# Direct limits for algebraic objects

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?

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.