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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!

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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 $f \times g$).

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  • $\begingroup$ Many thanks for this. Indeed it is not hard to show that a normal category has pullbacks of monos and this is what ends up being shown in the book. I would be happy to add products to the definition of normal, since this is what most of the other results depend on. Blyth gets this as a consequence of the existence of pullbacks (and the existence of a zero object). $\endgroup$ – José Figueroa-O'Farrill Sep 6 '10 at 2:10
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    $\begingroup$ It is not obvious at all that your category has no products. [The products in the bigger category are not in it, but this is not a proof] $\endgroup$ – Martin Brandenburg Sep 6 '10 at 6:34
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    $\begingroup$ I actually quite liked Peter's example. As for the objection, one might observe that the underlying functor from Peter's $Vect_n$ to $Set$ preserves any limits which exist in $Vect_n$ (since it is representable), and reflects them as well since it reflects isomorphisms (see the <a href="ncatlab.org/nlab/show/reflected+limit">nLab page</a>). So any product which exists in $Vect_n$ is the expected one. $\endgroup$ – Todd Trimble Sep 6 '10 at 12:37
  • $\begingroup$ ncatlab.org/nlab/show/reflected+limit $\endgroup$ – Todd Trimble Sep 6 '10 at 12:40
  • $\begingroup$ @Martin: the “dimension argument” I had in mind was looking at the dimensions of hom-sets, not the objects themselves, which as you say wouldn't be a proof. But Todd's argument gives a nicer big picture to see it in, I think. $\endgroup$ – Peter LeFanu Lumsdaine Sep 6 '10 at 14:32
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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".)

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    $\begingroup$ @Todd: Thank you very much for this answer and such a minimal example. I wish I could have accepted your answer as well. I accepted Peter's answer because his last paragraph has proved instrumental in giving me a way to fix (to my satisfaction) the rest of the chapter. $\endgroup$ – José Figueroa-O'Farrill Sep 6 '10 at 16:40
  • $\begingroup$ No problem, José -- Peter's example is probably better in that it is immediately apprehended and "in nature". I do recommend the book by Freyd and Scedrov to your attention, if you are not already familiar with it. :-) $\endgroup$ – Todd Trimble Sep 6 '10 at 17:06

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