=g_2$$, then $$m$$ is monic (and similarly for epimorphisms). I find this is not at all intuitive.

Is it OK to turn the argument around and talk about inequality between arrows instead of equality? (btw, this is for a blog post, not for a scientific paper)

### Monomorphisms:

$$m$$ is monomorphism if does not erase information: if $$g_1$$ and $$g_2$$ are different arrows, we can still tell them apart even if they are followed by $$m$$: $$g_1 \neq g_2 \quad \Rightarrow \quad m \circ g_1 \neq m \circ g_2\quad (\forall g_1,g_2)$$ In other words, the fact that $$m \circ g_1 = m \circ g_2$$ can't be because $$m$$ "hides" the differences between $$g_1$$ and $$g_2$$, it really must be because $$g_1 = g_2$$.

### Epimorphisms:

$$e$$ is an epimorphism if it does not restrict access to information: if $$g_1$$ and $$g_2$$ are different, we can still tell them apart if they are preceded by $$e$$ (e.g. $$e$$ does not hide the parts of the source where $$g_1$$ and $$g_2$$ would have differed): $$g_1 \neq g_2 \quad \Rightarrow \quad g_1 \circ e \neq g_2 \circ e\quad (\forall g_1,g_2)$$ In other words, the fact that $$g_1\circ e = g_2 \circ e$$ can't be because $$e$$ "sets things up" so that $$g_1$$ and $$g_2$$ look equal, it really must be because $$g_1 = g_2$$.

• Nothing wrong with that. Intuition might be subjective. – drhab Apr 30 '19 at 8:38