Direct limits for algebraic objects

Question

the Wikipedia article on direct limits says 'Unlike for algebraic objects, the direct limit may not exist in an arbitrary category.'.

This sounds really interesting: What do they mean with algebraic objects? In which categories do direct limits exist? Can one conclude that any of these categories has arbitrary (small)products?

Thanks for your help!

Answer 1

Well you have to read the whole article, not just cherry-pick one sentence out of it. In the previous subsection entitled "Algebraic objects", you can read:

In this section objects are understood to be sets with a given algebraic structure such as groups, rings, modules (over a fixed ring), algebras (over a fixed field), etc. With this in mind, homomorphisms are understood in the corresponding setting (group homomorphisms, etc.).

In all these examples (so: groups, rings, modules, algebras...), direct limits always exist. I sincerely doubt that the Wikipedia article is trying to say more than that. It's just a way to explain that while direct limits always exist in familiar categories, in general they may not exist. While there are some existence results (like "if $\mathcal{C}$ has all direct limits then the category of group objects in $\mathcal{C}$ has all direct limits") it would be disingenuous to pretend that this was what the Wikipedia article was saying.