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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!

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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.

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  • $\begingroup$ Even some structures that you might think of as algebraic objects don't have direct limits. For example, the category of fields does not have products. (By field I mean the algebraic structures such as rational numbers and real numbers, not field as in physics.) You need a precise definition of algebraic structure. The best one I know of is that categories of models of finite-limit theories always have limits. See Toposes, Triples and Theories, chapter 8.4 (TTT calls them left exact theories instead of finite limit theories). TTT is at tac.mta.ca/tac/reprints/articles/12/tr12.pdf $\endgroup$ – SixWingedSeraph Sep 28 '16 at 14:59

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