Do normal categories have pullbacks?

Question

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

1. Is the result true? And if so,

2. Can someone point me to a proof?

Thanks in advance!

Answer 1

If I'm not mistaken, the category “vector spaces of dimension $\leq n$” (for any $n > 0$) is a counterexample? The zero object, kernels, cokernels, and the normality of kernels can all be computed as they normally are for vector spaces; but products (and hence pullbacks) are missing for obvious reasons of dimension. [See comments for elaboration.]

The problem is, intuitively, that there's nothing in the definition of “normal” providing a way to build bigger things out of smaller.

On the other hand, I think one can prove “a normal category has pullbacks of monos”; it sounds like that might be what the book is proving here? Maybe that's all that it actually ends up using in the rest of the chapter, and this is just an omission in the statement of the theorem?

Alternatively, if one adds products to the definition of “normal”, then from that together with pullbacks of monos, one can build all pullbacks (the pullback of $f$ and $g$ is the pullback of the appropriate diagonal map (a mono) along $f \times g$).

Answer 2

It's false as stated. The following example is in Category, Allegories by Freyd and Scedrov, page 95. Consider a category with two objects, one a zero object, and the other an object x where $\hom(x, x)$ has just three morphisms: the identity, the zero morphism, and an idempotent $e: x \to x$. Kernels certainly exist (and so do cokernels since this category is isomorphic to its opposite), and all of the monos (which are maps out of the initial object or $1_x: x \to x$) are easily seen to be kernels. But the pullback of $e: x \to x$ against itself does not exist.

(Peter posted as I was writing this. The terminology "normal" may differ from one author to another; Freyd and Scedrov call this notion "left normal". The dual notion is "right normal".)