Doubt in Category Theory about dualization.

Question

I have to prove that the Category of Abelian Groups has pushouts.

I want to use a theorem that says that if a category has equalizers and products, then it has pulbacks. Then I want to dualize the statement and say that if a category has coequalizer and coproducts, then it has pushouts.

But I don't understand quite well why dualize works so I don't know if I'm using it the right way. Also, I want to know if my strategy is correct.

Thanks in advance.

Answer 1

Your strategy is correct; knowing that products and equalizers gives you pullbacks tells you that coproducts and coequalizers give you pushouts.

The reason for this is that for any category $\mathcal C$, you get $\mathcal C^{op}$ simply by having the new domain function of $\mathcal C^{op}$ be the codomain function of $\mathcal C$, and likewise for $\mathcal C^{op}$'s codomain function. If you draw the diagrams for various limits, and swap the directions of all the arrows (i.e. swap domain and codomain), you will see that you end up with colimit diagrams (because dualizing doesn't change commutativity, or unique commutativity, of diagrams).

Since a $\mathcal C$ has coproducts exactly when $\mathcal C^{op}$ has products, and it has coequalizers exactly when $\mathcal C^{op}$ has equalizers, the dual of a category with coequalizers and coproducts will always have pullbacks due to the theorem you know. But the dual of $\mathcal C$ has pullbacks if and only if $\mathcal C$ has pushouts, so you get your desired result.

Answer 2

You dualise by a statement $S$ in category theory by reversing each arrow in $S$ to get $S^{op}$.

The dual of a pullback is indeed a pushout and vice versa.

It looks like you're using it well.

Answer 3

It's difficult to know quite what you are looking for. But your strategy is good (as Malice V. explains). If you want to understand more about dualizing and see more examples, try e.g. in these notes

Category Theory: A Gentle Introduction.

Also, it is worth mentioning that

There are a lot of links to intro things about category theory here.

The links go to lecture notes and even freely-available books at various levels; there should be something there that will appropriate for you.