Talk:Direct limit

I would suggest to rewrite this article. IMHO it should be pointed out that a direct limit is a colimit and, that in the particular case of a direct limit of ab. groups etc. it can be calculated as written. Do you agree? Jakob.scholbach 19:25, 23 February 2007 (UTC)

Actually, I prefer the reverse approach which is presented here, mostly because it is more accessible. Chances are someone seeing direct limits for the first time will encounter them in an algebraic setting. They may not even be familiar with category theory let alone colimits. In fact, understanding the algebraic definition of a colimit and then generalizing to categories is a good motivation for the very general definition of a colimit. -- Fropuff 22:50, 23 February 2007 (UTC)

I think it would be nice to have it pointed out somewhere that SGA 4 doesn't require the indexing category to be directed in its use of the term inductive limit (c.f. SGA 4 I Definition 2.1). Shanekelly64 (talk) 20:19, 8 November 2009 (UTC)

"Any other such pair" (in section "Direct limit over a direct system in a category")

In the section "Direct limit over a direct system in a category", I find the sentence

The pair $\langle X,\phi _{i}\rangle$ must be universal in the sense that for any other such pair $\langle Y,\psi _{i}\rangle$ there exists a unique morphism $u:X\rightarrow Y$ making the diagram...

to be a bit vague, because such a pair $\langle Y,\psi _{i}\rangle$ is required to be a direct system (if I understand it correctly) and not just "any" pair of an object with morphism, so I would suggest replacing "any other such pair" with "any other direct system". Do you agree? Nielius (talk) 10:08, 13 April 2012 (UTC)

No, $Y$ is not assumed to be a direct system. The maps $\psi _{i}$ goes from $X_{i}$ to $Y$. Maybe for clarity, the maps $\psi _{i}$ needs to be collected into a single set, otherwise "pair" is slightly misleading. Mct mht (talk) 11:41, 13 April 2012 (UTC)

Of course, you're right and I was confused. The problem you address is even more apparent in the definition of a direct system: "Then the pair $\langle A_{i},f_{ij}\rangle$ is called a direct system over $I$." Nielius (talk) 18:23, 30 May 2012 (UTC)

Direct limit of "algebraic objects"

This section has mistakes and should be rewritten, the disjoint union is only valid for sets. The notion of "algebraic object" is vague and although they do possess an underlying set it is not true that direct limits in algebraic categories are disjoint unions (e.g. coproduct in Vect is direct sum not union, etc.) Either decline in examples, when the associated category has pushouts and limits, or restrict this section to Set. Olivier Peltre (talk) 16:48, 26 August 2022 (UTC)

If I remember correctly, I thought we use disjoint unions to construct direct limits for rings and modules. If not, how do you construct them? For coproduction, the indexing set is not directed so the situation is different? By the way, the construction of a coproduction (free product) of algebras is at Free product of associative algebras and is much less trivial. -- Taku (talk) 06:31, 27 August 2022 (UTC)