Constructing a counterexample in category theory

Question

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.

Answer 1

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 $\mathcal{C}^{\mathrm{op}}$.

Answer 2

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.)

Answer 3

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 $\times\neq \otimes$).

Answer 4

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?

Answer 5

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 $(A\times B)\times(A\times C)$.

So if your exercise was about categories with products and coproducts, then your example is not an answer on it.

The simplest way to construct a counterexample is to check some known categories. It will be quite useful for studying category theory, because you will check, what do abstract notions mean in concrete examples. In this case this works, you can check for example category of vector spaces over fixed field.

Another method is to use some method to construct new category by those you know (for example, as was it was shown by Robin Chapman).

Another method is just create some simple category --- it is the method you have tried.