The arrows from the initial object in a category are monomorphisms?

This statement is true if and only if its dual is true:

In any category, arrows to terminal objects are epimorphisms

We have an easy counter-example in $\mathbf{Set}$ to this claim: $\emptyset \to \{ \emptyset \} $

We can find a counter-example to the original statement in $\mathbf{Ring}$: $\mathbb{Z}$ is initial, but the maps $\mathbb{Z} \to \mathbb{Z} / p \mathbb{Z}$ are not monomorphisms.

What we do have, however, is that arrows from a terminal object are monomorphisms, and similarly arrows to an initial object are epimorphisms.


Let $\mathcal{C}$ be a category whose objects are $\mathrm{ob}(\mathcal{C})=\{x,y\}$ and whose morphisms are $$\begin{align} \mathrm{Mor}_{\mathcal{C}}(x,x)&=\{\mathrm{id}_x\} & \mathrm{Mor}_{\mathcal{C}}(x,y)&=\{f\}\\\\ \mathrm{Mor}_{\mathcal{C}}(y,x)&=\{g,h\} & \mathrm{Mor}_{\mathcal{C}}(y,y)&=\{\mathrm{id}_y,k\} \end{align}$$ where $g\circ f=h\circ f=\mathrm{id}_x$ and $f\circ g=f\circ h=k$, $\;k\circ k=k$, $\;g\circ k=g$, $\; h\circ k=h$.

Then $x$ is an initial object of $\mathcal{C}$ and $f\circ g=f\circ h$, but $g\neq h$.

(As Jim points out in a comment below his answer, the category I had used was not actually a counterexample; we need to include a non-identity morphism from $y$ to itself.)

Though you've already accepted my answer, I had been in the process of making a diagram of this category, and I might as well include it at this point:

                                       enter image description here

\documentclass{standalone}
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
x \ar[bend left=40]{r}{f}
\ar[loop left,out=220,in=140,distance=1cm]{}{\mathrm{id}_x}
& y \ar[bend left=40]{l}[swap]{g} \ar[bend left=70]{l}[pos=0.47]{h}
\ar[loop right,out=322,in=38,distance=1cm]{}{\mathrm{id}_y}
\ar[loop right,out=295,in=65,distance=2.5cm]{}[swap]{k}
\end{tikzcd}
\end{document}


This is false. I don't have a "natural" example for you, maybe someone else can come up with one. But here is a category for which this fails, there are exactly two objects, $\{A, I\}$, and there are exactly $6$ morphisms:

  • $f\colon A \to I$
  • $g\colon A \to I$
  • $a\colon I \to A$
  • $\mathrm{id}_A$
  • $\mathrm{id}_I = fa = ga$
  • $af = ag$

Composition is given according to the rules above. Then $I$ is initial but $a$ is not a monomorphism because $f \neq g$ but $af = ag$.

Tags:

Category Theory

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