CatDat

Relationships between epimorphisms and monomorphisms

There are several properties of morphisms, including various types of epimorphisms and monomorphisms. The implications establish various relationships between these types. Here we present a graphical overview of these relationships.

The various types of epimorphisms

In the diagram, an arrow $X \Longrightarrow Y$ means that every morphism with property $X$ also has property $Y$. If it is labelled with a category property $P$, the implication does not hold in general, but it holds in categories satisfying $P$. For example, in a category with pullbacks, every strict epimorphism is effective.

Fun fact: This describes a category in itself: We define the composition of $P : X \Rightarrow Y$ and $Q : Y \Rightarrow Z$ as $P \wedge Q : X \Rightarrow Z$.

The various types of monomorphisms

This diagram is just the dual of the previous diagram. The same notation applies.