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!


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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.