Jointly Monic Family

A family of morphisms $m_{i}:X\to Y_{i}$ is jointly monic if for all $g,h:A\to X$ with $\forall i.\;m_{i}\circ g=m_{i}=h$, $g=h$.

Properties

• If any $m_{i}$ is Monic, then the entire family is jointly monic.

• A singleton family $m$ is jointly monic if and only if $m$ is Monic.

• The empty family $\emptyset:X\to Y_{\emptyset}$ is jointly monic if there is at most one morphism $A\to X$ for every $A$. In other words, $X$ is a Weak Terminal Object.

• Piggybacking off the previous result, every family in a Preorder is jointly monic.

References

• Chapter 10 of Abstract and Concrete Categories: The Joy of Cats has a lot of nice results about jointly monic families, though it refers to such families as "mono-sources".