# Constructing a counterexample in category theory

Exercise $10$ in Geroch's Mathematical Physics asks whether direct products distribute over direct sums in arbitrary categories. (They do in the category of sets, which is what motivates the question). That is, (using $A \times B$ to mean the direct product and $A+$B to be the direct sum), whether there is an isomorphism between $A \times (B+C)$ and $(A\times B)+(A \times C)$.

An earlier question asked you to show that associativity of products holds. $(A \times B) \times C$ is isomorphic to $A \times (B \times C)$. This is true, even when not all objects have products (i.e. it is true that if $A \times (B \times C)$ and $(A \times B) \times C$ exist, then they are isomorphic...). So I take this question to not require that all objects have products...

I don't believe there is the relevant isomorphism, but I'm not sure what I've done proves it. So my question is: Is the following argument a good way of approaching category theory questions?

So I wrote out a $A,B,C,A \times B,A \times,C,B+C,A\times (B+C),(A\times B)+(A\times C)$ and drew in all the arrows I know exist from the definitions of products and sums (e.g. $A \times B$ guarantees that there is an arrow from $A \times B$ to $A$ and one to $B$...). Then I observed that this diagram has no arrows that go from $A\times (B+C)$ to $(A\times B)+(A \times C)$ or the other way. That is, even allowing for composition of arrows, there need not be a morphism between the two above mentioned objects. So, I said, this means that this diagram is a diagram of a category where $A\times (B+C)$ is not isomorphic to $(A\times B)+(A \times C)$, since if there are no morphisms between them, there can't be any isomorphisms.

Is this a good way of constructing counterexamples in category theory? Is there any rigorous discussion of "diagrams" used this way? (I've heard it mentioned, but I don't know where to look).

This is a question about whether the strategy I am using is a good one, not really about the actual truth value of the statement in question.

• As informal argument it is to some extent convincing but it's not a formal proof (maybe, for example, one can build a morphism using spaces like ABAAC as intermediate stages). One can try to make it rigorous (e.g. by considering a category freely generated by 3 objects and their (co)products...) but for the exercise you mention it's definitely easier to find a well-known category for which the two objects aren't isomorphic. Commented Aug 3, 2010 at 15:04
• (Compare with the following: if you have a presentation of a group it's usually fairly easy to prove that some identity holds but quite hard to prove that some doesn't — and to do the latter one usually has to construct some representation of the group.) Commented Aug 3, 2010 at 15:08
• You need to make your question more precise. Is the assumption that you have a category where all finite products and coproducts exist? Or are you just assuming that you can take the products and coproducts involved in just the statement Ax(B+C) = (AxB)+(AxC)? Commented Aug 3, 2010 at 16:43
• The question says "This is true for Set, is it true for arbitrary categories?" I took it to mean for all categories. But maybe it is supposed to mean "for arbitrary categories that have products and coproducts"... (I take coproduct to be a synonym for direct sum?) Commented Aug 3, 2010 at 17:49
• I've clarified the question. I appreciate that if I wanted to show something about categories with products and coproducts, then finding examples would be easier. But if it is really about genuinely arbitrary categories, then is my constructive approach valid? Commented Aug 4, 2010 at 16:43

The dual to the statement that $A\times(B+C)$ is isomorphic to $(A\times B)+(A\times C)$ is that $A+(B\times C)$ is isomorphic to $(A+B)\times(A+C)$. Now if this statement fails in some category $\mathcal{C}$ then the original fails in the opposite category $\mathcal{C}^{\mathrm{op}}$.

• While certainly true, looks like it answers Exercise 10 in Geroch's Mathematical Physics but not OP's question. Commented Aug 3, 2010 at 16:28
• This doesn't look like it's an answer to the question I asked Commented Aug 4, 2010 at 7:29

What you have done is a sensible start. You have tried proving the result and you suspect it isn't true. In this situation you want to look for a counter example (and just one is enough).

My suggestion is the category of non-commutative rings. You still have direct sum. However the product is the free product. (For commutative rings it is the tensor product.)

• But is the explicitly constructed category not already a counterexample? Commented Aug 4, 2010 at 7:28
• I don't think so. I have not looked up Geroch. You have defined a directed graph; have you defined composites? Even if you have you have not defined products and coproducts for all objects. This involves not just naming them but also establishing the universal property. This is the free category with products and coproducts on three objects. This may make sense and would involve the simplicial category. It is usually hard work to construct a universal example and for a counter example this work is unnecessary.
– BWW
Commented Aug 4, 2010 at 17:46

The category of of all groups is also a counterexample. Letting $A,B,C:=\mathbb{Z}$ , the LHS is $\mathbb{Z}\times (\mathbb{Z} * \mathbb{Z})$ and the RHS is $(\mathbb{Z}\times \mathbb{Z})*(\mathbb{Z}\times \mathbb{Z})$, which are not isomorphic, as their Abelianizations are $\mathbb{Z}^3$, resp. $\mathbb{Z}^4$.

Another example is the category of Vector spaces. If $A,B,C$ are just one dimensional vector spaces the dimension of the LHS and the RHS don't agree (Note $\times\neq \otimes$).

I am not an expert on this topic, so forgive me if this is obvious. I see what you have done, but isn't this a rather "vacuous" category? Products and coproducts satisfy universal properties in that we can factor through them with respect to other morphisms. For example, if there are arrows from B and C each to A, then suddenly you have a new arrow from B+C to A which is not in your diagram

Now here's where I lose the thread: if you assume products and coproducts exist for all objects A, B, and C, must not some other arrows exist between A, B, and C? If not, then you may be OK, but I just don't know if you can assume that. The definitions of (co)products reference such other arrows, so if there are none, then aren't all the (co)products vacuous?

• Yes it's a pretty vacuous category. That's partly my worry. But the question is whether the distributivity holds for arbitrary categories, and that includes these vacuous counterexamples [this is, after all, a legitimate category, if a rather uninteresting one]. If you assume you have sums and products for all objects, then you probably can show some sort of distributivity, but that would hold only for "direct product complete" categories (I'm making up terminology there...) Commented Aug 3, 2010 at 14:27

Note that in your example (as far as I understand it, you have a category with only 8 objects) your category does not have products and coproducts (you have them only for some special pairs of objects (for example, you don't have product $(A\times B)\times(A\times C)$.