### Background

I am reading Tom Blyth's book *Categories* as I am thinking of using it as a guide for a fourth-year project I'll be supervising this academic year. The books seems the right length and level for the kind of project we require of our final year single honours students in Edinburgh.

In the fourth chapter of the book, a *normal category* is defined as one with zero objects, in which every morphism has a kernel and a cokernel, and in which every monomorphism is a kernel. This last condition can be rephrased as saying that monomorphisms are normal.

My confusion stems from Theorem 4.6 in the book which states that *a normal category has pullbacks*. The proof in the book seems to use that in the diagram
$$\begin{matrix}
& B \cr
& \downarrow \cr
A \rightarrow & C \cr
\end{matrix}
$$
whose limit is the desired pullback, the morphism $A \to C$ is a kernel. This seems to me an unwarranted assumption, since it would seem to imply, in particular, that generic morphisms are monic.

Alas, I have not been able to find an independent proof of the theorem and I am starting to suspect that this may not be true. Googling seems not to be of much help. For one thing one has to wade through a surprising number of hits which have nothing to do with category theory.

Since much of the rest of the chapter seems to depend on this result, I am a little stuck. I have emailed the author, who is an emeritus professor across the firth in St Andrews, but so far no reply. So I thought I would try it here in MO, since I'm sure to get an authoritative answer to my

### Two questions

Is the result true? And if so,

Can someone point me to a proof?

Thanks in advance!